J~n –an –abha ßbDbBßbDbBßbDbBßbDbBßbDbB
VOLUME 38
2008
ISSN 0304-9892
Published by :
The Vij~n–ana Parishad of IndiaDAYANAND VEDIC POSTGRADUATE COLLEGE
(Bundelkhand University)ORAI, U.P., INDIA
J~N – AN – ABHA
EDITORSH. M. Srivastava R.C. Singh ChandelUniversity of Victoria AND D.V. Postgraduate CollegeVictorria, B.C., Canada Orai, U.P., India
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FINITE CAPACITY QUEUEING SYSTEM WITH QUEUE DEPENDENT SERVERS ANDDISCOURAGEMENT -Madhu Jian and Pankaj SharmaSOME GENERATING FUNCTIONS OF CERTAIN POLYNOMIALS USING LIEALGEBRAIC METHODS -M.B. El-KhazendarUNSTEDAY INCOMPRESSIBLE MHD HEAT TRANSER FLOW THROUGH A VARIABLEPOROUS MEDIUM IN A HORIZONTAL POROUS CHANNEL UNDER SLIP BOUNDARYCONDITIONS -N.C.Jain and Dinesh Kumar VijayREDUCIBILITY OF THE QUARDUPLE HYPERGEOMETRIC FUNCTIONS OF EXTON
-B. Khan, M. Kamarujjama and Nassem A. KhanCONVERGENCE RESULT OF (L,) UNIFORM LIPSCHITZ ASYMPTOTICALLY QUASINONEXPANSIVE MAPPINGS IN UNIFORMLY CONVEX BANACH SPACE
-G.S. SalujaON SPECIAL UNION AND HYPERASYMPTOTIC CURVES OF A KAEHLERIANHYPERSURFACE -A. K. Singh and A.K. GahlotMEROMORPHIC UNIVALINT FUNCTIONS WITH ALLTERNATING COEEFICIENTS
-K.K. Dixit and Indu Bala MishraA MATHEMATICAL MODEL FOR MIGRATION OF CAPILLARY SPROUTS WITHDIFFUSION OF CHEMOTTRACTANT CONCENTRATION DURING TUMORANGIOGENESIS -Madhu Jain, G. C. Sharma and Atar SinghDIFFUSION-REACTION MODELFOR MASS TRANSPORTATION IN BRAIN TISSUES
-Madhu Jain, G.C. Sharma and Ram SinghRELIABILITY ESTIMATION OF PARALLEL-SERIES SYSTEM USING C-H-AALGORITHM -A.K. Agarwal and Suneet SaxenaA NOTE ON PAIRWISE SLIGHTLY SEMI-CONTINUOUS FUNCTIONS
-M.C.Sharma and V.K.SolankiFIXED POINT THEOREMS FOR AN ADMISSIBLE CLASS OF ASYMPTOTICALLYREGULAR SEMIGROUPS IN LP- SPACES
-G.S. SalujaSOME INTEGRAL FORMULAS INVOLVING A GENERAL SEQUENCE OF FUNCTION, AGENERAL CLASS OF POLYNOMIALS AND THE MULTIVARIABLE H-FUNCTION
-V.G. Gupta and Suman JainMAXIMIZING SURVIVABILITY OF ACYCLIC MULTI-STATE TRANSMISSIONNETWORKS (AMTNs) -Raju Singh Gaur and Sanjay ChaudharyTHE INTEGRATION OF CERTAIN PRODUCTS INVOLVING H-FUNCTION WITHGENERAL POLYNOMIALS AND INTEGRAL FUNCTION OF TWO COMPLEX VARIABLES
-V.G. Gupta and Nawal Kishor JangidSTUDY OF VELOCITY AND DISTRIBUTION OF MAGNETIC FIELD IN LAMINARSTEADY FLOW BETWEEN PARALLEL PLATES
-Pratap SinghA NOTE ON THE ERROR BOUND OF A PERIODIC SIGNAL IN HOLDER METRIC BYTHE DEFERRED CESARO PROCESSOR
-Bhavana Soni andPravin Kumar MahajanAPPLICATIONS OF SYMMETRY GROUPS IN HEAT EQUATION
-V. G. Gupta, Kapil Pal and Rita MathurA GENERALIZATION OF MULTIVARIABLE POLYNOMIALS
-R.C. Singh Chandel and K.P. TiwariTHE DISTRIBUTION OF SUM OF MIXED INDEPENDENT RANDOM VARIABLES ONEOF THEM ASSOCIATED WITH H -FUNCTION
-Mahesh Kumar GuptaON GENERATING RELATIONSHIPS FOR FOX'S H-FUNCTION AND MULTIVARIABLEH-FUNCTION -B.B. Jaimini and Hemlata Saxena
CONTENTS
...1-12
...13-18
...19-32
...33-40
...41-48
...49-56
...57-64
...65-75
...77-84
...85-88
...89-94
...95-104
...105-112
...113-124
...125-134
...135-140
...141-146
...147-152
...153-160
...161-168
...169-179
J~n–an–abha, Vol. 38, 2008
FINITE CAPACITY QUEUEING SYSTEM WITH QUEUE DEPENDENTSERVERS AND DISCOURAGEMENT
ByMadhu Jian and Pankaj Sharma
Department of Mathematics, Institute of Basic ScienceDr. B.R. Ambedkar University, Agra-282002, Uttar Pradesh, India
E-mail : [email protected] and [email protected]
(Received : November 15, 2007)
ABSTRACTAn analytical study for optimal threshold policy for queueing system with
state dependent heterogeneous servers and discouragement is presented. In thisinvestigation the discouraging behavior of the customers has been considered dueto which the customers may balk or renege. To reduce the discouraging behaviorof the customers there is a provision of removable heterogeneous servers. Theservice rate of the servers are different and the number of servers in the systemchanges depending on the queue length. The first server starts service only whenN customers are accumulated in the queue and once he starts serving, continuesto serve until the system becomes empty. The jth( j=2,3...,r) server turns on whenthere are Nj–1 customers present in the system and are removed when the queuesize ceases to less than Nj–1. By employing the recursive method we derive thesteady state characteristics of the system such as queue size distribution, theaverage number of customers in the system, the average number of waitingcustomers, etc. Sensitivity analysis have been facilitated by taking numericalillustration.2000 Mathematics Subject Classification : Primary 90B15; Secondary 90B50Keywords and Phrases : Finite capacity, Queue dependent servers,Discouragement, Heterogeneous servers, Queue size, Threshold policy.
1. Introduction. In real life situations, it is common to use some extraservers to reduce the congestion by assuming that the number of servers changesaccording to the queue length. The decision makers often employ someheterogeneous removable servers in the system to reduce the discouraging behaviorof customers and waiting time. Such situations can be encountered in many day-to-day congestion situations including banks, check-out counters, super market,cafeterias, petrol pumps, etc.
There are several queueing systems for which based on cost criteria, it isrecommended that the number of servers should be increased one by one depending
2
upon the queue length. Garg and Singh (1993) investigated queue dependent serversqueueing system to determine the optimal queue length at which the next serveris provided in order to gain the maximum profit. Yamashiro (1996) considered asystem where the number of servers changes depending on the queue length. Aqueueing system with queue dependent servers and finite capacity was consideredby Wang and Tai (2000). Jain (2005) analysed finite capacity M/M/r queueing systemwith queue dependent servers. Processor-shared service system with queue-dependent processors was also studied by Jain et al. (2005).
In classical N-policy queueing system, when there are N customers presentin the system only then the server starts providing service to the customers. Manyresearchers have contributed to the studies on N-policy models in differentframewaorks. Larsen and Agrawala(1983) investigated optimal policy to minimizethe mean response time for M/M/2 queueing system. Medhi and Templeton (1992)analyzed the Poisson input queue under N-policy and with a general start up time.Kavusturucu and Gupta (1998) developed a methodology for analyzing finite buffertandem manufacturing system with N-policy. Jau (2003) and Jain (2003) consideredthe operating characteristics for a general input queue and redundant arrivalsystem, respectively under N-policy. Jain et al. (2004) analyzed N-policy for amachine repair system with spares and reneging. A two-threshold vacation policyfor multi server queueing system was given by Tian and Zhey (2006).
The discouraging behavior of the customers has also been incorporated byseveral researchers while developing the queueing models of real life congestionproblems. A prospective customer on arrival may join the queue or may balkdepending on the number of customers present in the queue. Ankar and Gafarian(1963) analyzed some queueing problems with reneging. Blackburn (1972) and Gupta(1994) discussed different types of queues with balking and reneging. Jain andVaidhya (1999) considered the multi server queue with discouragement andadditional servers. Jain and Sharma (2002) investigated a multi server queue withadditional server and discouragement.
To reduce the discouraging behavior of the customers, one of the importantattributes to be considered by system organizers is to increase the number of serverswhile analyzing the congestion situations in different frameworks. In order toreduce the balking/reneging, the service providers can make the provision of theremovable additional servers apart from some permanent servers from cost/spaceconstraint view point. Abou and Shawky (1992) considered the additional serversin the single server Markovian over flow queue with balking and reneging. Jain(1998) analysed M/M/m queue with discouragement and additional servers. Jainand Singh (2002) considered a M/M/m queue with balking, reneging and additionalservers. A multi server queueing model with discouragement and additional servers
3
was studied by Jain and Singh (2004) in order to facilitate a comparative study formulti servers queueing system with and without additional servers. Controllablemulti server queue with balking was investigated by Jain and Sharma (2005). Inthis model they have incorporated an additional server which is added and removedat pre-specified threshold level of queue size to control the balking behavior of thecustomers.
In this paper we consider a finite capacity Markov queueing system withqueue dependent heterogeneous servers and discouragement uder optimal thresholdpolicy. The concept that the heterogeneous servers may be employed in the systemone by one depending on the queue length can be helpful to reduce cost as well asthe discouraging behavior of the customers in the system. The steady state queuesize distribution by using recursive method is obtained which is further used todetermine various system characteristis. The remaining part of the paper isorganized as follows. In section 2, the model is described by stating requisitenotations and assumptions. Queue size distribution and other system metrics havebeen derived in section 3 and 4, respectively. Some special cases are deduced bysetting appropriate parameters, in Section 5. Sensitivity analysis is given in section6. Section 7 concludes the paper and highlights the future scope of the model.
2. The Model . Consider a Markov queueing system with finite capacityand queue dependent heterogeneous servers. The concept of discouragement andheterogeneous removable servers under optimal control policy are taken intoconsideration. We assume that queue dependent r(>1) servers offer services to thecustomers who arrive in Poisson fashion. By the queue dependent heterogeneousservers we mean to say that the servers turn on one by one depending upon thequeue length according to a pre-specified rule and renders service with differentrates, the service times taken by each server, are assumed to be exponentialdistributed. The arriving customers may balk with probability 1-bj, j(j=0,1,...,r)denotes the number of servers rendering service in the system. The customersmay also renege from the queue after waiting for some time; according to exponentialdistribution with parameter j, (j=0,1,2,...,r) here j indicates the number of serverspresent in the system. The customers are served according to first come first servediscipline and the capacity of the system including those in service is of size K.
The state dependent arrival rate of the customers are given as follows :
KnNbrjNnNb
Nnbn
rr
jjj
1
1
11
;1,...,3,2, ;
0 ;
...(1)
The effective service rates after incorporating the reneging concept, can be
4
given by :
r
irri
j
ijjji
KnNrn
rjNnNjn
NnnNnn
n
11
11
111
00
;
1,...,3,2, ;
N ;11 ;1
...(2)
The number of servers employed in the system depends upon the number ofcustomers present in the system according to the following threshold policy:✦ The first server turns on when N(>1) cutomers are present in the system
and turns off as soon as the system become empty.✦ When there are more than N customers waiting in queue he provides service
to the customers with the faster rate 1, otherwise serves the customerswith rate 0.
✦ The jth (2,3,4,...,r) server provides service to the customers when there aremore than Nj-1 customers present in the system, and as soon as the queuelength become less than Nj-1 the jth server is removed from the system.
Let P(j,n) denotes the steady state probability that there are n(n1) customerspresent in the system and j(j=1,2,...,r) heterogeneous servers are providing serviceto the customers. In our Model n denote the number of customers present in thesystem, i(i=0,1,2) denotes the level of the service state and j–1 (2,3,...,r) denotesthe number of removable heterogeneous servers employed in the system.
Let P(0,n) denote the steady state probability that there are 'n' customersin the system before start of the service 0 n N–1. P(j,Nj(i)) denotes the probabilitythat there are Nj customers present in the system being served by the newly addedserver or previously existing server when I takes value 1 or 2 respectively, andthere are j servers in the system. We denote the probability of Nj customers in the
system by 2,1,, jjj NjPNjPNjP . Here 1, jNjP is the probability that
the Njth customer in the system is being served by the jth server while 2,1 jNjP
denotes the probability that thjN customer is being served by the (j+1)th server,
while 2,1 jNjP denotes the probability that thjN customer is being served by
the (j+1)th server without having any reneging probability. We have also consideredthat when there are Nr customers waiting before the servers, then all the 'r'removable servers are available for service. The steady state equations for the
5
finite source multi server queueing system are given by
1,10,0 01 PPb ...(3)
11 ;2,0,0 11 NnnPbnPb ...(4)
2,11,1 0001 PPb ...(5)
NnnPnnPbnPnb 2 ; 1,11,1,11 001001 ...(6)
;1,11,11,0,11 1111001 NPNNPbNPbNPNb ...(7)
21;1,11,1,11 1111111 NnNnPnnPbnPnb ...(8)
1,112,11,12 11111111111 NPNNPbNPNb
;2,1 12 NP ...(9)
1,21,1,11 121111112 NPNPbNPNb ...(10)
1,212,1 1211122 NPNNPb ...(11)
,21;1,11, 1 jjjjjj NnNnjPnnPbnjPnb
1,...,2,1 rj ...(12)
1,2,1,1 jjjjjjjjj NjPNNjPbNjPNb
1,...,2,1; 2,11 rjNP jj ...(13)
)1(,....,2,1;1,11,1, 11 rjNjPNjPbNjPNb jjjjjjjj ...(14)
1,...,2,1; 1,112,1 1111 rjNjPNNPb jjjjjjj ...(15)
KnNnrPnnrPbnrPnb rrrrr 1; 1,11,, ...(16)
1,, KrPbKrP rr ...(17)
3. Queue Size Distribution. In this section, we derive the mathematicalexpressions for the queue size distribution and average queue length for finitecapacity model by employing recursive methed.For brevity of notaion, we denote
j
ijij jnn
1
; 1,....,3,2,1 rjNnN jj
;1
r
r
iir rnn
KnNr 1
6
;1,...,2,1,, ;
1 ;1
01
1
0
1
rjNNNnNjn
b
Nnnb
rjjj
iji
j
o
n
1
0
1111
11
0
.1,...,3,2,,,1
,1
N
i j
jj
j
jjj
ijji
jN
i
jjN rj
bR
br
jN
brA
j
The steady state probability P(1,n) is obtained by solving equations (3) to (8) as :
Nn
iiN
n
i
i
jjn
NnNPrA
NnPrnP
01
1
0 0
; 0,0
1 ; 0,0,1
...(18)
For ,1,...,3,2 rj we obtain probabilities using equations (9)-(15) as
;
1
0,01, 1
1
1
1
1
1
1
j
K
j
K
KKN
KKN
KN
j
K
Nn
iNiiN
KKKN
RrrrrRr
PrRrrAnjP
KKK
K
1,...,3,2,1 rjNnN jj ...(19)
Also solving equations (16) and (17), we get
KnN
RrrrrRr
PrRrrAnrP rr
K
r
K
KKN
kKN
kN
r
K
Nn
iNiiN
KKkN
KKK
K
11
1
1
11
1
1 ;1
0,01,
...(20)
0,0P can be obtained by using normalizing condition
1
0 1
1
2 1 1
1
1 1
1,, ,1,0N
n
N
n
r
j
N
Nn
K
Nn
j
j r
nrPnjPnPnP .
7
At the threshold level the probabilities are obtained as :
1,...,2,1;,1
11, 1
rjNjP
RrrrrR
rrRrNjP jjj
Njj
Nj
jjN
jN
j
jj
jj
...(21)
and
1,...,2,1;,1
2, 1
2
rjNjPRrrrrR
rRrNjP jjj
Njj
Nj
jjN
j
jj
j
...(22)
Also
.1,...,2,1,2,1,, rjNjPNjPNjP jjj
4. Performance Metrics. This section is devoted to some performancecharacteristics by using steady state queue size distribution as follows:Pr (1 server is rendering service in the system)=P(1)=Pr ob{1<nN1}
N
n
N
Nn
Nn
iN
n
i
i
jjn PrAPr
1 1 01
1
0 0
1
0,00,0 ...(23)
Pr (j servers are rendering service in the system)=P(j)=Prob.{Nj-1 n Nj}
;
1
0,01
1
1
1
11
1
1
1
1
j
K
j
K
KKN
KKN
KN
j
K
N
Nn
Nn
iNiiN
KKKN
RrrrrRr
PrRrrA
KKK
j
j
K
1,...,3,2,1 rjNnN jj (24)
Pr (all r servers are rendering service in the system)=P(r)=Prob{Nr-1nK}
;
1
0,01
1
1
1
11
1
1
1
1
j
K
j
K
KKN
KKN
KN
j
K
N
Nn
Nn
iNiiN
KKKN
RrrrrRr
PrRrrA
KKK
j
j
K
1,...,3,2,1 rjNnN jj (25)
The average number of customers in the system having r heterogeneous queuedependent removable servers which are employed at thresholds
121 ,...,,, rNNNNN is obtained using:
K
nnnPNrL
0
:
8
N
n
N
Nn
Nn
iiN
n
i
i
jjn rAr
0
1
1 0
1
0 0
1
1
2 11
1
1
11
1
1
1 1
1
r
j
N
Nnj
K
KKN
KKN
Kj
KN
j
K
Nn
iNiiN
KKKNj
jKKK
K
RrrrrRr
rRrrA
K
Nnr
K
r
K
KKN
KKN
KN
r
K
Nn
iNiiN
KKKN
rKKK
K
PRrrrrRr
rRrrA
1
0,01
1
1
1
1
11
1
1
...(26)
The throughout of the system is given by
K
nnnP
1. ...(27)
5. Special Cases. In this section, we discuss some special cases that canbe deduced from analytical results derive in the previous sections.Case I: M/M/R Finite Capacity Queueing System With Queue Dependent
Servers. In this case we set bj=1 and rjj ,...,2,1,00 , and avoid the first
threshold level i.e. we set N=1 then our results tally with those of Jain's (2005)model for M/M/r queue with heterogeneous queue dependent servers. In particular,we come across the following special situations:i. If we set r=3, then our results coincide with the results obtained by Wang
and Tai (2000).ii. When the system capacity is infinite i.e. (K), then the results correspond
to the finite capacity queueing system with three queue dependentheterogeneous servers model which was discussed by Wang and Tai (2000).
iii. We can obtain the results for model of queue dependent homogeneous servers
by setting r ...10 for this model.
Case II : When N=1, N1=2, N3=3,...,Nr=r, bj=1 and j=0(j=0,1,2,...,r), then ourmodel converts to M/M/r queueing model with heterogeneous server.
Case III: When N=1, N1=2, N3=3,....,Nr=r, bj=1, j=0(j=0,1,2,...,r), j then
model provides results for classical M/M/r queueing model with homogeneous
9
server.5. Sensitivity Analysis. In this section, the sensitivity analysis is
performed to examine the effect of various system descriptors on the average queuelength and throughout of the system. A computer program is developed by usingthe mathematical software MATLAB on Pentium IV. Various performance indicesare computed and summarized in tables 1 to 3 by setting the default parameters as
N=1, K=50, r=3, b1=0.4, , 5.1,5.1 00 .3,2.00 Some more results are
also obtained and displayed in figures 1-4 using the default parameters as N=1,
K=50, r=3, b1=1, , 5.1,5.1 00 .5.1,2.00 The following three policies
having different threshold parameters are taken into consideration:Policy 1: In this policy, the homogeneous servers turn on one by one with the
arrival of each customers i.e. 1,1 jN j .
Policy 2: In this case, the heterogeneous servers activate one by one with the
arrival of customers i.e. 1 jN j also we set 011.01 jj .
Policy 3: For this policy, we consider the heterogeneous removable three servers
who starts the service according to rule jN j 3 and we chosen 011.01 jj .
Table 1 shows the effect of N, and arrival rate (). Both, average queuelength (L) and through put () of the system increase with the increase in and b1.
Tables 2 and 3 depict the effect of (N, 0 ) v (N, ) on the average queue length (L)
and through put (). We note that the average queue length (L) and through put ()
decrease as 0 increase but both L and increases as b1 increases. Effect of N and
0 are demonstrated in table 3. When we increase the 0 , L decreases and
increases. but effect is not much significant.From figures 1 and 2, we see that the average queue length (L) increases
(decreases) with arrival rate (service rate ) for the different policies, which iswhat we expect in the real life situations. Also it is noted from figures 3 and 4, thatthe average queue length (L) decreases (increases) with the increase in reneging
parameter 0 (joining parameter b1) for the different policies. If homogeneous
servers are taken into consideration i.e. for policy 1, the average queue length ishigher in comparison to the policies 2 and 3, where heterogeneous servers areemployed. The queue length is slightly higher in policy 3 where heterogeneousservers turn on with the additional workload of 3 customers, in comparison topolicy 2 where heterogeneous servers turn on with the addition of one customer;this pattern matches with physical situations.
10
11
Over all, we conclude that the average queue length (L) reduces (increases)
with the increase in N, 0 and 0 (and b1). The through put () shows the
increasing trends with the parameters N, , b1, 0 and 0 which is quite obvious.
By incorporation of additional removable servers, there is remarkable decrementin the queue length which promotes the provision of a pool of additional serversalong with permanent servers.
6. Discussion. In this paper, we have studied a finite capacity multi serverqueueing system with queue dependent removable servers and discouragementunder N-policy. The assumption of discouraging behavior of customres makes ourmodel more versatile as it deals with more realistic congestion situations. Toreduce the cost, the decision makers are suggested to employ heterogeneous servesthat may further be removed after pre specified threshold level. The costrelationship established is helpful to determine the optimal queue level to introducethe removable servers in order to gain the maximum net profit.
12
REFERENCES[1] M.O. Abou-El-Ata and A.I. Shawky; The single server Markovian overflow queue with balking,
reneging and an additional server for longer queues, Microelectron. Reliab., 32 (1992), 1389-1394.
[2] C.J. Anker and A.V. Gafarian, Queueing with reneging and multiple heterogeneous servers, NavalRes. Log., Quart, 10 No. 1 (1963), 125-149.
[3] J.D. Blackburn, Optimal control of single server queue with balking and reneging, Manage Sci.,19 No.8 (1972), 297-379.
[4] R.L. Garg and P. Singh, Queue dependent servers queueing system, Microelectron. Reliab. 33(1993), 2289-2295.
[5] S.M. Gupta, Interrelationship between queueing models with balking and reneging and machinerepair problem with warm spares. Microelectron. Reliab., 34 No. 2, (1994), 201-209.
[6] M. Jain and P. Singh, A multi server queueing model with discouragement and additional servers,Recent Advances in Operations Research, Information Technology and Industry, (M. Jain and G.C.Sharma (Eds)), S.R. Scientific Pub., Agra India (2004).
[7] M. Jain, M/M/m queue with discouragment and additional servers, Journal of G.S.R., 36 No. 1-2,(1998), 30-42.
[8] M. Jain and V. Vaidhya, M/M/m/K queue with discouragement and additional servers, Opsearch36 No.1 (1999), 73-80.
[9] M. Jain, N-policy for redundant repairable system with additional repairmen, Opsearch, 40 No.2, (2003), 97-114.
[10] M. Jain, Finite capacity M/M/r queueing system with queue dependent servers, Comput. Math.Appl., 50 (2005), 187-199.
[11] M. Jain and G.C. Sharma, M/M/m/K queue with additional servers and discouragement, Int. J.Engg., 15 No. 4 (2002), 349-354.
[12] M. Jain and P. Sharma, Controllable multi server queue with balking, Int. J. Engg., 18 No.3(2005), 263-271.
[13] M. Jain and P. Singh, M/M/m queue with balking, reneging and additional servers, Int. J. Engg.,15 No. 2 (2002), 169-178.
[14] M. Jain, Rakhee and S. Maheshwari, N-policy for a machine repair system with spares andreneging, Appl.Math. Model., 28 No. 6 (2004), 513-531.
[15] M. Jain, G.C.Sharma and C.Shakhar, Processor shared service system with queue dependentprocessors, Comput. Oper. Res., 32, (2005), 629-641.
[16] C. Jau, The operating characteristics analysis on general input queue with N-policy anda start uptimes, Math. Oper. Res., 57 No.2 (2003), 235-254.
[17] A. Kavusturucu and S.M. Gupta, A methodology for analyzing finite buffer tandem manufacturingsystems witn N-policy, Comput. Indust. Engg., 34 issue 4, (1998), 837-843.
[18] R.L. Larcen and A.K. Agrawala, Control of a heterogeneous two server exponential queueingsystem, IEEE Trans. Soft. Engg., 9 (1983), 522-529.
[19] J. Medhi and J.G.C. Templeton; A poisson input queue under N-policy and with a general setuptime, Comput. Oper. Res., 19 No.1 (1992) 35-41.
[20] N. Tian and Z.G. Zhang, A two threshold vacation policy in multi server queueing systems, Eur. J.Oper. Res., 168 (2006), 153-163.
[21] K.H. Wang and K.Y. Tai, A queueing system with queue dependent servers and finite capacity.Appl. Math. Model., 24 (2000), 807-814.
[22] M. Yamashiro, A system where the number of servers changes depending on the queue length,Microelectron. Relab., 36 (1996), 389-391.
J~n–an–abha, Vol. 38, 2008
SOME GENERATING FUNCTIONS OF CERTAIN POLYNOMIALSUSING LIE ALGEBRAIC METHODS
ByM.B. El-Khazendar
Department of MathematicsAlazhar University-Gaza, P.O. Box 1277, Gaza, Plestine
e-mail :[email protected]
(Received : November 15, 2007)
ABSTRACTIn ths present paper a group theoretic method is applied to the differential
eqation whose solution is xfvnF ;;,12 . Then we study in details the case
xexf . We consider six-parameter Lie group for this hypergeometric function,
which does not seem to appear earlier. By means of this group theoretic methodsome new generating functions are obtained from which several new specialgenerating functions can be easily derived.2000 Mathematics Subject Classification : 22E30, 22E80, 33C05Keywords and Phrases : Hypergeometric function, Lie group, generatingfunctions
1. Introduction. The hypergeometric function of one variable is introducedby Gauss in (1822) and defined by
...
1..2.11.1
.1
.1
!;;,
2
012
cczbbaa
zcba
nz
cba
zcbaFn
n n
nn(1.1)
where (a)n denote the Pochhammer symbol defined by
...3,2,1 if 1...1
0 if 1nnaaan
a n (1.2)
The hypergeometric polynomial (1.1) is a solution of the following equation :
vxfn
xfxf
xfxf
xfxfdx
yd
xf
xfxf1
'1
'
"
'
13
2
2
2
2
2
dxdy
0 yxfn (1.3)
Several generating relations for hypergeometric polynomials have been derivedby different methods e.g. classical, theory of Lie-group etc. In a recent paper A.K.Chongdar [1] has derived some generating functions for the said polynomials byLie algebraic method. See also [2] and [3].
14
Now, we can apply the group theoretic method on (1.3) by replacing
dxd
by x
, ,
' yyfyf
n
zzfzf
' and zfyfxfuy ,, in (1.2).
We get the following partial differential equation :
xzu
zfzf
xfxf
xyu
yfyf
xfxf
xu
xf
xfxf 2222
2
2
2
2
'''.
''
1
0'
1"'''" 3
222
xu
xf
xfxfxfxfxf
vxfxf
zyu
zfzf
yfyf
xf (1.4)
Thus zfyfxvnFzfyfxfu n;,,,, 121 is a solution of the differential
equation (1.3) since xfvnF ;,,12 is a solution of (1.3) we now defined the
infinitesimal operator 6,...,2,1iAi
6,...,2,1;0321
jAz
Ay
Ax
AA ijijijiji
as follows :
zzfzf
xxfzfxf
A
zvfzzfyyfzf
yfxfxxfzf
xfxfA
yvfzzf
zfyfxfyyf
yfxxf
yfxfxfA
yyfxxfyfxf
A
zzfzf
A
yyfyf
A
''
'1
''1
1'1
'
'
'
2
6
15
2
4
3
2
1
(1.5)
2. Application. Now, we study in details the case when xexf , then the
differential equation (1.3) can be rewritten as
011 2
2
wendxdw
vendx
wde xxx , (2.1)
where xevnFW ;;,12
Let xdxd
zyn
,, and , ,, zyx eeeuW
15
Then from (1.3), we get
011222
2
2
xu
vzy
ue
xzu
exy
ue
xu
e xxxx(2.2)
which has the solution
aznyxzyx eeevnFeeeu ;;,,, 12 . (2.3)
So we define
zxeAvezeyexeeA
vezeyexeeAyxeA
zAyA
z
zzzxxz
yyxyyx
y
6
5
4
3
2
1
11 (2.4)
such that
112126
112125
12124
112123
12122
12121
;;1, ;;,
;;1, ;;,
;;,1 ;;,
;;,1 ;;,
;;, ;;,
;;, ;;,
nyxznyx
nyxznyx
znyxznyx
znxznyx
znyxznyx
znyxznyx
eevnFeevnFA
eevnFveevnFA
eevnFnveevnFA
eevnFneevnFA
eevnFeevnFA
eevnFneevnFA
. (2.5)
Now we have the following commutate or relations. Using the relation
,, uBAABuBA we get
;0,2,;,;0,
;0,;,;,0,;0,;0,;,0,;2,;0,;0,
61
26566251
63552441
645342331
541433221
AAvAAAAAAAA
AAAAAAAAAAAAAAAAAAAVAAAAAAA
(2.6)
Hence we get the following theorem :Theorem. The set (I,Ai (i=1,2,...,6)) when I stands for the indentity operator,generates a Lie-algebra L and each of the sets
4,3,2,1, iAI i and 6,5,2,1, jAI j
forms a subalgebra of L.It can be easily shown that the partial differential operator L given by :
16
x
vzy
exz
exy
ex
eL xxxx
11222
2
2
(2.7)
can be related to each Ai in the following two different ways, i.e.
2256
1134
1and1
AvAAALvAAAAL
. (2.8)
From which it follows that L commutes with each of the Ai
i.e. 6,...,2,1 0, iLAi . (2.9)
The extended form of the groups generated by 6,...,2,1iAi given by;
zyaxueeeue zyxAa ,,,, 111 (2.10)
zayxueeeue yyxAa 2,,,,22 (2.11)
zaeaexyueeeue 3yyyxAa ,In, In,, 3333 (2.12)
yyyxAa eaveeeue 41In,,44
xy
yyxy
4eea
eazeyeea1xu
11
1In, 1In, 1 In
4
44 (2.13)
xzz
zyyxAa eaeyxu
ae
eveeeue
1InIn,, 5
5
55
55
5 In, 1
In aeae
eaee z
z
xzy
(2.14)
zzyyxAa eazyeaxueeeue 66 1In,,1In,,66 (2.15)
Therefore we easily get
yyxAaAaAaAaAaAa eeeueeeeee ,,112233445566
pnueaaeaaaaezv yxyz ,,11In 544565 (2.16)
where
424442565
446
111
11In
aaeaaaaeaaae
eaeaeazyx
yxyz
yxyz
yxyz
yxzy
eaaeaaaae
aaeaaaaeaaaea
544565
4355653431
11
111
17
yxyz
yxyz
eaeaeaeaaeaaaae
a
446
5445652 11
11.
3. Generating Functions. From (2.2) znyxzyx eevnFeeeu ;;, ,, 12 is
a solution of the systems :
0 0
; 0
0 ;
0 0
2121 unAALu
uALu
unALu
and from (2.9), we easily get
,0;;,;;, 1212 znyxznyx eeevnFLSeeevnFSL
where 112233445566 AaAaAaAaAaAa eeeeeeS .
Therefore the transformation znyx eevnFS ;;,12 is also annulled by L.
By putting a1=a2=0 in (2.16) and then using it, we get
znyxAaAaAaAa eevnFeeee ;;,1233445566
yxyzzyxyz eaeaeaeaaeaaaeanv 444544655 11In11In
435655343 111In aaeaaaeaeaaan yxzy
eaaaaaeaeaeaea
ev
nF
yzyxyz
x
243655446
121111;
;, In
zy
aaea zx
425 1
(3.1)
Again
aznyxAaAaAaAa evnFeeee ;;,1233445566
0 0 0 0
2456
!!!!In
p m l kk
k
l
l
m
m
p
p
nk
aknv
la
vm
am
pa
pmlknevpmlkn
F x
;;,
In 12 (3.2)
18
Equating (3.1) and (3.2), so we have
yxyzzyxyz eaeaeaeaaeaaaeanv 442544655 11In11In
425655242 111In aaeaaaeaeaaan zxzy
243655446
121111;
;, In
eaaaaaeaeaeaea
ev
nF
yzyxyx
x
425 1 aaea zx
0 0 0 0
2456
!!!!p m l kk
k
l
l
m
m
p
p
nk
aknv
la
vm
am
pa
pmzkyevpmlkn
F x
1
;;,
In 12 (3.3)
The above generating function does not seen to appear before where from a largenumber of different generating relations (near and known) may be easily obtainedby attributing different values to ai's of which the generating relations.Derivation of some generating functions involving Jacobi polynomialsfrom the relation (2.19)Now putting .
,1 n ,1 v2
1 yx e
e
in (3.3) and then
Case 1. Letting ,0642 aaa 15 a , and te z 2 we get,
te
teePett y
yy
nny
11
11121 ,
0
, !
2
m
mxmnm
mtePn
m (3.4)
Case 2 : Putting 1,0 6542 aaaa , and tez , we obtain
0
,1 1!
11
1p
pypnp
y
nn tePn
ptte
Pt (3.5)
REFERENCES[1] A.K. Chongdar, Group study for certain generating functions. Cal. Math. Soc., 77 (1985), 151-157.
[2] E.B. Mc Bride, Obtainging Generating Functions. Springer Verlag, Berlin (1971).
[3] L. Weisner, Group theoretical origins of certain generating functions, Pacific J. Math., 5 (1955),1033-1039.
J~n–an–abha, Vol. 38, 2008
UNSTEDAY INCOMPRESSIBLE MHD HEAT TRANSER FLOWTHROUGH A VARIABLE POROUS MEDIUM IN A HORIZONTALPOROUS CHANNEL UNDER SLIP BOUNDARY CONDITIONS
ByN.C.Jain
Department of Mathematics, University of Rajasthan Jaipur-302004 ,Rajasthan, india
andDinesh Kumar Vijay
Department of Mathematics, Kautilya Institute of Technology & EngineeringJaipur-302022 ,Rajasthan, india
E-mail:[email protected](Received : December 30, 2007)
ABSTRACTThe paper examines the problem of unsteady MHD flow and heat transfer
of a viscous incompressible fluid in slip flow regime with variable permeabilitybounded by two parallel porous plates. Using perturbation technique theexpressions are obtained for velocity and temperature distributions, skin frictionand Nusselt number. The Effects of slip parameters, Hartmann number, Reynoldsnumber, permeability parameter are discussed on velocity and temperaturedistributions. Important parameters, the skin friction and Nusselt number at boththe plates are also calculated.2000 Mathematics Subject Classification : Primary 76D, Secondary 76S05Keywords : Unsteady, Variable permeability, Slip parameters, MHD, PorousMedium, Heat source/sink.
1. Introduction. Fluid flow and heat transfer in porous media is animportant subject in hydrology. It is of vital interest in petrolium and chemicalengineerings. To study the underground water resources and seepage of water in adam, one needs to investigate the flows through a porous media. Flow in a channelis very fundamental problem and attracted the attention of many research workers.Schlichting and Gersten [10], Bansal [4] and some others have considered theproblem in their books. Considering magnetic effect Attia and Kolb [1], Yen andChang [15] and Singh [11] studied the flow between two parallel plates. Jain andBansal [5] considered temperature dependent viscosity in the Couette flow. Attia[2] studied hall current effects on the velocity and temperature fields of an unsteadyHartmann flow. Numerical solution of free convection MHD micropolar fluid flowbetween two parallel porous vertical plates is also discussed by Bhargava et al. [3].
20
Pulsatile flow in a channel or in a tube has application in Biofluid flow inthe dialysis of blood in artificial kidneys. Sharma and Mishra [12] , Sarangi andSharma [14] and Sharma et al. [13] have solved the problems with time dependentpressure gradiant.
In geothermal region situation may arise when the flow becomes unsteadyand slip at the boundary may take place as well. At high altitude, the study of slipflow becomes very important. Kepping this in mind Soundlegekar and Arnake [9].Johri and Sharma [6], Jain [7] and Jothimani and Anjali Devi [8] considered theslip flow boundary conditions in their problems.
The aim of present paper is to investigate the MHD flow and heat transferof a viscous incompressible fluid in slip flow regime with variable permeabilitybounded by two parallel porous plates. It is found that the skin friction at both theplates increases with increase of slip parameter h1.
2. Formulation of the Problem. Let us consider the unsteadyincompressible MHD heat teransfer flow through a porous medium of variable
permeability nteKtK 10 in slip flow regime bounded by two infinite long
parallel thin porous plates. The parallel plates are placed at a distance h apart.Let x*-axis be taken along the direction of plates and y*-axis is taken in normaldirection to the plates.
The equations of continuity, motion and energy for unsteady flow of anincompressible viscous fluids are
nteAvvtv
10 0 ...(1)
nteK
uuB
y
uxp
yu
tv
10
202
2
...(2)
yp
tv
...(3)
sp TTQyu
yT
kyT
tT
C
2
2
2
...(4)
The boundary conditions are
::0
hyy
,1
,0
10 dydu
LeAuu
unt
dydT
LTT
TT
20
0
...(5)
where u, v are the components of velocity along x-axis and y-axis respectively, t thetime, v0 is the cross flow velocity, p the pressure, is the density, Cp the specfic
heat at constant pressure, k the thermal conductivity, the coefficient of viscocity,
21
Q the volumetric rate of heat generation, T the fluid temperature, Ts the statictemperature, coefficient of the electrical conductivity, B0 the coefficient of
electromagnetic induction, here LLm
mL ,
2
1
11
being mean free path and m1
the Maxwell' s reflection coefficient.
A and n are real positive coefficients such that 1ANow introducing the following non-dimensional quantities
,*hx
x ,*hy
y ,*
uh
u ,* 2ht
t
,Prk
C p ,* 2
2
vph
p
,RevhVo ,* 2
0
hK
K ,0 s
s
TTTT
spc TTCh
vE
02
2
, ,2
pvCQh
,22
02
vhB
M
vnh
n2
* .
Equations (3) and (4) reduce to the following form after dropping theasterisks over them
ueAK
uMyu
xp
yu
eAtu
ntnt
11
1Re 22
2
...(6)
PrPr1RePrPr2
2
2
cnt E
yu
yyeA
t ...(7)
with corresponding boundary conditions
,1
0
10 dydu
heARu
unt
dyd
h
21
1
1 0
yatyat ...(8)
Here v
huR
hL
hhL
h 00
22
11 ,, .
3. Solution of the Problem. We assume
yeyyueAyuu
eAxp
nt
nt
nt
10
10
1
...(9)
Using equation (9) in the equations (6) and (7) and equating the coefficients of thesome powers of , we get the following set of ordinary differential equations after
neglecting the coefficients of 2o .
22
1 1Re 02
00 uKMuu ...(10)
0012'
11 1Re1 1Re uKuunKMuu ...(11)
2'00
'0
''0 PrPrRePr uEc ...(12)
'1
'0
'011
'1
''1 Pr2RePrPrPrRePr uuEAn c ...(13)
with corresponding boundary conditions becomes
:1:0
yy
,,0
'0100
0
uhRuu
,,0
'1101
1
uhRuu
,1,1
'020
0
h
'121
1 0h ...(14)
Equations (10) to (13) are second order linear differential equations withconstant coefficients, the solution of which are
65443121214321 AeAeAeCeCeAAeCeCu ymymymymntymym ...(15)
ymymymymymym
ntymmymymymym
eAeAeAeAeCeC
eeAeAeAeCeC216587
212165
213
212111087
92
82
765
ymmymymmymmymm eAeAeAeAeA 21141312118
217161514
ymymmymmymm eAeAeAeA 2214232 2
22212019 ...(16)where
KMA 121
,
2
14ReRe 22
1
KMm
,
2
14ReRe 22
2
KMm
,11
1
1121
012111
12
2
mhemhe
RAmheAC mm
m
,11
11
1121
01112
12
1
mhemhe
RmheAC
mm
m
,2
4ReRe 12
3
Anm
,
2
4ReRe 12
4
Anm
, Re 1
112
KC
mCA
,Re 2223
KC
mCA ,Re 1121
24 Anmm
AA
,Re 1222
35 Anmm
AA
,
1
1
16 An
AA
,65443 AAACC
3141
65431215114604 11
11134
321
mhemheAAAmhemheAmheAAR
C mm
mmm
,
23
,2
422
5
PrRePrPrRe
m ,2
422
6
PrRePrPrRe
m
,Re24 1
21
21
21
7
PrmPrm
CmPrEA c ,
Re24 222
22
22
8
PrmPrm
CmPrEA c
,
PrRePr
2
212
21
21219
mmmm
CCmmA ,1 98765 AAACC
,11
11221
5262
987529212
822
7126
56
52121
mhemhe
AAAmheeAmmeAmeAmhC mm
mmmmm
,
2422
7
nPrRePrPrRe
m
,2
422
8
nPrRePrPrRe
m
,
525
5510
nPrPrRemm
CPrReAmA
,
626
6611
nPrPrRemm
CPrReAmA
,
24
2
121
7112
nPrPrRemm
APrReAmA
,
24
2
222
8213
nPrPrRemm
APrReAmA
,21
221
21914
nPrmmPrRemm
mmPrReAAA
,
2
312
31
313115
nPrmmPrRemm
mmCCPrEA c
,
2
412
41
414116
nPrmmPrRemm
mmCCPrEA c
,
24
2
121
4211
17
nPrPrRemm
AmCPrEA c
,
2
212
21
215118
nPrmmPrRemm
mmACPrEA c
,
2
322
32
323219
nPrmmPrRemm
mmCCPrEA c
,
2
422
42
424220
nPrmmPrRemm
mmCCPrEA c
,
2
212
21
214221
nPrmmPrRemm
mmACPrEA c
,
24
2
222
2252
22
nPrPrRemm
mACPrEA c
, 2221201918171615141312111087 AAAAAAAAAAAAACC
2121651421
2132
21211161052 22 mmmmmm eAmmeAmeAmeAmeAmh
24
21141311821
217116411531 2 mmmmmmm eAmmeAmeAmmeAmm
2214231 2222212120421931 2 mmmmmmm eAmeAmmeAmmeAmm
4131212165
1615142
132
121110mmmmmmmmmm eAeAeAeAeAeAeA
2214232211 22221201918
217
mmmmmmmmmm eAeAeAeAeAeA
,11
1
7282
22212019
18171615141312111072
87
7
mhemheAAAA
AAAAAAAAAmhe
C mm
m
s
4. Skin-Friction. The coefficient of skin friction at the lower plate is
0
0
yat
yf yu
C
10 '' ueAu nt
524144332211 AmAmCmCmeACmCm nt
The coefficient of skin friction at the lower plate is
1
1
yat
yf yu
C
214321524144332211
mmmmntmm eAmeAmeCmeCmeAeCmeCm
5. Nusselt Number. The rate of heat transfer in terms of Nusselt numberat the lower plate is
0
0
y
y yNu
105887792182716655 22 AmCmCmeAmmAmAmCmCm nt
164115311221132121116 22 AmmAmmAmmAmAmAm
2222121204219321821171 22 AmAmmAmmAmmAmmAm
The rate of heat transfer in terms of Nusselt number at the upper plate is
1
1
y
y yNu
ntmmmmmm eeAmmeAmeAmeCmeCm 212165921
282
2716652 22
25
216587 2132
21211161058877 22 mmmmmm eAmeAmeAmeAmeCmeCm
413121164115311421
mmmmmm eAmmeAmmeAmm
121712 meAm 3221
19321821mmmm eAmmeAmm
422042
mmeAmm 221 22222121 2 mmm eAmeAmm .
6. Result and Discussion. In order to understand the solutions physically,we have calculated the numerical values of the velocity distributions [Figure 1.0],temperature distributions [Figure 2.0], skin friction [Figure 3.0 and Figure 4.0]and Nusselt number [Figure 5.0 and Figure 6.0] for different values of h1 (slipparameter), h2 (temperature jump), M (Magnetic parameter), Re (Reynolds number),(volumetric rate of heat generation) and K (Permeability parameter).
In Figure 1.0, the velocity distribution (u) is plotted against y. It is beingobserved that velocity increases with the increase in h1, Re and Ro but decreaseswith increase in M and K. It is also observed that increase in velocity with theincrease in injection parameter takes place for both the cases slip flow or no slipflow.
In Figure 2.0, temperature distribution () is plotted against y. It is beingseen that temperature increases as M, K, Pr and increase but phenomena reversesfor the case of h1, h2 and Ro. It is interesting to note that for the case of Pr=7.0(water as a fluid), the temperature first increases and after some channel width itdecreases asymptotically, while for the case of Pr=0.71 (air as a fluid) thetemperature decreases continuously.
Skin friction which is plotted in Figures 3.0 and 4.0 against K is having aworth noting observation. It is being observed that skin friction at lower plate
0yfC increases with the increase in Re and h1 but decreases with the increase in
M and t while skin friction at the upper plate 1yfC increases with increase in M
and h1 but decreases with the increase in Re and t. Moreover, increase in K has
very small increasing effect on 0yfC but increase in K decreases
1yfC sharply
for lower values of K than for higher values.Nusselt number at the lower and upper plates are plotted against t in Figures
5.0 and 6.0 respectively. From the figures it is observed that Nusselt number atboth the plates increases with the increase in and K. Moreover, Nusselt number(Nu) at the lower plate remains positive for Pr=7.0 but for the Pr=0.71 it isnegative. As expected Nusselt number decreases for sink tha source at both theplates.
26
27
28
29
30
31
32
REFERENCES[1] H.A. Attia and N.A. Kotb, MHD flow between two parallel plates with heat transfer, Acta Mech,
117 (1996) 215-220.
[2] H.A. Attia, Hall current effects on the velocity and temperature fields of an unsteady Hartmannflow, Candian Jounral of Physics., (1998), 739-746.
[3] R. Bhargava, L. Kumar and H.S. Takhar, Numerical solution of free convection MHD micropolarfluid flow between two parallel porous vertical plates, International Journal of EngineeringScience, 41(2) (2003), 123-136.
[4] J.L. Bansal, Viscous Fluid Dynamics, OXFORD and IBH publishing company (2004).
[5] N.C. Jain and J.L. Bansal, Couette flow with transpiration cooling when the viscosity of the fluiddepends on temperature, Proc. Ind. Acad. Sci., 77(A)(4) (1973), 184-200.
[6] A.K. Johri and J.S. Sharma, MHD fluctuating flow of Maxwell viscoelastic fluid past a porous platein slip flow regime, Acta Cinecia Indica, 5 (1979), 101.
[7] N.C. Jain, Viscoelastic flow past an infinite plate in slip flow regime with constant heat flux, TheMathematics Education, 24 (1990), p.3.
[8] S. Jothimani and S.P. Anjali Devi, MHD Couette flow with heat and slip flow effects in an inclinedchannel, Indian J. Math., 43(1), (2001), 47
[9] V.M. Soundalgekur and R.N. Arnake, free convection effects on the oscillation flow of an electricallyconducting rarefied gas over an infinite vertical plate with constant suction, Appl. Sce. Res., 34(1978), 49.
[10] H. Schlichting, K. Gersten, Boundary Layer Theory, Springer Verlage Berlin (1999).
[11] K.D. Singh, An oscillatory hydromagnetic coquette flow in rotating system, ZAMP, 80 (2000),429-432.
[12] P.R. Sharma and U. Mishra, Pulsatile MHD flow and heat transfer through a porous channel, Bull.Pure and Appl. Sciences, 21E(1), (2002), 93-100.
[13] P.R. Sharma, R.P. Sharma, U. Mishra and N. Kumar, Unsteady flow and heat transfer of a viscousincompressible fluid between parallel porous plates with heat source/sink, Applied SciencePeriodical, 6 (2) (2004).
[14] K.C. Sarangi, V.K. Sharma, Unsteady MHD flow and heat transfer of a viscous incompressible,fluid through a porous medium of variable permeability bounded by two parallel porous plateswith heat source/sink, Ganita Sandesh, 19(2) (2005), 179-188.
[15] J.T. Yen and C.C. Chang, Magneto hydrodynamics Couette flow as affected by wall electricalcondition, ZAMP, 15 (1964), 400-407.
33
J~n–an–abha, Vol. 38, 2008
REDUCIBILITY OF THE QUARDUPLE HYPERGEOMETRICFUNCTIONS OF EXTON
ByB. Khan
Department of Basic SciencesHamelmalo Agricultural College, Keran P.O. Box 397, Eritrea.
e-mail : [email protected]
M. KamarujjamaDepartment of Mathematics
Yanbu University College, Royal Commission of Yanbu, P.O. Box 31387, YanbuAl-Sinayah Kingdom of Saudi Arabia
e-mail:[email protected]
Nassem A. KhanDepartment of Applied Mathematics, Z.H. College of Engineering and
Technology, Aligarh Muslim University, Aligarh-202002, Indiae-mail:[email protected]
(Received : February 05, 2008)
ABSTRACTSome reducible cases of the quadruple hypergeometric functions K1,K2 and
K3 of Exton are obtained. Our main results transform a quardruple function intoKampé de Fériet's double or Srivastava's triple hypergeometric F(3) functions ortheir combinations. Many reduction formulae involving Saran's functions EE, EF
Lauricella's function 3CF , Kampé de Fériet's function ,::
;:rqp
tsF generalized
hypergeometric series qp F and Appell's function F4 are obtained. It is shown that
Exton's result [7] and [8] are not correct. These results in their correct forms areobtained as special cases of our main reduction formulae.2000 Mathematics Subject Classification : 33C70.Keywords : Generalized hypergeometric functions, Kmapé de Fériet's & Appell'sfunctions, Quadruple hypergeometric functions, and Lauricella's function.
1. Introduction. In the year 1972, Exton [6]] defined and examined a fewproperties of quadruple hypergeometric functions. In his notations K1, K2 and K3are given by [8;p.78]
34
0,,,
1 !!!!,,,;,,,;,,,;,,,
qpnm pnqm
qpnmqpnmqpnm
qpnmfed
tzyxcbatzyxdfedcbbbaaaaK (1.1)
0,,,
2 !!!!,,,;,,,;,,,;,,,
qpnm qpnm
qpnmqpnmqpnm
qpnmgfed
tzyxcbatzyxgfedcbbbaaaaK (1.2)
and
0,,,
3 !!!!,,,;,,,;,,,;,,,
qpnm pnqm
qpnmqpnmqpnm
qpnmed
tzyxcbatzyxdeedccbbaaaaK (1.3)
respectively, where
.a
kaa k
The convergence of the series K1, K2 and K3 are investigated by Exton, by mean ofthe general theory obtained in [8, p. 65(2.9)].
In this paper, various reducible cases of the functions K1, K2 and K3 areobtained. Our results transform a function of four variables into a double or tripleseries or their combinations. The main insterst of the result obtained in Section 2
is that the series of Lauricella's 3CF is tranformed into a combinations of Kampé
de Fériet's functions. In Section 3 a reduction of FE in terms of generalized
hypergeometric series 34 F is given. In Section 4 a reduction of K3 into a
combination of four F(3)'s is obtained which further gives a transformation ofSaran's FF into a combination of two Kampé de Fériet's functions.
In a paper of Exton [7, p. 66(3.2)], the reduction formula for Saran's functionFE [11] is given in the form
,4,;;1,,;,,:2
1,
2,;,,:,,;,,;,,;,,
32322
3232213:3:1
3,3:0321221
yx
ccccBAb
ccccbBAbaFyyxcccbbbaaaFE (1.4)
where as in a recent book of Exton [8, p.134 (4.7.8)], the reduction of FE is given by
,4,;;1,,;,,;2
1,
2,;,,:,,;,,;,,;,,
32312
3232213;3:1
3;3:0321221
yx
ccccBAb
ccccbBAbaFyyxcccbbbaaaFE (1.5)
where 3;3:13;3:0F is Kampé de Fériet's function in contracted notation of Burchnall and
Chaundy [2;p. 112-113]. In equation (1.4) and (1.5), A, and B are arbitrary
35
parameters in which numerator parameters are cancelled by denominatorparameters A and B.
It is to be noted that (1.4) and (1.5) are incorrect. In fact, their correct formis obtained in Section 2 and follows as a special case of our reduction formula ofK1.
2. Reducibility of K1-Function. From (1.1), we have
0,
1 !!,,,;,,,;,,,;,,,
qm qm
qmqmqm
qmd
txcbatzyxdfedcbbbaaaaK
zyfembqmaF ,;,;;4 , (2.1)
where F4 is Appell's function of fourth kind [1,p.14(14)].Putting z=y in (2.1) and using a result of Burchnall [3; p.101 (37)], we get
0,
1 !!,,,;,,,;,,,;,,,
qm qm
qmqmqm
qmd
txcbatyyxdfedcbbbaaaaK
y
fefe
fefembqmaF 4
; 1,,
;2
1,
2,,
34 , (2.2)
where 4F3 is generalized hypergeometric function [10; p. 73(2)]. Now expressing
4F3 in power series form and interpreting the result in the form of Srivastavafunction F(3) [12; p.428], we get
tyx
fefed
cfefe
baFtyyxdfedcbbbaaaaK ,4,; ;1,,; :; ; ::
;;2
1,
2; : ; ;::,,,;,,,;,,,;,,, 3
1 (2.3)
when t=0 or c=0, (2.3) reduces to
,4,;1,,;:–
;2
1,
2–;:,,,;,,; 2;0:2
3;1:03
yx
fefed
fefebaFyyxfebaFC (2.4)
where 3CF and 2;0:2
3;1:0F are Lauricella [9,p.114] and Kampé de Fériet [14;p.23 (1.2)]
functions, respectively.When x=0 in (2.3), we get the following correct form of the reduction formula (1.4)and (1.5) of Exton [8,p. 134 (4.7.8)], for Saran's function FE [11].
yt
feedBA
fefebCBAaFyytfedbbcaaaFE 4,
;1,;,,:
;2
1,
2,;,,:,,;,,;,,;,, 3;3:1
3,3:0 . (2.5)
36
Setting f=e, z=–y and t=x in (2.1) and using a result of Srivastava [13; p.296 (9)].we get
0,1 !!
a,,,;,,,;,,,;,,,
qm qm
qmqmqm
qmd
xcbxyyxdeedcbbbaaaaK
.4;,,4 2;2
1,
2
21
,2
1
2
,2
3
yFmbmb
e
qma
e
qma
e ...(2.6)
Now expressing 4F3 in power series form and using a result [10;p. 22(lemma 5)],we get
xyyxdeedcbbbaaaaK ,,,;,,,;,,,;,,,1
0
12
222 ,;;,2;2
!n nn
nnn xxdcnbnaF
neeyba
, ...(2.7)
where F1 is Appell's function of first kind [1;p. 14 (11)].Again, using a result [5; p. 239(11) for F1 in (2.7), and a result of Carlson [4;
p. 234(10)] for the resulting Gauss function 2F1 [10, p. 45 (1), we get the followingreduction of K1 into a combination of two Kampé de Fériet's functions.
xyyxdeedcbbbaaaaK ,,,;,,,;,,,;,,,1
d
xcbayxcbcbee
edd
bbcbcaaa
F
222,0:45,3:0 4,
;2
1,
2,
21
,2
,;2
1,
2,
21
:
;2
1,
2;:
21
,2
,2
1,
2
.4,;
21
,2
,2
1,
2,;
22
;2
1,,
23
:
;2
1,
2;:
22
,2
1,
22
,2
1222,0:4
5,3:0
yxcbcbeee
dd
bbcbcbaa
F...(2.8)
when c=0 in (2.8), we get
yyxeedbaFC ,,;,,;;3
dabx
yxeee
dd
bbaa
F
220,0:43,3:0 4,
;2
1,
2,;
21
,2
,21
:
;;:2
1,
2,
21
,2
37
.4,;
21
,2
,;2
2;
21
,23
:
;;:2
2,
21
,2
2,
21
220,0:43,3:0
yxeee
dd
bbaa
F...(2.9)
3. Reducibility of K2 Function. Writing (1.2) in the form
0,
2 !!
a,,,;,,,;,,,;,,,
qp qp
qpqpqp
qpgf
tzcbtzyxgfedcbbbaaaaK
yxedpbqpaF ,;,;;4 . ...(3.1)
Putting y=x in (3.1) and using [3; p. 101(37)], we get
tzxxgfedcbbbaaaaK ,,,;,,,;,,,;,,,2
. ,,4;;,1,,:;;::
;;,2
1,
2:;;::3
tzxgfeded
ceded
baF ...(3.2)
Setting. z=–x and f=d in (3.1) and using a result of Srivastava [13; p. 296 (9)], weget
0,
2 !!
a,,,;,,,;,,,;,,,
qn qn
qnqmqn
qnge
tycbtxyxgdedcbbbaaaaK
.4; 2;2
1,2
;2
1
,2
1
, 2
,2, 34
xFnbnb
d
qna
d
qna
d ...(3.3)
Now expanding 4F3 into series form and using [10; p. 22 (Lemma 5)], we get
txyxgdedcbbbaaaaK ,,,;,,,;,,,;,,,2
,,;,;,2;2
! 20 2
222 tygecmbmaF
mddxba
m mm
mmm
...(3.4)
where F2 is Appell's function of second kind [1; p. 14(12)]. Putting g=c in (3.4),making use of a result [5;p.238(2)] together with a result of Carlson [4;p.234 (10)],we get
txyxcdedcbbbaaaaK ,,,;,,,;,,,;,,,2
38
22
0,0:43,3:0 1
,12
;2
1,
2,
21
;2
1,
2,:
;;:2
1,
2,
21
,21
ty
tx
eee
ddd
bbaa
Ft a
.
1,
12
;2
2,
21
,23
;2
1,
2,:
;;:2
2,
21
,2
2,
21
1
220,0:4
3,3:01
ty
tx
eeddd
bbaa
Fte
abya ...(3.5)
When y=0, (3.5) yields a reduction FE into 4F3 in the form
xxtddcbbcaaaFE ,,;,,,,,;,, .1
4
;2
1,
2,
:2
1,
2;
21
,21
2
34
tx
ddd
bbaa
Ft a ...(3.6)
4. Reducibility of K3- Function. Expressing (1.3) in the form
tzyxdeedccbbaaaaK ,,,;,,,;,,,;,,,3
0,
1 .,;,;;!!qm qm
qmqmqm zyeqcmbqmaFqmd
txcba...(4.1)
Putting z= y in (4.1) and using [5; p. 239 (11)], we get a reduction formula of K3intoF(3)in the form
tyyxdeedccbbaaaaK ,,,;,,,;,,,;,,,3 .,,
;;:,;;::;;;:;;::,3
tyx
ecbdcbcba
F ...(4.2)
When t=0, (4.2) reduces to
yyxeedbcbaaaFF ,,;,,;,,;,, , ,;:,,:
;;;,0;1:21;2:0
yx
ecbdbcba
F ...(4.3)
where FF is another Saran's function[11].Furthermore for d=b,(4.3) reduces to a reduction formula of FF into F4 in
the form
. ,;,;;,,;,,;,,;,, 4 yxccbcbaFyyxeebbcbaaaFF ...(4.4)
Again, writing
xzyxcbaeecbaccbbaaaaK ,,,,1,,,1;,,,;,,,3
39
0,
1 , ,;1,;;!!pn pn
pnpnpn xxcbapcnbpnaFpne
zycba...(4.5)
and making use of [5, p. 239(11)] and Goursat's quadratic transformation [5, p.113(34)], we have
xzyxcbaeecbaccbbaaaaK ,,,,1,,,1;,,,;,,,3
0,
212 .1
4
;1
;2
1,
2!!111
pn pn
pn
pnpna
x
x
cba
pnapnaF
pnex
zx
ycba
x
(4.6)Now using a double series identity of Srivastava [15; p. 196 (23)]
0, 0,0,
2,122,2,pn pnpn
pnApnApnA
0,0,
12,1212,2pnpn
pnApnA ...(4.7)
in (4.6), we get a reduction of K3 in terms of combination of four F(3)'s in the form
xzyxcbaeecbaccbbaaaaK ,,,;1,,,1;,,,;,,,3
22
23
1,
1,
1
4
;21
;21
;1:;2
1,
2;::
;2
1;
2;
21
,2
;:;;::2
1,
21x
zx
y
x
x
cbaee
ccbbaa
Fx a
22
23
1 1,
11
4
;21
;23
;1:;2
2,
21
;::
;2
1;
2;
22
,2
1;:;;::
22
,2
1
1 xz
xy
x
x
cbaee
ccbbaa
Fxe
abya
22
23
1 1,
11
4
;23
;21
;1:;2
2,
21
;::
;2
2;
21
;2
1,
2;:;;::
22
,2
1
1 xz
xy
x
x
cbaee
ccbbaa
Fxe
acza
22
23
2 1,
11
4
;23
;23
;1:;2
3,
22
;::
;2
2;
21
;2
2;
21
;:;;::2
3,
22
11
1x
zx
y
x
x
cbaee
ccbbaa
Fxee
bcyzaaa
...(4.8)When y=0 in (4.8), we get a reduction for Saran function FF involving thecombination of the two Kampé de Fériet's functions
40
xxzcbacbacbcaaaFF ,,;1,1;,,;,,
2
22,0:2
3,1:0 1,
1
4
21
,2
;21
;1:
;2
1;
2;:
21
,21
xz
x
xee
cba
ccaa
Fx a
.
1,
1
4
22
,2
1;
23
;1:
;2
1;
2;:
22
,2
1
1
2
22,0:2
3,1:01
xz
x
xee
cba
ccaa
Fxe
acza ...(4.9)
REFERENCES[1] P. Appell, et J. Kampé de Fériet, Fonctions Hypergeometriques et Hyperspheriques. Polinomes
d'Hermite, Gauthier-Villars, Paris, 1926.[2] J.L. Burchnall and T.W. Chaundy, Expansions of Appell's double hypergeometric function (II),
Quart. J. Math., 12 (Oxford series) (1941), 112-128.[3] J.L. Burchnall, Differential equations associated with hypergeometric functions, Quat. J. Math.,
(Oxford series), 13 (1942), 90-106.[4] B.C.Carlson, Some extensions of Lardner's relations between 0F3 and Bessel function, SLAM J.
Math. Anal., 1(2) (1970), 232-242.[5] A. Erdélyi, et al., Higher Transcendental functions, Vol.1 McGraw-Hill, New York (1953).[6] H. Exton, Certain hypergeometric functions of four variables, Bull. Greek Math. Soc. N.S., 13
(1972), 104-113.[7] H. Exton, On the reducibility of the triple hypergeometric functions FE, FG and HB, Indian J.
Math., 18(2) (1976), 63-69.[8] H. Exton, Multiple Hypergeometric Functions and Applications, Ellis Horwaood Limited,
Chichester (1976).[9] G. Lauicella , Sulle Funzioni Ipergeometriche a piú variabili, Rend. Circ. Mat. Palermo, 7
(1893), 111-158.[10] E. D. Rainville, Special Functions, The Macmilan Comp., New York , 1960, Reprinted Chelsea
Publ. Co. Bronx., New Yrok, 1971.[11] S. Saran, Hypergeometric functions of three variables, Ganita, 5 (1954), 77-91.[12] H. M. Srivastava, Generalized Neumann expansions involving hypergeometric functions, Proc.
Camb. Philos. Soc., 63(1-2) (1967), 425-429.[13] H.M. Srivastava, On the reducibility of Appell's function F4, Canad. Math. Bull., 16(2) (1973),
295-298.[14] H.M. Srivastava and M.A. Pathan, Some bilateral generating functions for the extended jacobi
polynomials-I, Comment. Math. Univ. St. Pauli., 28(1) (1979), 23-30.[15] H.M. Srivastava, A note on certain identities involving generalized hypergeometric Series, Nederal.
Akad. Wetersch. Proc. Ser. A. 82(2)=Indag. Math., 41(2) (1979), 191-201.
J~n–an–abha, Vol. 38, 2008
CONVERGENCE RESULT OF (L,) UNIFORM LIPSCHITZASYMPTOTICALLY QUASI NONEXPANSIVE MAPPINGS IN
UNIFORMLY CONVEX BANACH SPACEBy
G.S. SalujaDepartment of Mathematics and I.T.
Government College of Science, Raipur-4920101, ChhattisgarhE-mail : [email protected]
(Received : February 25, 2008)
ABSTRACTIn this paper, we have proved common fixed point of the modified Ishikawa
iterative sequences with errors for two (L,) uniform Lipschitz asymptoticallyquasi nonexpansive mappings in a unifromly convex Banach space. Our resultextends and improves the corresponding result of Khan and Takahashi [1], Tanand Xu [5] and many others.2000 Mathematics Subject Classification : 47H10.Keywords : Asymptotically quasi nonexpansive mapping, common fixed point,(L,) uniform Lipschitz mapping, the modified Ishikawa iteration scheme witherrors, uniformly convex Banach space.
1. Introduction and Preliminaries. Let C be a subset of a normed spaceE, and let T be a self mapping of C. The mapping T is said to be an asymptotically
nonexpansive mapping, if there exists a sequence {kn} in [0,) with 0lim
nn
k
such that
yxkyTxT nnn 1
for all Cyx , and 1n .
The mapping T is said to be an asymptotically quasi nonexpansive mapping,
if there exists a sequence {kn} in [0,) with 0lim
nn
k and F(T) such that
pxkpxT nn 1
for all Cyx , , for all TFp and 1n .
It there are constant L>0 and >0, such that
yxLyTxT nn
42
for all x,yC and nN, then T is (L,) uniform Lipschitz.In 1994, Tan and Xu [5] had proved the problem on convergence of Ishikawa
iteration for asymtotically nonexpansive mapping on a compact convex subset ofa uniformly convex Banach space. In 2001, Qihou [2], presents the necessary andsufficient conditions for the Ishikawa iteration of asymptotically quasi-nonexpansive mapping with an error member on a Banach space converging to afixed point. Again in 2002, the same author has proved the convergence of Ishikawaiteration of a (L,) unifrom Lischitz asymptotically quasi nonexpansive mappingwith an error member on a compact convex subset of a uniformly convex Banachspace based on some results of [2].
Recently, Khan and Takahashi [1] considered the problems of approximatingcommon fixed point of two asymptotically nonexpansive self mappings S and T ofC through weak and strong convergence of the iterative sequence {xn} defined by
Cx 1
1 ,11 nySaxax nn
nnnn
1 ,1 nxTbxby nn
nnnn (A)
where {an} and {bn} are some sequences in [0,1].Motivated and inspired by Khan and Takahashi [1] and others we study the
following iteration scheme: Let C be a nonempty subset of a Banach space E andS,T:CC be two asymptotically quasi nonexpansive mappings. Consider thefollowing iterative sequence {xn} with errors defined by
Cx 0
,1 nnnn
nnnn ucySbxax
,nnnn
nnnn vcxTbxay (B)
where ,, Cvu nn ,1,,,,0, nnnnnn cbacbaCn ,1 nnnnnn cbacba for
all
11. ,,
n nn n ccNn
In this paper, we have proved convergence of common fixed point of modifiedIshikawa iteration scheme with errors of (L,) uniform Lipschitz asymptoticallyquasi nonexpansive mappings on a compact convex subset of a uniformly convexBanach space. Our result extends and improves the corresponding result of Tanand Xu [5], Khan and Takahashi [1] and many others.
We need to following result and lemmas to prove our main result:Theorem LQ [2, Theorem 3]: Let E be a nonempty closed convex subset of a
43
Banach space, T is an asymptotically quasi-nonexpansive mapping on E, and F(T)
nonempty. Given 1
,n nu Ex 1 , defined 1nnx as
,1 nnnn
nnnn mcyTbxax
,, NnlcxTbxay nnnn
nnnn
where mn, lnE, and , 1nnm 1 nnl are bounded, nnnn acba 1
.1,,,,,0, nnnnnnnn cbacbacb Then 1 nnx converges to some fixed point p of
T if and only if there exists some infinite subsequence 1 knkx of
1 nnx which
converges to p.Lemma 1.1 [J.Schu's Lemma]: Let X be a uniformly convex Banach space,
,10 nt ,, Xyx nn ,suplim axnn
,suplim aynn
and
,1lim aytxt nnnnn
0a . Then 0lim
nn
nyx .
Lemma 1.2 [2, Lemma 2] : Let nonnegative series ,1nna ,1
nn 1nn satisfy
,1 nnnn ,Nn and 1
,n n
1
;n n then n
n
lim exits.
Lemma 1.3: Let C be a nonempty convex subset of normed space E, S,T: CC are
two asymptotically quasi nonexpansive mappings, and F=F(S)F(T) nonempty,
1 ,n nk for all Cx 0 , let
,1 nnnn
nnnn ucySbxax
,nnnn
nnnn vcxTbxay
where ,1,,,,,0,,, nnnnnnnn cbacbaCnCvu nnnnn bacba 1
,nc for all
11 .,, n nn n ccNn Then
(a) ,,,1 21 TFSFFpNnmpxkpx nnnn
where pucpvckbm nnnnnnn 1 .
44
(b) There exists a constant M>0 such that pxMpx nmn
1,,,
mn
nkk TFSFFpNmnmM .
Proof of (a) For any pF, we have from (B)
pxn 1 pucpySbpxa nnnn
nnn
pucpykbpxa nnnnnnn 1 (1)
and
pyn pucpxTbpxa nnnn
nnn
pvcpxkbpxa nnnnnnn 1
pvcpxkba nnnnnn 1 (2)
From (1) and (2), we have
pxn 1 pucpvcpxkbakbpxa nnnnnnnnnnnn 11
pxkbbpxakbpxa nnnnnnnnnn 211
pucpvkcb nnnnnn 1
nnnnnnnnn capxakcapxa 111
nnnn mpxkb 21
nnnnn mpxkbapnxnanknapnxna 21111
nnnnnnnnnnn mpxkbapxakapxa 22 1111
nnnnnnnnn mpxbakapxka 22 111
nnnnnnn mpxkapxka 22 111
nnn mpxk 21
where pucpvkcbm nnnnnnn 1 .
45
Proof of (b) Since xex 1 for all 0x . Therefore from (a) it can be obtained
that
px mn 112
11 mnmnmn mpxu
112 1
mnmn
u mpxe mn
12121 22
2
mnmnmnmnmn mmumn
uu epxe
122
22 121
mnmnu
mnuu mmepxe mnmnmn
... ... ...
1
2211 mn
nkk
un
u mepxemn
nk kmn
nk k
1mn
nkkn mmpxM
where
nk kueM 2 .
This completes the proof of (b).2. Main Result.
Theorem 2.1 : Let C be a nonempty compact subset of uniformly convex Banachspace X, and S,T:CC be two (L,) uniform Lipschitz asymptotically quasi
nonexpansive mappings with sequence {kn}[0,) such that 1n nk . Let
Cx 0
, and
,1 nnnn
nnnn ucySbxax
nnnn
nnnn vcxTbxay ,
where ,1,,,,,0,,, nnnnnnnn cbacbaCnCvu ,1 nnnnnn cbacba
Nnbbaaa nnnn ,1,10,0,10 , ,0lim
nn
b
1
,n nc
1n nc . If TFSFF . Then
1 nnx converges to some
common fixed point p of S and T.Proof. For any pF, we have from Lemma 1.3(a)
46
nnnn mpxkpx 2
1 1 ,
where .1 pucpvkcbm nnnnnnn Since 1 ,n nk
1 ,n nc
1
,n nc C is bounded, thus we know from Lemma 1.2 that pxn
n
lim exists.
On the other hand, we have
pyn pvcpxTbpxa nnnn
nnn
pvcpxkbpxa nnnnnnn 1
Thus
.limsuplimsuplim pxpxpy nn
nn
nn
Note that
.lim1suplimsuplim pxpykpyS nn
nn
nnn
n
and
pxnn
1lim pucySbxa nnnn
nnnn
lim
pu
bc
pySbpuac
pxa nn
mn
nnn
n
nnn
n 22lim
pxnn
lim .
Thus from J. Schu's Lemma, we have
022
lim
pu
bc
ac
ySx nn
n
n
nn
nn
n .
Note that 022
lim
pu
bc
ac
nn
n
n
n
n.
Therefore we have
0lim
nn
nn
ySx . (3)
47
Since C is compact, the sequence 1 nnx has a convergent subsequence 1 knk
x .
Let
.lim pxkn
k
...(4)
Thus from (3) and 0lim
nn
c , we have
kk nn xx 1 kkkk
k
kkk nnnnn
nnn xucySbxa
kkkk
k
kkkk nnnnn
nnnn xucySbxcb 1
0kkkkk
k
k nnnnnn
n xucxySb as k . (5)
Note that
;0lim,0lim
nn
nn
cb
therefore, we have
nn xy nnnnn
nnn xvcxTbxa
nnnnn
nnnn xvcxTbxcb 1
0 nnnnnn
n xvcxxTb as n (6)
Thus from (3) and (4), we have
.lim pySk
kn
n
k
(7)
Thus pxkn
k
1lim . Similary, px
knk
2lim and
.lim 11 pyS
k
kn
n
k
(8)
Now
0 nn
nnnnn
n yTyxyyTx as n . (9)
Therefore from (4) and (9), we have
.lim pyTk
kn
n
k
(10)
48
Thus pxkn
k
1lim and since px
knk
2lim so
pyTk
kn
n
k
1
1lim (11)
From (3) – (8), we have
k
k
k
k
k
k
k
k
k
kn
nn
nn
nn
nn
n xSxSxSySySpSpp 11
11
11
11
1
Pnn
nn
nn SySySxS
k
k
k
k
k
k 111
kkkkkkk
knnnnnnn
n yxLxxLxyLySp 11111
0
pySLk
kn
n as k
Similarly we can show that
0Tpp as k .
Thus p is a common fixed point of S and T. Since the subsequence 1knk
x of 1nnx
converges to p from (4), we have TFSFFpxnn
lim from Theorem LQ.
This completes the proof.Remark 2.2 . Theorem 2.1 improves and extends the corresponding result of Khanand Takahashi [1] in several aspects.Remark 2.3 . Theorem 2.1 also extends the corresponding result of Tan and Xu[5] to the case of more general class of asymptotically nonexpansive mappings.Remark 2.4 . Theorem 2.1 also generalizes the result of Qihou [3].
REFERENCES[1] S.H. Khan and W. Takahashi, Approximating common fixed points of two asymptotically
nonexpansive mappings, Sci. Math. Jpn., 53 (2001), 143-148.[2] L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mapping with error member,
J. Math. Anal. Appl., 259 (2001), 18-24.[3] L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mapping with an error
member of uniformly covex Banach space, J. Math. Anal. Appl. 266 (2002), 468-471.[4] J. Schu, Iterative construction of fixed points of strictly quasicontractive mappings, Appl. Anal.
40 (1991), 67-72.[5] K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically non-expansive mappings,
Proc. Amer. Math. Sco., 122 (1994), 733-739.
J~n–an–abha, Vol. 38, 2008
ON SPECIAL UNION AND HYPERASYMPTOTIC CURVES OF AKAEHLERIAN HYPERSURFACE
ByA. K. Singh and A.K. Gahlot
Department of Mathematics, H.N.B. Garhwal UniversityCampus Badshahi Thaul, Tehri Garhwal-249199, Uttarakhand
(Received : December 25, 2007, September 2, 2008)
ABSTRACTSpecial, union and hypersymptotic curves of a Riemannian hypersurface
have been studied by Singh [3]. The object of this paper is to investigate thesecurves in a Kaehlerian hypersurface.2000 Mathematics Subject Classification : Primary 55H38; Secondary 58H32Keywords: Hypersymptotic curves, Riemannian hypersurface, Kaehlerianhypersurface.
1. Introduction. In an (n+1) dimensional real space Rn+1 referred to anallowable co-ordinate system,
,,...,, 121 na xxxx
Let us introduce the metric defined by the positive Hermitian form
.22 dxdxgds ...(1.1)
If the tensor g also satisfies the Kaehler's condition
,
xg
x
g akk ...(1.2)
then the space with metric satisfying the condition (1.2) is called Kaehler space.We always assume the self adjointness of the indices [2].
The angle between two self adjoint vector u and v , whose components
are S and S is given by
.cosRS
Srg ...(1.3)
Let us consider an analytic hypersurface Kn of Kn+1.If ui (u1,u2,...,un) denote the co-ordinates of a point in Kn, the equations of theanalytic hypersurface Kn may be written in the form
iuxx ...(1.4)
50
We qnote below some fundamental formulae form [4] which will be used inthe later part of this paper. Suppose that gij is the fundamental metric tensor ofKn, then we have
, jiij BBgg ...(1.5)
where jjii ux
Bux
B
,
If N be the component of unit normal vector to the hypersurface, then
12 NNg ...(1.6)
and ;0 jBNg 0
jBNg ...(1.7)
The unit vector orthogonal to ds
dx is given by
0
dsdx
g ...(1.8)
and .12 g ...(1.9)
Consider a curve sxxC : of Kn.
The components ds
dx and
dsdui
of the unit tangent vector of real space,
w.r.t. the enveloping space and the hypersurface are related by
dsdu
Bds
dx i
i
...(1.10)
If q and Pi are the components of the first curvature vector w.r.t. Kn+1 and Kn
respectively, then
, NKBpq nii ...(1.11)
where nK is the component of the normal curvature of Kn in the direction of the
curve C and
dsdu
dsdu
dsdu
pds
dxds
duds
dxq
ijii
j
,;, ...(1.12)
dsdu
dsdu
KNBji
ijniji ,
51
ij are the components of the second fundamental tensors of the
hypersurface, q and pi are the components of first curvature vector of the curve
with respect Kn+1 and Kn respectively2. Special and Union Curves in Kn. Let an n-dimensional hypersurface
Kn given by equation
ninxyy i ,...,2,1,1...,2,1;
be immersed in a Kaehlerian space Kn+1.The first two Frenet's formulae of a curve xi=xi (S) are given by
11
0 ks
nand
,22011
kk
s
n ...(2.1)
where
210 ,,
dsdy
are components of the unit tangent, principal normal
and first binormal vector and k(1) and k(2) are the first and second curvatures of
the curves. The components q and pi of the first curvature vector w.r.t. Kn+1 and
Kn are related by
, NKBpq nii ...(2.2)
where, iidxdy
B
, N are the components of unit normal vector and nK is the
normal curvature of the hypersurface in the direction of the curve.
Consider two congruence of the curve given by the unit ventor field and
which are such that the point of Kn, we have
CNBr ii ...(2.3a)
and . DNBs ii ...(2.3b)
The special K and union curves relative to have been defined by Tsagas
[5] and Springer [4] respectively.
Let be a K curve relative to congruence and a union curve relative
to the congruence , then
, wqBvp ii ...(2.4a)
52
and ,
yqds
dys ...(2.4b)
Using (2.2), (2.3) and (2.4), we get
,ii pwvr ;nwKC ...(2.5)
,ii
i ypdsdx
xs nyKD ...(2.6)
Defining jiij rrgR 2
,2 jiij ssgS ,2
1ji
ij ppgk
RS
srg iiijcos
and using equation (2.5) and (2.6), we have
.cos 1DKSKn This equation and the facts
,1 22 DSg ...(2.7a)
22
121 nKkK ...(2.7)b
yields
2/122
2/121
sin1
1
S
SeKKn and K(1)
2/122
1
sin1
cos
S
SeK...(2.8)
where, e=1 or -1 in order that cose be non-negative. Since S and cos depend
upon the congruence and , we have the following propositions from equation
(2.8):
Theorem 2.1. If a special curve relative to a fixed congruence is an union curve
relative to another fixed congruence , then the normal and first curvature (w.r.t.
Kn) at a given point of the curve are proportional to its first curvature (w.r.t.enveloping space).
Theorem 2.2. If the component of the vector fields and , tangent to the
hypersurface are in the same direction, then the ratio of the two first curvature is
equal to magnitude of the tangential (to the hypersurface) component .
53
Proof. Since ri and si are along the same direction 1cos and .0sin Therefore,
from (2.8), we have(k(1)/K(1))= S .
3. Hyperasymptotic Curves A hyperasymptotic curve (of order one)
related to is characterized by Mishra ([1], [3])
2101 yx . ...(3.1)
From the first two Frenet's formulae (refer equation(2.1)), we deduce
2211021 log KKqK
dsd
Ks
q . ...(3.2)
Another expression for s
q
can be obtained with the help of (2.2) and the
relations (Mishra [2]).
Ndsdx
sB i
iji
and ds
dxBg
sN k
iij
jk
This later expression for s
q
and (3.1), (3.2) may be used in the elimination
of o . In view of these equations and equations (2.2), (2.3b), we get
dsdx
KKdsd
pds
dxgK
sp
Zdsdx
xsi
ik
jijkn
iii 2
111 log ,...(3.3)
,log 1
K
dsd
KKdsd
dsdx
pZD nn
ji
ij ...(3.4)
where
.
21
1
KK
yZ
Let iii210 ,, be the unit tangent, unit principal normal, unit first
binormal vectors and k(1), k(2) be the first and second curvature of the curve w.r.t.the hypersurface. The first two Frenet's formulae w.r.t. Kn yield
,log 221121
iiii
kkkdsd
pdsdx
ksp
...(3.5)
54
From equations (2.7b), (3.3), (3.4), (3.5) and the definition
,
/cos
S
dsdxsg jiij
we deduce cos1 Sx and
dsdx
SsKdsd
KKdsd
dsdx
pi
inn
jk
kj cos.log 1
.log 221
1
12
dsdx
gKkkK
k
dsd
pdsdx
KDk
ijjkn
iii
n ...(3.6)
Theorem 3.1. A hyperasymptotic curve relative to is characterized by equation
(3.6).proof : Multiplying (3.6) by gil p
l and simplifying, we have
,logcoslog/cos
11
111
KK
dsd
SKK
k
dsd
DkkDKSds
dxp n
nn
ji
ij
where we have defined
.cos1 Sk
spg jiij
4. Hyperasymptotic and Special Curves. Let a hyperasymptotic curve
w.r.t. be a special curve related to . This implies that . Denoting
1
1
K
kX
and ,/ 1KKY n writing ii kp 11 and ddifferentiating the well known identity
,122 YX we get
,cos..cos 1 DYXSdsds
dsdY
SdsdX
Dj
iij
...(4.1)
0dsdY
YdsdX
X ...(4.2)
We shall discuss the solution of the above simultaneous equations in the
55
following cases:
Case 1. The vector i1 is not conjugate w.r.t.
0.,. 1 ds
dsei
dsdx j
iij
i
and the
matrix
yS
XD
Acos
is non-singular. Equation (4.1) and (4.2) yield
Xdsds
dsdY
Ydsds
dsdX j
iij
ji
ij
11 ; .
The quantities
1
1
K
kX and 1/ KKY
n are obtained as solution of the
above simultaneous equations.
Case 2. Let i1 be conjugate w.r.t.
dsdxi
and the matrix A be non-singular. Equation
(4.1) and (4.2) have a unique solution given by the following
Theorem 4.1. If a special curve related to is a hyperasymptotic curve relative
to and the conditions mentioned above (in case 2) are satisfied, then the ratios
11 / Kk and 1/ KKn are the same at each point of the curve.
Case 3. The condition stated in case1 and 2 are not satisfied, i.e., the matrix A is
singular, dsds j
iij 1 may or may not be zero.
We have ,0cos1 SkDKn which in view of (2.7) yields the following
Theorem 4.2. If a K curve is hyperasymptotic relative to and the condition
mentioned in case 1 and 2 are not satisfied, then
2/122
1
2/12
2/121
1
sin1
cos
,sin1
1
S
SKK
S
SKk
n...(4.3)
56
Comparing (4.3) and (2.8), we have the following
Theorem 4.3. If two special curves and relative to are respectively the
union and hyperasymptotic curves relative to another congruence and the
conditions stated in case 1 and 2 are not satisfied, then the modulus of the normal
curvature in the direction of is equal to the first curvature of and the first
curvature of is equal to the modulus of the normal curvature of .
REFERENCE[1] R.S. Mishra , A Course in Tensor with Appllication to Riemannian Geometry, Pothishala Private
Limited (1965).[2] R.S. Mishra , Hyperasymptotic curves on a Riemannian hyper-surface, Math. Student, 20, (1952),
63-65.[3] U.P. Singh, On sheeial union and hyherasymptatic curves is Riemannian spaces, Tensor(N.S.), 22
(1971), 15-18.[4] C. Springer, Union curves of a hypersurface, Can J. Math, 2 (1950), 457-460.
[5] G. Tsagas , Special curves of a hypersurface of Riemannian space, Tensor, N.S., 20, (1969), 88-90.
57
J~n–an–abha, Vol. 38, 2008
MEROMORPHIC UNIVALINT FUNCTIONS WITH ALLTERNATINGCOEEFICIENTS
ByK.K. Dixit and Indu Bala Mishra
Department of Mathematics,Janta College, Bakewar, Etawah-206124, Uttar Pradesh, India
Emaial : [email protected](Received : March 8, 2008)
ABSTRACT
Let 0,, zBATM denote the subclass of functions
1
11n
nn
n zaza
zf
0,1 naa regular and univalent in the disk 10: zzD with a simple
pole at z=0 the conditions
Dz
zfzf
BzA
zfzzf
,1
'
1'
and 20
01
'z
zf where 0<z0<1.
Sharp coefficients estimates, radius of meromorphic convexity, integral transformof functions for this class have been obtained. It is also seen that the class
0,, zBATM is closed under convex linear combination and in the last, certain
convolution properties of functions have been studied,2000 Mathematics Subject clasification : Primary 32H04; Secondary 32Exx.Keywords : Meromerphic Univalent function, meromorphic cenvexity.
1. Introduction. Let be the class of functions of the form
...1 221 zazazzf that are regular and univalent in punctured disk
10: zzD with a simple pole z=0. A function f in is said to be starlike of
order 10 denoted by af * if zfzzf '
Re for .1z The class
* has been extensively studied by Bajpai [1], Cluni [2], Goel and Sohi [4],
Juneja and Reddy [5], Silverman [6] and others.
58
Let m be the subclass of consisting of functions of the form
0...,1 3
32
21 nazazazaz
zf
1
1 ,11
n
nn
n zaz 0na ...(1.1)
Uralgaddi and Ganigi [7] have obtained certain results for meromorphicfunctions with alternating coefficients that are starlike of order .
Let BAM , denote the class of functions of the form (1.1) regular and
univalent in the punctured disk D satisfying the condition
. ,1'
1'
Dz
zfzzf
BA
zfzzf
Dixit and Misra [3] have determined certain interesting results for the class
BAM , , where A and B are fixed numbers .10,11 BBA
Let TM denote the class of functions
1
11 0,11
nn
nn
n aazaza
zf
regular and univalent in the disk D. Let TM(A,B) denote the subclass of functionsin TM satisfying the condition.
. , 1'
1'
Dz
zfzzf
BA
zfzzf
...(1.2)
Also TM(A,B,z0) denote the subclass of functions in TM(A,B) satisfying
200 1' zzf where .10 0 z
In this paper, we obtain sharp coefficient estimated, radius of meromorphicconvexity, integral transform of functions in TM(A,B,z0). It is also shown that theclass TM(A,B,z0) is closed under convex linear combination. In the last, convolutionproblem of functions have been studied.
We begin by recalling the following lemma due to Dixit and Misra [3].
59
Lemma: A function f in M is in M (A,B) if and only if
1
.11n
n ABaABn ...(1.3)
2. Coefficient Estimates
Theorem 2.1. Let
1
11 .0,11
2 nn
nn
n aazaa
zf If f is regular in D and
satisfies 200 1' zzf , then 0,, zBATf M if and only if
1
110 . 111
nn
nn ABazABnABn ...(2.1)
The result is sharp.
Proof. We know from lemma (1.3) that a function
1
1 0,11
nn
nn
n bzbz
zg
regular in D, satisfies
. , 1'
1'
DzA
zgzg
Bz
zgzzg
if and only if
1
.11n
n ABbABn ...(2.2)
Applying the result to the function ,azfzg we find that f satisfies
(1.3) if and only if
1.11
nn ABaaABn ...(2.3)
Since ,1' 20zzf we also have from the representation of f(z) that
1
10
111n
nn
n znaa
Putting the value of a in inequality (2.3), we have the required result.For attaining the equality (2.1), we choose the function
60
1
01
1
111
11
11
nn
nn
zABnABn
zABz
ABnzf ...(2.4)
From (2.4), we have
10
1111
nnn
zABnABn
ABa
or ,111 10
1 ABzABnABn nn
and
1
10
111n
nn
n znaa
1
01
10
1
111
11
nn
nn
zABnABn
zABn
.1111
111
01
nn zABnABn
ABn
3. Radius of Meromorphically Convexity
Theorem 3.1. If ,,, 0zBATf M then f is meromorphically convex of order
10 in the disc ,Rz where
11
inf1 2
111
n
n ABnnABn
R . ...(3.1)
This result is sharp for each n for functions of the form (2.4).Proof. In order to determine the required result, is suffices to show that
Rzzfzzf
1'''
2 for a function f(z) belonging to the class ,,, 0zBATM where
R is defined by (3.1). The details involved are fairly straight forward and may beomitted.
4. Integral Transform.
Theorem 4.1. If ,,, 0zBATf M then the integral transform
1
0 duuzfuczF c for ,0 c is in , ,',' 0zBATM where
61
.
22'1'1 2
0
cz
ABZccABcBA
AB
The result is sharp for the function
2
02/12
zABBAzABzBA
zf
.
Proof. Let ,,,1 01
1 zBATzaza
zf Mn
nn
n
then
1
0 1
11 1 duzuaza
uczFn
ncnn
nc
.1
11
n
nnn
cnza
za
It is sufficient to show that
1
10 .1
1''''1'1'1
nn
nn
acnAB
zABnABn...(4.1)
Since 0,, zBATf M implies that
1
10
1
.1111
nn
nn
aAB
zABnABn
From Theorem 2.1, (4.1) will be satisfied if
1
011
1 111
'''1
nn z
cn
cABncABcnABn
ABB
...(4.2)
The right hand side of (4.2) is an increasing function of n, therefore putting n=1in (4.2) we have
.
222
'''1 2
0
cz
ABccABcBA
ABB
Hence the theorem.5. Closure Theorems
Theorem 5.1. Let be a real number such that .1 If ,,, 0zBATf M then
the function F defined by
62
z
dttftz
zF
0 11
.
Also belong to 0,, zBATM .
Proof . The proof of Theorem 5.1 is similar to that of Dixit and Misra [3].
Since the class 0,, zBATM is convex, it does have some 'extreme points' given by
Theorem 5.2.
Theorem 5.2. Let z
zf1
and
.
111
11
11
10
1
1
nn
nn
nzABnABn
zABz
ABnzf
Then 0,, zBATh M if and only if it can be expressed in the form
1n
nn zfzfzh
where 0 and
1
1n
n .
Proof : Let us suppose that
1n
nn zfzfzh
1
11n
nn
n zaza
where
11
01111
11
nnn
n
zABnABn
ABna
and
10
1111
nn
nn
zABnABn
ABa
Then, it is easy to see that 200 1' zzh and the condition (2.1) is satisfied. Hence
0,, zBATh M . Conversely, Let 0,, zBATh M and
.11
1
n
nn
n zaza
zh
63
Then, from (2.1), it follows that
1
01111
nnnzABnABn
ABa ....3,2,1n
Setting
n
nn
n aAB
zABnABn
1
01111
and
1
,1n
n
we have
1.
nnn zfzfzh
This completes the proof of the theorem.The following inclusion property is an easy consequence of the Theorem 2.1.
Theorem 5.3. Let
1
1 .,...,2,1,1n
nnj
njj mjza
z
azf If 0,, zBATf Mj for
each j=1,2,...,m, then the function
1
11n
nn
n zbzb
zh also belongs to 0,, zBATM , where
m
jjjab
1
,
1
,j
njjn ab mn ,...,2,1
0 j and
1
1j
j .
Theorem 5.4. If 01
1 ,,1 zBATzaza
zf Mn
nn
n
and
1
11n
nn
n zbZb
zg
with 1nb for n=1,2,3..., then 0,,* zBATgf M .
Proof. For convolution of functions f and g; we can write
64
1
10
1
1
10
1 111111n
nnn
nnn
nn azABnABnbazABnABn
1 Since nb
B-A, by (2.1)
Hence, by theorem (2.1), 0,,* zBATgf M .
Note. It will be of interest to find other convolution results analogous to thoseof Juneja Reddy [5].
REFERENCES[1] S. K., A note on a class of meromorphic univalent functions, Rev. Roumanie Maths Puse Appl., 22
(1977). 295-297.[2] J. Clunie, On meromorphic Schlicht functions, J. London Math. Soc., 34 (1959) 215-216.[3] K.K. Dixit and Y.K. Misra, On a class of meromorphic starlike functions with alternating coefficients,
Acta Cientia Indica, XXVII M, No(1) (2001) 065.[4] R.M. Goel and N.S. Sohi, On class of Meromorphic function, Glasnik Mathematicki, 17 (1981), 19-
28.[5] O.P. Juneja and T.R. Reddy, Meromorphic starlike univalent functions with positive coefficients,
Annales Universitatis Mariae Curie- Sklodawska Lubin-Polonia, 39 (9) Section A (1985) 55-75.[6] H. Silverman , Univalent functions with negative coefficients, Proc. Amer. Math Soc. 51 (1975)
109-116.[7] B. A. Uralegaddi and M.D. Ganigi, Meromorphic starlike functions with alternating coefficients,
Rendiconti di Mathematica, Seril VII, Vol. II, Roma (1991), 441-446.
65
J~n–an–abha, Vol. 38, 2008
A MATHEMATICAL MODEL FOR MIGRATION OF CAPILLARYSPROUTS WITH DIFFUSION OF CHEMOTTRACTANTCONCENTRATION DURING TUMOR ANGIOGENESIS
ByMadhu Jain, G. C. Sharma
Department of Mathematics, Institute of Basic Science,Dr B. R. Ambedkar University, Khandari, Agra-282002, Uttar Pradesh, India
Email : [email protected], [email protected]
Atar SinghDepartment of Mathematics, Bipin Bihari College, Jhansi-284128, U.P., India
(Received : March 12,2008)
ABSTRACTIn this paper, we develop a simplified mathematical model for tumor
angiogenesis, which is the process of sprout growth, forming new blood vesselsfrom already existing micro vessels. The diffusion of tumor angiogenesis factor inextra cellular matrix is explored and the effects of TAF, chemottractantconcentration, on tip density and vessel density have also been discussed. Theanalytical solution has been obtained. The numerical experiments have beenperformed to justify the same and to facilitate sensitivity analysis.2000 Mathematics Subject Classification : Primary 92C10 Secondary 92C37.Keywords: Angiogenesis, Tumor growth, Diffusion, TAF, Neovascularization.
1. Introduction. The studies on cancer are being made due to its prevalenceall over the world. Very recently a known scientist Sasi Sekharan from MIT, USAhas tested the molecular size bomb on the skin or lung cancer of the mice. Basically,solid tumors are the collection of tumor cells, which are the core factors in cancerresearch. Other factors like neovascularization and the interstitium are alsoaccountable for solid tumor growth. The aim of all the studies in this area is tofind a drug, which can penetrate deep into tumors, cut of the blood supply anddetonate a lethal dose of anticancer toxins without harming healthy cells. Solidtumors are of two types, avascular and vascular. Initially solid tumors are smallmass cells within the tissue. But this mass multiplies rapidly due to the process ofmultation. At initial stage, avascular tumor growth leads to non-metastatic tumor,which may remain dormant for long time. At avascular stage, tumor growth maynot be possible beyond 1-2 millimeters.
For tumor growth, vascularization to tumor is one of the important factors.This process occurs both in non-malignant conditions. Angiogenesis is the process
66
of sprouting new blood vessels from already exsting micro vessels. To make thetumor vascularized, solid tumor secrets some chemical compounds, which stimulateendothelial cells (EC) of the tissue. Some enzymes secreted by EC, break downbasement of membrane, allowing endothelial cells to proliferate across the disruptedmembrane into extra cellular matrix. On the basis of this process, new capillarysprouts form and migrate towards a vascular solid tumor. Finally, a network ofcapillaries is formed and penetrates into the tumor. Then a solid tumor getsvascularization. Many researchers developed mathematical models related to tumorgrowth in different frame works.
Muthu et al. (1982) studied tumor induced-neovascularization in mouseeye. Chaplain and Sleemen (1990) developed a mathematical model for produtionand secretion of tumor angiogenesis factor in tumors. Adam and Maggelakis (1990)explored diffusion regulated growth characteristics of spherical pre-vascularcarcinoma. Lauffenburger and Stokes (1991) analysed the roles of microvesselendothelial cell random motility and chemotaxis in angiogenesis. Chaplain andStuart (1993) provided a mechanism for chemotrctic response of endothelial cellsto tumor angiogenesis. Byrne and Chaplain (1999) developed a model of vascularsolid tumor growth. Cui and Friedman (2001) developed a mathematical model ofthe growth of necrotic tumors. Friedman and Reitch (2001) worked on the existenceof spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumor. Levine et al. (2001) discussed onset of capillary formationinitiating angiogenesis. Petted et al. (2001) studied the migration of cells inmulticells tumor angiogenesis. Petted et al. (2001) studied the migration of cellsin multicells tumor spheroids. McDougall et al. (2002) examined mathematicallythe flow through networks: in order to study tumor-induced angiogenesis andchemotherapy strategies. Bazaliy and Friedman (2003) prepared a model of tumorgrowth. Cui and Friedman (2003) studied a hyperbolic free boundary problemsmodeling tumor growth. Owen et al. (2004) did work on mathematical modeling ofthe use of macrophages as vehicles for drug delivery to hypoxic tumor sites. Theyinvestigated the role of chemotaxis and chemokine production and the efficacy ofmacrophages as vehicles for drug delivery. Plank et al. (2004) formulated amathematical model of tumor angiogenesis, regulated by vascular endothelialgrowth factor and the angiopoietins. this work attempted to provide a mathematicaldescription of the role of the angiopoietins in angiogenesis. Tarabolette andGiavazzi (2004) suggested modeling approaches for angiogenesis and designed foranti-angiogenic compounds.
The paper mathematical modeling is capable of providing deep insight intothe ways by which one can think of producing some medicine to destroy the blood
67
vessels feeding the tumor. In this paper, we have modelled sprout growth duringtumor angiogenesis and diffusion of TAF in extra cellular matrix. Our focus inthe present investigation is on capillary sprout growth, diffusion of TAF and roleof angiogenesis in tumor growth. The simplified model is solved analytically. Byrneand Chaplain (1996) have not considered the diffusion of tumor angiogenesis factorin their studies. Our effort in this paper is to extend their studies for morecomplicated cases by incorporating this factor. We examine the diffusion ofchemottractant concentration , the role of angiogenesis in tumor growth. thenumerical experiments have also been performed to visualize the effects of variousparameters on tip and vessel densities. The present investigation is organized inthe following way. In section 2, we describe mathematical model by stating requisiteassumptions and notations. The analysis part of the paper has been discussed insection 3. The numerical illustrations are presented to validate analytical resultsin section 4. The section 5 provides descussion part and concluding remarks.
2. The Model. It is well known that vessels within a sprout have a velocityforming the continuity of the tip-vessels structure. Tumor angiogenesis factor(TAF) concentration is governed by reaction-diffusion equation based on theexperimental observation and earlier studies. We consider the finite and uni-directional model for sprout growth and diffusion to tumor. This model isformulated in terms of tip density, vessel density and TAF concentration. Thetumor is at x=0 and limbus is at x=1, (0<=x<=1). Following Byrne and Chaplain(1996) model , that vascular front with tips stimulated by TAF and sprouting fromcapillary vessels. The proliferation of tip occurs at the vascular edge (13) and istriggered when exceeded threshold concentration c'. The vessels within the sproutsmigrate to tumor and decay linearly. For TAF concentration, reaction-diffusionequation with linear-decay has been used. We use the following notations formathematical formulation of the problem.
The set of partial differential equations governing the model (Byrne andChaplain, 1996) is given by
nncccHcxc
nxx
ntn
'102
2
...(1)
xc
nxn
t...(2)
cxc
Dtc
2
2
, ...(3)
where n(x,t), (x,t), c(x,t) are the density, vessel density and TAF concentration,
68
respectively. D and H are the diffusion coefficient of TAF and Hevinside unit
function. 10 ,,,,,,, c are rate of TAF decay, rate of vessel density decay,
threshold TAF concentration, rate of tip to branch anastomoses, random tipmotility, chemo taxis coefficient, first tip proliferation rate, second proliferationrate respectively.
At the tumor, c is at constant value 1 and n=0, if tips penetrate the tumor,then our model is not applicable. At the limbus, c=0, while n and decay
exponentially to zero. If tip and capillary once formed then the capillary buds'production at limbus is stopped.Initial and boundary conditions are
,0,0 tn ,,1 ktcentn ,00, xn for ,10 x cnn 0,1 ...(4)
,1,1 minminktet ,00, x for ,10 x 10,1 ...(5)
0,1,1,0 tctc 00, cxc for ,10 x ...(6)where k is decay rate of tip and vessel density.
3. The Analysis. The TAF concentration for unsteady profile with thehelp of the equations (3) and (6) can be obtained as:
1
0 cos1
22
rn
tD
n
nt xeecc n ...(7)
where
,...,3,2,1 and
212
nn
n
Chaplain and Stuart [9], estimated motility coefficient for cells chemicals
Dcell/Dchemical= 010 3 .
Then vessel density within the sprout is given by
xc
nt
...(8)
Solving the equation (8) with the help of the equation (5), we have
1
0 ,sin122
r
tDn
rt dttxnexec n ...(9)
The non-linear wave equation for tip density can be reduced to
nncccHcx
cn
xc
xn
tn
'102
2
...(10)
With the help of equation (9), the equation (10) in terms of n(x,t) and c(x,t) can bewritten as
69
ncncccHxc
xn
tn
n
02
1 '
dttxnexeC tDn
tr n ,sin212
0 ...(11)
We simplify the model with the concept that the vessels form tips for a very short
time and in view of this ,0
tn
so that ., ndxdc
tx
Now we have
2021 ' n
dxdc
cndxdc
ccHxc
xn
tn
n
...(12)
With the condition
n(0,t)=0 n(x,0)=H(x–x*), n(1,t)=1 for some 1,0*x
we further simplify the model in respect of TAF concentration and tip creationrates. For more simple approximation, if we consider,
,0 ,00 1' ccH .
The reduced equation is given as follows:
221 n
dxdc
ncncxn
xc
tn
n
...(13)
We use perturbation method for approximation of n(x,t), which provides
..., 22
210 titi exnexnxntxn ...(14)
where is a small quantity.Now
tieintn
1 ,xn
exn
xn ti
10 ...(15)
From equations (14) and (15), we get
xn
exn
xeCein tin
trti 1001 sin12
nti
ntr ennxec /cos12 1010
tin
trn ennxeC 100 cos12
titintr
ennennxeC
10
221
220
0 2sin12
...(16)
where 2nD .
70
Comparing terms free of on both sids, we have
0200
10 .cot.cot nxnnxxn
nnnn
or 200
0 '.cot nFnxFxn
n
where
nn
F 1 and
'F .
On solving, equation (17) with the help of (4), we get
dxxF
xn
Gn
Gn
sin'
sin0
where2
21
n
nG
.
Now comparing the coefficients of on both sides. we have
n
nn
trn
tr xeC
xn
xeCin
cos12.sin12 1
20
101
100 .sin12
nnxeC n
tr
or 11
2
01 .cossinsin nixpxn
pdxdn
xp nn
nnn
...(18)
On solving the differential equation (18), we get
CxP
xdxdxx
xn n
n
GnG
n
Gn 2/tanlogsinlog
sin
sinlog 1 ...(19)
With the help of the (4), then we haveC=0,
2/tanlogsinlogsin
sinlog 1 x
Pxdx
dxx
xn n
n
GnG
n
Gn
...(20)
2/tanlogsinlog
sinsin
1
xP
xdxdxx
xn
n
GnG
n
Gn
enBe ...(21)
where
2/tanlogsinlogsin
sinx
Pxdx
dxx
xB n
n
GnG
n
Gn .
71
Now tn eCp 12 0 ,
pi
P
..., 0 Bti eentxn ...(22)
Substituting the values of n0 and n1 in (22), we have
tiG
n
Gn e
dxxF
xtxn
sin'
sin,
...2/tanlogsinlogsin
sinexp
xP
xdxdxx
xn
n
GnG
n
Gn
.
4. Numerical Results. The effect of various parameters on TAF, vesseldensity and tip density has been examined by considering a numerical illustration.
We fix default parameters as ,1,4.,1,8.,01. 10 rc ,100 ,001.
,000002. to obtain numerical results, which are depicted in figures 1-4.
Figs. 1(a-c) depict that chemottractant concentration (c) varying x fordifferent rate of diffusion coefficients (D) for t=0.8, 1,2, 1.6, respectively. It is noticedthat the concentration decreases by increasing x and it is highest where tumor issituted (i.e. x=0) and is almost zero at limbus (i.e. x=1). The chemottractantconcentration decreases as the rate of diffusion increases. Initially the concentrationdecreases slightly upto x=0.2 but beyond this, it decreases sharply. On comparingthe figs. 1(a-c) we see that the concentration decreases as time increases; differenceof concentration for different values of D increases as time increases butdiminishes towards limbus.
Figs. 2(a-c) demonstrate the chemottactant concentration profile fordifferent values of x. We see that concentration decreases as time increases anddistance of limbus from tumor (x) increases. It is also clear from the figures thatthe concentration, at D=0.3 for t=2.8 and for different values of x, is almost zero;however for lower value fo D depicted in figures 2 (a)& 2 (b) for D=0.01 and D=0.1,respectively, the value of c becomes almost zero much later. Thus the diffusioncoeficient has significant effect in the growth of the tumor.
It figures 3 (a-c); the effects of different parameters on vessel density ( )
are depicted. Fig.3(a) illustrates vessel density ( ) vs. x (distance of limbus from
tumor) at different time. We observe that at limbus (x=1) the vessel density ishighest and at tumor site (x=0) the vessel density is zero; the vessel densityincreases with time but the effect is more prevalent for higher values of x. In fig3(b), we see that the vessel density increases initially sharply with x upto x=0.6,but beyond that, the density increases gradually with x. It also increases as ,
72
increases. In fig. 3(c), which shows the vessel density profile for different values of , initially the vessel density increases slightly with time but later on it increases
sharply. Upto t=0.4, the vessel formation for all values of is very slow and beyond
that, the vessel formation increases sharply as increases.
Figs. 4(a-c) demonstrate tip density (n) vs. for different values of t and .
From fig. 4(a) we see that tip density increases steeply with x upto x=0.7 but beyond,the tip formation is static upto x=0.8 and onwards tip formation decreases. Thetip density does not change with time; the path of tip formation is of wave type.Fig. 4(b) exhibits that tip density (n) increases as and x increases upto x=0.7,
beyond this, its increasing trend slows down and converges at limbus (at x=1) fordifferent values of . From the fig. 4(c), it is clear that tip density is free of time
and increases as increases.
From numerical experiment performed concluding we infer that-The chemottractant concentration decreases as x,D and t increase; higherfalling trends for lower rate of diffusion is noticed. For very long time, theconcentration is static. Thus higher rate of diffusion and increasing trend intime decrease the chemottractant concentration, which is quite natural inreal life situation.The effect of diffusion coefficient is more prevalent on chemottractantconcentration.The vessel density is highest at limbus and decays to zero from limbus to thetumor. Also it decays as time increases and x decreases.Initially, the tip density at limbus is highest but it sharply decreases towardstumor and is almost zero at the tumor. It is also static with time.The effect of on the tip density and vessel density is quite remarkable . The
path of tip density is in the form of wave propagation.5. Conclusion. We have formulated a simplified mathematical model of
capillary sprout growth with diffusion for TAF concentration for cancer cells toexamine its ability to induce vascularization of the tumor in order to receive oxygenand nutrients. We make some simplifications to find out the effects of tip densityand vessel density on tumor angiogenesis. The tip density and vessel density dependson TAF concentration. The diffusion is considered and its effects are discussed onTAF. Also the effects of various parameters on tip density and vessel density havebeen explained. It is very clear from the result that the TAF concentration dependson the rate of diffusion.
73
74
75
REFERENCES[1] A. Adam and , S.A. Maggelais, Diffusion regulated growth characteristics of spherical pre-vascular
carcinoma, Bull. Math. Biol., 52 (1990), 549-582.[2] B. V. Bazaliy, A. Friedman , A free boundary problem for an elliptic- parabolic system: Application to
a model of tumor growth, Comm. PDE, 28 (2003), 627-675.[3] H.M. Byrne and M.A.J. Chaplain, : A weakly non-linear analysis of model of vascular solid tumar
growth, J.Math. Biol., 39 (1999). 59-89.[4] S. Cui and A. Friedman, Analysis of matheamatical model of the growth of necrotic tumors, J. Math.
Anal., 255 (2001) 636-677.[5] S. Cui and A. Friedman,: A hyperbolic free boundary problem modeling tumor growth, Int-Faces and
Free Bounds, 5 (2003), 159-182.[6] A. Friedman, and F. Reitich: On the existence of spatially patterned dormant malignancies in a
model for the growth of non-necrotic vascular tumor, Math. Models Mach. Appl., 77 (2001), 1-25.[7] H.M. Byrne and M.A.J. Chaplain, Explicit solution of a simplified model of capillary sprout growth
during tumor angiogenesis, Appl. Mah.Lett., 9 (1996), 69-74.[8] M.A.J. Chaplain and B.D. Sleeman, A matematical model for producton and secretion of tumor
angiogenesis factor in tumors, M.A.J. Maths. Appl. Med. Biol.,7 (1990).[9] M.A.J.Chaplain and A.M. Stuart, A model mechanism for the chemotactic response of endothelial
cells to tumor angiogenesis factor, M.A.J. Math. Appl. Med. Biol.,10 (1993), 149-168.[10] H.A. Levine and B.D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of onset of capillary
formation initiating angogenesis, J. Math. Biol. 42 (2001), 1432-1466.[11] D.A.J. Lauffenburger and C.L. Stokes, Analysis of the roles of micro vessles endothelial cell random
motility and chemo taxis in angiogenesis, J. Theo. Biol., 152 (1991) 377-403.[12] S.R. Mc Dougall, A.R.A. Anderson, M.A.J. Chaplain and J.A. Sheratt, Mathematical modeling of flow
through vascular networks: Implications for tumor-induced angiogenesis and chemotherapystrategies, Bull. Math. Biol., 64 (2002), 673-702.
[13] V.R. Muthukkaruppen, L. Kubai and R. Auerbach. Tumor-induced neovascularization in mouse eye,J. Natl. Cancer Inst., 69 (1982) 699-705.
[14] M.R. Owen, H.M. Byrne and C. E. Lewis, Mathematical modeling of the use of macrophages asvehicles for drug delivery to hypoxic tumor sites, J. Thio. Biol. 226 (2004), 377-391.
[15] G. Pettet, C.P. Please, M. J. Tindall and D. McElwain, The migration of cells in multicell tumorspheroid, Bull. Math. Biol., 03 (2001), 231-257.
[16] M.J. Plank, B.D. Sleeman, and P.F. Jones, A mathematical model of tumor angiogenesis, regulated byvascular endothelial growth factor and the angiopoitins, J. Theo. Biol., 229 (2004), 435-454.
[17] G. Tarabolette and R. Giavazzi, Modeling approaches for angiogenesis, Europ. J. Cancer, 40 (2004),881-889.
76
J~n–an–abha, Vol. 38, 2008
DIFFUSION-REACTION MODELFOR MASS TRANSPORTATION INBRAIN TISSUES
ByMadhu Jain, G.C. Sharma
Department of MathematicsInstitute of Basic Science, Khandari, Agra-282002, Uttar Pradesh, India
E-mail :[email protected], [email protected]
Ram SinghDepartment of Mathematics,
St. John's College, Agra, 282002, Uttar Pradesh, IndiaE-mail: [email protected]
(Received : April 12, 2008)
ABSTRACTThis investigation deals with the study of mass transport in brain tissues.
The tissues in the human body are considered as porous medium by keeping thefact in mind that these are made up of dispersed cells seperated by connectivevoids through which nutrients and other menerals are available to each cell in thetissues. The general diffusion-reaction equations have been used to elucidate thedeveloped bi-layer model. The mass concentrations have been obtained at bothlayers. The importance of porosity and tortuosity factors has also been discussed.The diffusion-reaction process is more realistic one .2000 Mathematics Subject Classification : Primary 92C18; Secondary 92C37.Keywords : Brain tissues, Mass transport, Diffusion-reaction model, Porosity,Tortuosity,
1. Introduction. Diffusion of molecules in the brain and other tissues isimportant in wide range of biological processes and measurements ranging fromthe delivery of drugs to diffusion- weighted magnetic resonance imaging. Thediffusion molecules may have different diffusion co-efficients and concentrationsin the different domains, namely within the tubes’ inner core, membrane and withinthe outer medium. Biological tissues are multi compartmental heterogeneous mediacomposed of cellular and subcellular domains. Diffusion of mass transport is verysensitive to the local environment in tissues, and is affected by the packinggeometry of the cells and thier membrane permeability that controls the exchangeof the nutrients molecules across the membranes.
The main aim of our investigation is to find out the concentrations of thenutrients at two different layers of the brain tissues by using diffusion-reaction
78
model. The diffusion-reaction in a geometrically complicated environment is , ona microscopic level, an extremely difficult process. But in biological application,we are often satisfied with macroscopic level. Several researchers have studieddiffusion-reaction models in different frameworks.
Vafai and Tien [14] studied a numerical scheme to evaluate the velocityand temperature fields inside a porous medium near an impermeable boundary.They presented a new concept of momentum boundary leyer central to numericalroutine. Puri et al. [10] developed a mathematical model to predict the steadystate transport of a conservative, neutrally bouyant tracer injected along thecenterline into a fully developed turbulent pipe flow. A mathematical model inwhich, steady state transport of a decaying contaminant in a fractured porousrock matrix by two dimensional diffusion and vertical advection which is treatedby Fourier Sine Transform technique, has been studied by Fogden et al. [4]. Tompsonand Dougherty [13] studied a two-step particle-in-cell model for reactive masstransport problems in subsurface porous formation and then applied this modelto non-lineat diffusion-reaction system. Kangle et al. [7] presented a generalanalytical solution for one-dimensional solute transport in heterogeneous porousmedia with scale dependent dispersion. McDougall et al. [8] used flow modelingtools and tecniques in the field of petroleum engineering. They examined the effectsof fluid viscosity, blood vessels size and theoretical networks geometry upon (i)the rate of flow through network (ii) the amount of fluid present in the completenetwork (iii) the amount of fluid reaching the tumor.
A mathematical model for the oxygen transport in the brainmicrocirculation in the presence of blood substitues has been developed by Sharanand Popel [12]. Bertuzzi et al. [12] investigated a mathematical model for theevolution of a tumor cord after treatment by using extensive numerical simulation.A mathematical model combined with a physical model to simulate the growthcharacteristics of a single bubble in liquid by the process of rectified diffusion hasbeen developed by Meidani and Hasan [9]. Their model is besed on the coupledmomentum energy and mass transport equations. Water diffusion model withinthe structure of a brain extracellular space for various diffusion parameters ofbrain tissue namely extracellular space, porosity and tortuosity has numericallybeen analyzed by Vafai et al. [15], Hrabe et al. [5] studied a mathematical model foreffective diffusion and tortuosity in the extracellular space of the brain . Theyused a volume-averaging procedure to obtain a general expression for the tortusityin a complex envirnment. Jain and Sharma [6] developed a time dependentmathematical model for oxygen transport in peripheral nerve tissues by usingKrogh cylinder symmetry. Sen and Basser [11] presented a mathematical modelfor diffusion of white matter in brain by using diffusion tensor imaging method to
79
characterize neuronal tissue in the human brain. Arifin et al. [1] used mathematicalmodeling and simluation to provide a comprehensive review of drug release frompolymeric microspheres and of drug transport in adjacent tissue. An experimentalinvestigation of the effect fo water diffusion exchange between compartments onthe paramagnetic relaxation enhancement of paramagnetic agent compartmenthas been presented by Zhang et al. [16]. El-Kabeir et al. [3] studied a mathematicalmodel for combined magneto-hydrodynamic (MHD) heat and mass transport ofnon-Darcy natural convection about an impermeable horizontal cylinder in a nonNewtonian power law fluid embedded in porous medium under magnetic field andthermal radiation effects.
In this investigation, we construct diffusion-reaction equations tounderstand the concept of mass transport in brain tissues. We shell develop amathematical model by considering solute concentration at two different layers.The rest of the paper is arranged as follows. The model description, notations andgoverning equations have been provided in the Section 2. Section 3 is to devote theanalysis of the model. Finally, the conclusions are drawn in the Section 4.
2. Model Description. The present investigation is concerned with themass transport in porous medium by using diffusion-reaction equations inbiological tissues particularly in brain tissues. Two concepts porosity and toruosityhave been incorporated in the model. Porosity determines what percentage of thetotal tissue volume is accessible to the diffusing molecules and on the other handtortuosity describes the average hindrance of a complex medium relative toobstacles-free medium. One-dimensional transient diffusion-reaction equationshave been constructed to find out mass concentration at different situations. It isalso assumed that the chemical reaction coefficient at both layers is constant andequal.Followings are the notations used to formulate the model mathematically:
txCi , Concentration of mass transport in ith (i=1,2) layer
Di Diffusivity of ith layer, i=1,2
i Tortouosity of ith layer, i=1,2
Si Mass source density of ith layer , i=1,2
i Porosity of ith layer, i=1,2
K Chemical reaction co-efficientt Time.
Governing Equations. The equations governing the mass transport in the braintissues due to diffusion-reaction process are constructed as follows:Then, the deffusion-reaction equation for the concentration in layer-1 is
80
,,
,
,
1
112
121
11
S
txKCx
txCDt
txC...(1)
The diffusion-reaction equation for the concentration in layer-2 is
,,
,
,
2
222
222
22
S
txKCx
txCDt
txC...(3)
The interface between two layers is at x=a. The perfect contacting is assumed atinterface and concentration at that position is same.The initial and boundary conditions for equations (2)-(3) are given as
0, at0
,
0,0 at0
,0,0 at,0,0 at,
2
1
02
01
tbxt
txC
txt
txCbxtxCtxCaxtxCtxC
...(4)
The matching conditions are
0, at,
0, at
,
,0, at,,
212
2
1
1,21
22
11
21
taxSS
DD
taxt
txCD
ttxC
D
taxtxCtxC
...(5)
3. Mathematical Analysis: Taking Laplace Transform of the equation(2), we obtain
11212
12
,
,YpxC
dx
pxCd ...(6)
where pKD
1
212
1 and . 1
10
1
21
1
p
SC
DY
There fore, 21
11
11,
YBeAepxC xxa . ...(7)
Again talking the Laplace Transform of the equation (3), we get
22222
22
,
,YpxC
dx
pxCd ...(8)
where pKD
2
222
2 and
p
SC
DY
2
20
2
22
2 .
81
Therefore, 2222 22, YDeCepxC xxa . ...(9)
Transformed boundary and machining conditions are
0, at
,
,0, at,,
0, at0
,
0,0 at0
,
22
11
21
2
1
taxdx
pxCdD
dxpxCd
D
taxpxCpxC
tbxdx
pxCd
txdx
pxCd
...(10)
Using these conditions, the solutions of eqs (7) and (9), are:
pxC ,1
n
r
n
r
m n
n
r
nnn
r
rrnm
rnn
rnn
SSDD
rnn
rnn
0 0 12
0 0 0 2
2
1
1
2
12
2
1
10
2 3
1
1
1
21
122
21
Yp
e
p
e pKpK
...(11)
pxC ,2
n
r
n
r
m n
n
r
nnn
r
rrnm
rnn
rnn
SSDD
rnn
rnn
0 0 12
0 0 0 1
1
2
2
1
22
1
2
10
2 3
1
211
22
222
43
Yp
e
p
e pKpK
...(12)
Taking inverse Laplace Transform of eqs (11) and (12), we get
82
txc ,1
n
r
n
r
m n
n
r
nn
n
r
rrnm
rnn
rnn
SSDD
rnn
rnn
0 0 32
0 0 0 2
2
1
1
2
12
2
1
210
2 3
1 2
31
t u
KuKt u
KuK
duuteu
duuteu
0 4
32
0 4
31
22
...(13)
KtKt eK
SeC
1
1
10
txc ,2
n
r
n
r
m n
n
r
nnn
r
rrnm
rnn
rnn
SSDD
rnn
rnn
0 0 12
0 0 0 1
1
2
2
1
22
1
2
10
2 3
1
211
t u
KuK
uK
uKduute
uduute
u
0 4
341
0 4
33
22
(14)
KtKt eK
SeC
1
2
20
where
.02
1222 312
22
1
11
rrab
Dxrrma
D
.02
1222 312
22
1
12
rrab
Dxrrma
D
83
.02
12 21
131
2
23
ar
Daxrrmab
D
.02
12 21
131
2
24
ar
Daxrrmab
D
4. Conclusion. In this study we have developed a diffusion-recation modelfor the mass transport in biological tissues, pariticularly in the brain tissues. Thetransporation of nutrients, oxygen, glucose etc. in the brain tissues from vascularsystem through diffusion-reaction process has been investigated. The analyticalexpressions for mass concentration in two layers have been obtained. It has alsobeen obserbed that the porosity and tortuosity affect the mass transport signifi-cantly. Our investigation may be helpful in the treatment for brain tumor men-tally ratarded patients, in particular when clemical reaction is taken in to consid-eration.
REFERENCES[1] D.Y. Arifin, L.Y. Lee, and C.H. Wang, Mathematical modeling and simulation of drug release from
microspheres: Implication to drug delivery systems, Adv. Drug Del. Rev., 58, No. 12-13 (2006)1274-1325.
[2] A Bertuzzi, A. D’onofrio, A.Fasano and A.Gandoffi, Regression and re-growth of tumour cordsfollowing single-dose anti-cancer treatment, Bull Math. Bio.,65, No.5.(2003), 903-931.
[3] S.M.M. El-Kabier, M.A. El-Hakim and A.M. Rashad, Group method analysis of combined heat andmass transfer by MHD non-Darcy non-Newtonion natural convection adjacent to horizontal cylin-der in a saturated porous medium, Appl. Math. Model., 2007 (In Press).
[4] A. Fogden, K.A. Landman, and L.R. White, Contaminant transport in fractured porous media:steady state solutions by Fourier Sine Transform method, Appl. Math. Model.,13, No. 3 (1989),160-177.
[5] J. Hrabe, S. Hrabetova and K. Segeth, A model of effective diffusion and tortuosity in the extracel-lular space of the brain, Biophy., 87 (2004), 1606-1617.
[6] M. Jain and G.C. Sharma, A computational solution of mathematical model for oxygen transport inperipheral nerve, Comput. Bio. Med., 34 (2004), 633-645.
[7] H. Kangle, T.V. Genuchten and Z. Renduo, Exact solutions for one dimensional transport withasymptotic scale dependent dispersion, App. Math. Mod., 20 No.4 (1996), 298-308.
[8] S.R. MeDougall, A.R.A. Anderson, M.A.J. Chaplain and J.A. Sherratt, Mathematical modeling offlow through vascular networks: Implication for tumor-induced angiogenesis and chemotherapystrategies, Bull. Math. Bio., 64, No. 4 (2002), 673-702.
[9] A.R.N. Meidani and M. Hasan, Mathematical and physical model of growth due to ultrasound, App.Math. Mod., 28 No.4 (2004), 333-351.
[10] A.N. Puri, C.Y. Kou and R.S. Chapman, Turbulent diffusion of mass in circular pipe flow, App.Math. Mod., 7, Issue 2 (1983), 135-138.
[11] P.N. Sen, and P.J. Basser, A model for diffusion in white matter in the brain, Biophy., 89 (2005),2927-2938.
84
[12] M. Sharan, A.S. Popoel, A compartmental model for oxygen transport in brain microculation in thepresence of blood substitutes, J. Theo. Bio., 216 (2002), 479-500.
[13] A. F. B. Tompson and D.E. Dougherty, Particle-grid method for reacting flows in porous media withapplication to Fisher’s equation, App. Math. Mod., 16 Issue7 (1992), 374-383.
[14] K. Vafai and C.L. Tien, Boundary and inertia effect on flow and heat transfer in porous media, Int.J.Heat Mass Tranhsfer, 24 (1980), 195-203.
[15] K. Vafai, K. Khanafer and A. Kangarlu, Computational modeling of cerebral diffusion-application ofstroke imaging, Meg. Res. Img., 21 Issue 6 (2003), 651-661.
[16] H. Zhang, Y. Xie and T. Ji, Water diffusion-exchange effect on the paramagnetic relaxation enhance-
ment in off-resonance rotating frame, J. Mag. Res., 186 Issue 2 (2007), 259-272.
J~n–an–abha, Vol. 38, 2008
RELIABILITY ESTIMATION OF PARALLEL-SERIES SYSTEM USINGC-H-A ALGORITHM
ByA.K. Agarwal
Department of MathematicsHindu College, Moradabad, Uttar Pradesh, India
andSuneet Saxena
Department of MathematicsBhagwat Institute of Technology, Muzaffanagar, Uttar Pradesh, India
E-mail: [email protected](Received : March 15, 2008)
ABSTRACTC-H-A algorithm is based on based on minimal path set to evaluate system
structure function. Once one obtain the expression for the structure function, thesystem reliability computation becomes straight-forward. In this paper, we havestudied C-H-A algorithm and applied it to estimate the reliability of parallel-seriessystem.2000 Mathematics Subject Classification: Primary Secondary.Keywords: C-H-A-algorithm, structure function, Reliability, Parallel-seriessystem.
1. Introduction. A very general alternative approach for analyzing thereliability of complex system is through the use of the system structure function.Once one obtains the expression for the structure function, the system reliabilitycomputation becomes straight-forward. Such attempts have been made in theclassical 1975 book by Barlow and Proshan [3]. Various algorithms have beendeveloped to evaluate structure function. Aven Algorithm [1] is based on minimalcut sets. It depends on the initial choices of 2 parameters. Recently Chaudhari, Huand Afshar [4] proposed new algorithm based on minimal path set to evaluatesystem structure function. They named it C-H-A algorithm.
In this paper, we study C-H-A algorithm and apply it to estimate thereliability of parallel- series system.
2. Notations and Definitions(2.1) Notations
n : Number of components.x1 : State of ith component.x : (x1, x2,..., xi,...,xn) states of the components.
86
x : Structure function.
P1 : Probability{x1=1} : reliability of ith component.R : Reliability of system.
(2.2) Definitions(2.2.1)Structure Function
Let
01
1xifif
componentcomponent
ii hasoperates
.failed
Then the system structure function is defined as
01
xifif
SystemSystem
hasoperates
.failed
(2.2.2) Reliability of SystemReiability of system in terms of structure function is defined as
R=probability 1 x
xE
where .0 xE probability .0 x +1. Probability .1x
(2.2.3) Coherent SystemA system is coherent when a component reliability improvement does not
degrade the system reliability. A coherent system has a structure function that ismonotonically increasing.
if ,ii xy for i=1 to n.
then . xy
(2.2.4)Relevant ComponentIf the inequality is strict for a given component i, then that component is
said to be relevant.(2.2.5)Path Set
A path set is a set of components whose functioning ensures that the systemfunctions.(2.2.6)Minimal Path set
A minimal path is one in which all the components within the set mustfunction for the system to function.(2.2.7) (Cut Set)
A set is a cut of components whose failure will result in a system failure.(2.2.8)Minimal Cut Set
A minimal cut is one in which all the components must fail in order for the
87
system to fail.(2.2.9)OR Operaton (Binary)
1.1=1, 1.0=1, 0.1=1, 0.0=0.3. Reliability of system Using C-H-A Algorithm. Let us consider two
components in parallel with two components in series.
Step 1. Find out minimal path setsMinimal path sets are : {1,3,4}, {2,3,4}.
Step 2. Construct matrix P using minimal path sets. Each column of P representminimal path. Assign 1 for the component present in path set and 0 for thecomponent not present in path set.
1101
P
1110
Steps 3. Construct the design matrix D using the columns of P matrix. Startwith two columns of P matrix and apply OR operation on a respective rows andresultant column will be appended in P matrix. Perform same operation on allremaining columns. Process will be extended to three columns, four columns andso on. Once process will stop will obtain design matrix D.
1101
D
1110
.
1111
Step 4. Construct a vector S whose number of columns are same that of designmatrix D. First m elements are 1's, where m is number of columns in P matrix.Next elements(1 or -1) are determined according to rule (-1)i-1, where i is the numberof columns of P that are taken at a time to be OR' ed in particular step.
S=[1 1 -1].Step 5. Let m be the number of columns in P matrix. Construct the structurefunction of the system.
n
i
jiDi
jxjSx
m
1
,12
1.
2
3 4
88
where, D(i,j)= element (i,j) of DS(j) =element j of SHere, m=2, n=4.
Structure function (x) is
4
1
,12
1
.2
i
jiDi
J
xjSx
4
1
,3
1
.i
jiDi
J
xjS
1,44
1,33
1,22
1,111 DDDD xxxxS
2,44
2,33
2,22
2,112 DDDD xxxxS
3,44
3,33
3,22
3,113 DDDD xxxxS
14
13
12
11
14
13
12
01
14
13
02
11 .1.1.1 xxxxxxxxxxxx
.4321432431 xxxxxxxxxxx
Reliability is
xER
4321432431 xxxxExxxExxxE
.4321432431 PPPPPPPPPP
.111 4321 PPPpR
4. Disussion. For last four decades, various methods have been developedto evaluate reliabiality of system. The C-H-A method has got its own importance.Being an algorithmic approach, one can write computer program and use computerto evaluate reliability of complex system. Further, the important relibility measuressuch as Birnbaum Reliability - Importance, Chaudhari bounds can also be evaluatedeasily.
REFERENCES[1] T. Aven, Reliability/availability of inherent system based on minimal cut sets, Reliability
Engineering, 13 (1986), 93-104.[2] E. Balaguruswami, Reliability Engineering, Tata McGraw-Hill Publishing Co. Ltd., New Delhi
(2003).[3] R.E. Barlow and F. Prokshan, Statistical Theory of Reliability and Testing, Holt, Winston Richart
(1975).[4] Gopal Chaudhari, Hu, K. and A. Nadar, A new approach to system reliability, IEEE Trans. on
Reliability, 50.No 1 (2001), 75-84.
[5] E. Charles Ebeling, Reliability and Maintainability Engineering, Tata McGraw-Hill, (2000).
J~n–an–abha, Vol. 38, 2008
A NOTE ON PAIRWISE SLIGHTLY SEMI-CONTINUOUS FUNCTIONSBy
M.C.Sharma and V.K.Solanki Department of Mathematics
N.R.E.C. Collage Khurja, 203131, Uttar Pradesh, India
(Received : March 15, 2007)
ABSTRACTIn this paper we introduce concept of pairwise slightly semi-cintinuous
function in bitopological spaces and discuss some of the basic properties of them.Several examples are provided of illustrate behaviour of these new classes offunctions2000 Mathematics Subject Classification : 54E55.Keywords and Phrases : (i,j) clopen set, pairwise slightly continuous, pairwiseslightly semi-continuous, pairwise almost semi-continuous, pairwise semi -
continuous, pairwise weakly semi-continuous, pairwise s- closed, pairwise ultraregular.
1. Introduction. J.C. Kelly [5] initiated the systematic study ofbitopological spaces. A set equiped with two topologies is called bitopological spaces.Continuity play an important role in topological and bitopological spaces. In 1980,R.C. Jain [4] introduced the concept of slightly continuity in topological spaces.Recently T.M. Nour [10] defined a slightly semi-continuous functions as ageneralization of slightly continuous function using semi-open sets andinvestigated its properties. In 2000, T. Noiri and G.I. Chae [9] introduce a note onslightly semi-continuous functions in topological spaces.
The object of the present paper is to introduce a new class of function calledpairwise slightly semi-continuous functions. This class contained the class ofpairwise continuous functions and that of pairwise semi continuous function.Relation between this class and other class of pairwise continuous functions areobtained.
Throughout the present paper the spaces X and Y always representbitopological spaces (X,P1,P2) and (Y,Q1,Q2) on which no seperation axioms are
assumed. Let S X. Then S is said to be (i, j) semi-open [8] if ClPS j – (Pi-
Int(S)) (where Pj–Cl(S) denoted the closure operator with respect to topology Pjand Pi–Int(S) denoted the interior operator with respect to topology Pi, (i, j=1,2,ij) and its complement is called (i, j) semi-closed. The intersection of all (i, j)semi-closed sets containing S is called the (i, j) semi-closure of S and it will be
90
denoted by (i, j) s Cl(S). A subset S is said to be (i, j) semi-regular if S is both (i,j) semi-open and (i , j) semi closed. A subset S is said to (i, j) semi -open if S isthe union of (i, j) semi-regular sets and the complement of a (i, j) semi -open set iscalled (i, j) semi - closed. A subset S is said to be (i, j) clopen if S is Pi–closedand Pj–open set in X.
In this note we will denote the family of all (i, j) semi-open (resp. Pi–open,(i, j) semi-regular and (i, j) clopen of (X,P1,P2) by (i, j)SO(X)(resp. Pi-open(X),(i, j)SR(X) and (i, j)CO(X)), and denote the family of (i, j) semi-open (resp. Pi-open, (i, j)semi-regular and (i, j) clopen) set of (X,P1,P2) containing x by (i, j)SO(X,x)(resp. Pi(X,x), (i, j)SR(X,x) and (i, j)CO(X,x). i, j=1,2. ij.
2. Preliminaries.Definition 2.1. A function f:(X,P1,P2)(Y,Q1,Q2) is said to be pairwise-semicontinuous [8] (p.s.C) resp. pairwise almost semi continuous (p.a.s.C)[12] pairwisesemi -continuous (p.s.C.) [12] and pairwise weakly semi continuous (p.w.s.C.)
[12]) if for each Xx and for each xfyQV i , there exists xXSOjiU ,,
such that Qint resp. j VClQUfVUf i
VClQUfVClQUsCljif jj and , .
Definition 2.2. A function f:(X,P1,P2)(Y,Q1,Q2) is called to be pairwise almostcontinuous (p.a.C) [2] (resp. pairwise -continuous (p..C.)[1], pairwise weaklycontinuous (p.w.C)[2] if for each xX and for each VQi–(Y, f(x)), there is UPi(X,x)such that f(U) Qi–Int(Qj–Cl(V)) (resp. f(Pj–Cl(U) Qj–Cl(V), f(U) Qj–Cl(V)).Definition 2.3. [11] A function f:(X,P1,P2)(Y,Q1,Q2) is called slightly semi-continuous (p.sl.s.C.)(resp. pairwise slightly continuous (p.sl.C) if for each xXand for each V(i, j)CO(Y, f(x)), there exists U(i, j)SO(X,x) (resp. UPi(X,x)) suchthat f(U) V, i, j=1,2 and ij.
A function f:(X,P1,P2)(Y,Q1,Q2) is said to be pairwise slightly semi-continuous (resp. pairwise slightly continuous) if inverse image of each (i, j)-clopenset of Y is (i, j) semi-open (resp. Pi–open) in X; i, j=1,2, ij.
The following diagram is obtained :p. C p.a.C p..C. p.w.C p.sl.C
p.s.C p.a.s.C p..s.C p.w.s.C p.sl.s.C
diagram 1Remark 2.4. It was point out in [11] that pairwise slightly continuity impliespairwise slighltly semi-continuity, but not conversely. Its counter examples are notgiven in it.
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Example 2.5. Let X={a, b, c}, P1={,X, {a},{b},{a,b}}, P2={,X, {a},{a,b}} andlet Q1={, X, {a}}, Q2={, Y, {b,c}}. Then the mapping f:(X,P1,P2)(Y,Q1,Q2) ispairwise slihtly semi-continuous but not pairwise slightly continuous for f–1({a})is (j, i) semi-open and (i, j) semi-closed, but not Pi –closed in (X,P1,P2).Theorem 2.6. The pairwise set-connectedness and the pairwise slightly continuityare equivalent for a surjeetive function.Proof. A surjection f:(X,P1,P2)(Y,Q1,Q2) is pairwise set-connected if and only iff–1(F) is (i, j) clopen in X for each (i, j) clopen set F of Y. It is easy to prove that afunction f:(X,P1,P2)(Y,Q1,Q2) is pairwise slightly continuous if and only if f–1(F)is P1–open in X for each (i, j) clopen set F of Y. Therefore, the proof is obvious.Theorem 2.7. For a function f:(X,P1,P2)(Y,Q1,Q2) the following are equivalent:
(a) f is pairwise slightly semi-continuous,(b) f–1(V)(i, j)SO(X) for each V(i, j)CO(Y),(c) f–1(V) is (j, i) semi-open and (i, j) semi-closed for each V(i, j)CO(Y).3. Properties of pairwise slightly semi-continuity.
Theorem 3.1. The following are equivalent for a function f:(X,P1,P2)(Y,Q1,Q2):(a) f is pairwise slightly semi-continuous,(b) For each xX and for each (V)(i, j)CO(Y,f(x)), there exists U(i, j)
SR(X,x) such that f(U) V,(c) For each xX and for each (V)(i, j)CO(Y, f(x)), there is U(i, j)
SO(X,x) such that f(i, j)sCl(U)) V.Proof. (a) (b). Let xX and V(i, j)CO(Y, f(x)). By Theorem 2.7, we have f–1(V)(i,j)SR(X,x). Put U=f–1(V)), then xU and f(U) V.
(b) (c). It is obvious and is thus omitted.(c) (a). If U(i, j)SO(X), then (i, j)sCl(U)(i, j)SO(X).
Definition 3.2. A bitopological space (X,P1,P2) is called(a) Pairwise semi-T2[6] (resp. pairwise ultra Hausdorff or pairwise UT2) iffor each pair of distinct points x, y of X, there exists a P1–semi-open (resp. P1–clopen) set U and a P2–semi-open (resp. P2–clopen) set V such that xU, yV andUV=.(b) Pairwise s-normal[7] (resp.pairwise ultra normal) if for every Pi-closedset A and Pj-closed set B such that AB=, there exist USO(X) (resp Co(X,Pj)and VSO(X,Pi)(resp.CO(X,Pi)such that A U, B V and UV=where i, j=1,2,ij(c) Pairwise s-closed (resp.pairwise mildly compact) if every (i,j) semi regular(resp.(i, j) clopen) cover of (X,P1,P2) has a finite subcover. i, j=1,2 and ij.Theorem 3.3. If f : (X,P1,P2)(Y,Q1,Q2) is a pairwise slightly semi-continuousinjection and Y is pairwise UT2 ,then X is pairwise semi-T2.
92
Proof. Let x1,x2X and x1x2. Then since f is injective and Y is pairwise UT2, f(x1)f(x2) and there exist, V1,V2(i, j)CO(Y) such that f(x1)V1, f(x2)V2 and V1V2=.By Theorem 2.7, xif-1(Vi)(i, j)SO(X) for i=1,2 and f-1(V1)f-1(V2)=. Thus X ispairwise semi-T2.Theorem 3.4. If f:(X,P1,P2)(Y,Q1,Q2) is a pairwise slightly semi-continuous,P2-closed injection and Y is pairwise ultra normal, then X is pairwise s-normal.Proof. Let F1 and F2 be disjoint (P1,P2)-closed subsets of X. Since f is P2-closedand injective, f(F1) and f(F2) are disjoint (Q1,Q2)-closed subsets of Y. Since Y ispairwise ultra normal, f(F1) and f(F2) are separeted by disjoint Pi–clopen sets V1and Pj-clopen V2, respectively. Hence Fif–1(Vi), f
-1(Vi)(i, j)SO(X) for i=1,2 fromTheorem 2.7 and f-1(V1)f-1(V2)=. Thus X is pairwise s-normal.Theorem 3.5. If f:(X,P1,P2)(Y,Q1,Q2) is a pairwise slightly semi-continuoussurjection and (X,P1,P2) is pairwise s-closed, then Y is pairwise mildly compact.Proof. Let {V|V(i,j )CO(Y), } be a cover of Y. Since f is pairwise slightly
semi-continuous, by the Theorem 2.7 {f-1(V)| } is a (i, j) semi-regular cover
of X and so there is a finite subset 0 of such that 0
1
VfX .
Therefore,
0
VY
since f is surjective. Thus Y is pairwise mildly compact.Theorem 3.6 If f:(X,P1,P2)(Y,Q1,Q2) is pairwise slightly semi-continuous andY is pairwise UT2, then the graph G(f) of f is (i, j) semi -closed in the bitopologicalproduct space X×Y.Proof. Let (x,y) G(f), then y f(x). Since Y is pairwise UT2, there exist V(i, j)CO(Y, y) and W(i, j) CO(Y, f(X)) such that VW=Since f is pairwise slightlysemi-continuous, by the Theorem 2.7 there exist U(i, j) SR (X, x) and V(i, j)CO(Y, y), (x,y) U V and U V (i,j)SR(X Y). Hence G(f) is (i, j) semi - closed.Theorem 3.7. If f : (X, P1, P2)(Y, Q1, Q2) is pairwise slightly semi-continuousand (Y, Q1, Q2) is pairwise UT2, then A={(X1,X2) f(x1)=f(x2) } is )(i, j)semi -closed in the bitopological product space X X.Proof. Let (X1,X2) A. Then f(x1)f(x2). Since Y is pairwise UT2, there exist V1(i, j) CO(Y, f(X1)) and V2 (i, j) CO(Y, f(X2)) such that V1V2=. Since f is pairwiseslightly semi-continuous, there exist U1,U2 (i, j)SR(X) such that X1U1 andf(U1)V1 for i=1,2. Therefore, (X1,X2)U1U2 , U1U2(i, j) SR(XX), and (U1U2)A=. So A is (i, j) semi -closed in bitopological product space X×X.Definition 3.8. [3] A bitopological(X, P1, P2) is said to be pairwise extremally
93
disconnected if P2 -closure of each P1-open set of (X, P1, P2) is P1-openLemma 3.9. Let (X, P1, P2) be pairwise exteremally disconnected space, then U(i, j)SR(X) if and only if U(i, j)CO(X),i,j=1,2 and ij.Proof. Let U(i, j)SR(X). Since U(i, j)SO(X),Pj-Cl (U)=Pj-Cl(Pi-Int (U) and soPi-Cl(U) Pi(X). Since U is (i, j) semi-closed , Pi-Int(U) =U= Pj-Cl (U) and hence Uis (i, j) clopen. The convers is obvious.Theorem 3.10. If f:(X,P1,P2)(Y,Q1,Q2) is pairwise slightly semi-continuous,and X is pairwise extremally disconnected then f is pairwise slightly continuous.Proof. Let xX and V(i, j)CO(Y,f(x)). Since f is pairwise slightly semi-continuousby Theorem 2.7, there exists U(i, j)SR(X,x) such that f(U) V, since X is pairwiseextremally disconnected by the Lemma 3.9,U(i, j) CO (X) and hence f is pairwiseslightly continous.Difinition 3.11. A function f : (X, P1, P2)(Y, Q1, Q2) is called pairwise almoststrongly -semi continuous (p,a..s.C.) if for each x X and for each V Qi(Y,f(x)),there exists U(i, j)SO(X,x) such that f((i, j) sCl(U)) (i, j)sCl(V) (resp.f((i, j)sCl(U))V).Theorem 3.12. If f:(X,P1,P2)(Y,Q1,Q2) is pairwise slightly semi-continuous and(Y, Q1, Q2) is pairwise extremally disconnected, then f is pairwise almost strogly-semi-continuous.Proof. Let xX and VQi(Y,f(x)), then (i, j) sCl(V) = Qi -Int(Qj-Cl(V)) is (i, j) regularopen in (Y, Q1, Q2). Since Y is pairwise extremally disconnected, (i, j) sCl(V)(i, j)CO(Y). Since f is pairwise slightly semi-continuous, by Theorem 3.1, there existsU (i, j) SO(X, x) such that f((i, j)sCl(U)(i, j)sCl(V)). So f is pairwise almoststrongly -semi-continuous.Corollary3.13. [11] If f:(X,P1,P2)(Y,Q1,Q2) is pairwise slightly semi-continuousand (Y, Q1, Q2) is pairwise extremally disconnected. Then f is pairwise weaklysemi-continuous.Difinition 3.14. A bitopological space (X, P1, P2) is called pairwise ultra regularif for each UPi(X) and for each xU, there exists O(i, j) CO (X) such that xOU.Theorem 3.15. If f : (X, P1, P2)(Y, Q1, Q2) is pairwise slightly semi-continuousand (Y,Q1,Q2) is pairwise ultra regular, then f is pairwise stroghly -semi-continuous.Proof. Let xX and VQi(Y,f(X)). Since (Y,Q1,Q2) is pairwise ultra regular, thereis W(i,j)CO(Y) such that f(x)WV. Since f is pairwise slightly semi-continuous,bythe Theorem 3.1 there is U(i,j)SO(X,x) such that f((i, j)sCl(U))W and sof(i,j)sCl(U))V. Thus f is strongly -semi-continuous.We have to the following diagram:
p.C
94
P.C P.a.C P..C P.w.C P. sl.C P.s.C P.a.s.C P.s..C P.w.s.C P. sl.s.C P.st..s.C P.st..s.C
REFERENCES[1] S.Bose, Almost open sets, almost closed, -continuous and almost quasi compact mapping in
bitopological spaces, Bull, Cal. Math.Soc; 73 (1981),345-354.
[2] S.Bose and D. Sinha, Pairwise almost continuous map and weakly continuous map in bitopologicalspaces, Bull, Cal. Math.Soc., 74 (1982), 195-206.
[3] M.C.Datta, J.Austr. Math. Soc, 14(1972) 119-128.
[4] R.C.Jain, The Role of Regularly Open Sets in General Topology, Ph.D. Thisis, Meerut University,Institute of Advanced Studies, Meerut, India(1980).
[5] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13 (1961),71-89.
[6] S.N. Maheshwari and R. Prasad, Some new separation axioms in bitopological spaces, Mathematica;12 (27) (1975), 159-162.
[7] S.N. Maheshwari and R. Prasad, On pairwise s-normal sapces, Kyungpook Math. J., 15 (1975), 37-40.
[8] S.N. Maheshwari and R. Prasad, On pairwise irresolute functions, Mathematica, 18 (1976), 169-172.
[9] T.Noiri and G.I.Chae, A note on slightly semi-continuous functions, Bull. Cal. Math. Soc., 92(2)(2000), 87-92.
[10] T.M.Nour,Slightly semi-continuous functions, Bull. Cal. Math. Soc, 87 (1995), 187-190.
[11] M.C. Sharma and V.K.Solanki, Pairwise slightly semi-continuous function in bitopological spaces.[communicated]
[12] V.K.Solanki, Almost semi-continuous mapping in bitopological spaces .The Mathematics EducationXLII, No. 3 (To appear)
J~n–an–abha, Vol. 38, 2008
FIXED POINT THEOREMS FOR AN ADMISSIBLE CLASS OFASYMPTOTICALLY REGULAR SEMIGROUPS IN LP- SPACES
ByG.S. Saluja
Department of Mathematics and I.T.,Government College of Science, Raipur-492010, Chhattisgarh
E-mail: [email protected](Received : June 15, 2008)
ABSTRACTThe aim of this paper is to prove existence of fixed point of an admissible
class of asymptotically regular semigroup Ts in LP-space 21 p satisfying the
condition:
GsxTyyTxcyTyxTxbyxayTxT sssssssss , 22
where as, bs and cs are non-negative constants satisfying certain conditions. Ourresult extends and improves the result of Ishihara [10], Ishihara and Taka-hashi[11,12] and many others.2000 Mathematics subject classification No : 46B20,47H10.
Keywords: Asymptotic regularity, fixed point, LP-space 21 p
1. Introduction. Let K be a nonempty subset of a Banch space E and
KKT : be a nonlinear mapping. The mapping T is said to be Lipschitzian if
there exists a positive constant kn such that
, yxkyTxT nnn
for all Kyx , and for all .Nn A Lipschitzian mapping is said to be nonexpansive
if 1nk for all .Nn uniformly k-Lipschitzian if kkn for all .Nn , and
asymptotically nonexpansive if ,1lim
nn
k respectively. These mappings were first
studied by Goebel and Kirk [7] and Goebel, Kirk and Thele [9]. Lifschitz [14] proved
that in a Hillbert space a uniformly k-Lipschitzian mappings with 2k has a
fixed point. Downing and Ray [5] and Ishihara and Takahashi [12] prove that in a
Hilbert space a uniformly k-Lipschitzian semigroup with 2k has a common
fixed point. Casini and Maluta [4] and Ishihara and Takahashi [11] proved that auniformly k-Lipschitzian semigroup in a Banach space E has a common fixed pointif k<N(E), where N(E) is the constant of uniformly normal structure.
96
In these results, the domain of semigroups were assumed to be closed andconvex. Ishihara [10] gave the fixed point theorem for Lipschizian semigroups inboth Banach and Hilbert spaces in which closedness and convexity of domin werenot needed.
The concept of asymptotic regularity is due to Browder and Petryshyn [2].A mapping EET : is said to be asymptotically regular if
,0lim 1
xTxT nn
n
for all ., Eyx
It is well known that if T is nonexpansive then TttITt 1 is asymptoti-
cally regular for all 0<t<1.Now we consider the following class of mappings, whose nth iterate Tn
satisfies the following condition:
xTyyTxcyTyxTxbyxayTxT nnn
nnnn
nn . 22
for all Cyx , and n=1,2,... where nn ba , and nc are nonnegative constants
satisfying certain conditions. This class of mappings is more general than theclass of nonexpansive, asymptotically nonexpansive, Lipschizian and uniformlyk-Lipschitzian mappings. The above facts can be seen by taking bn=cn=0. The aimof this paper is to prove a fixed point theorem for the above said class of
asymptotically regular semigroups in LP-space .21 p Our result extends and
improves the results of Ishihara [10], Ishihara and Takahashi [11,12] and manyothers.
2. Preliminaries. Let G be a semitopological semigroup, that is, G is a
semigroup with a Hausdorff topology such that for each Ga the mapping ass
and sas from G to G are continuous. A Semitopological semigroup G is leftreversible if any two closed right ideals of G have nonempty intersection. In this
case, ,G is a directed system when the binary relation" " on G is defined by a
b if and only if .bGbaGa Examples of left reversible semigroups include
commutative and all left amenable semigroups.
Let K be a nonempty subset of a Banach space E. Let GtTS t : be a
family of mappings from K into itself. Then S is said to be an admissible class ofasymptotically regular semigroup on K if it satisfying the following:
(2) for each ,Kx the mapping xTxs s, from KG into K is
97
continuous when KG has the product topology,
(3) for each ,Kx ,Gh
,0lim xTxT tht
t
(4) for each Gs
(2.0.1) GsxTyyTxcyTyxTxbyxayTxT sssssssss , 22
for all ,, Kyx where ss ba , and sc are non-negative constants satisfying certain
conditions.
Let AB : be a decreasing net of bounded subsets of a Banach space E.
For a nonempty subset K of E, define
;:supinf, ByyxxBr
;:,inf, KxxBrKBr
.,,:, KBrxBrKxKBA
We know that ,.Br is a continuous convex function on E which satisfies
the following:
yBrxBryxyBrxBr ,,,,
for each ., Eyx It is easy to see that E is reflexive and K is closed convex, then
KBA , is nonempty, and moreover, if E is uniformly convex, then it consists
of a single point (cf. [15]).Let p>1 and denote by the number in [0,1] and by Wp( ) the function
.11 pp
The functional p. is said to be uniformly convex (cf.[25]) on the Banach
space E if there exists a positive constant cp such that for all ]1,0[ and Eyx ,
following inequality holds:
.11 ppp
ppp yxcWyxyx
In Hilbert space H, the following equality holds:
,111 2222 yxyxyx ...(**)
for all Hyx , and ].1,0[
98
If ,21 p then we have for all x,y in LP and ],1,0[ the following
inequality holds:
(2.0.2) 2222 1111 yxpyxyx
(The inequality (2.0.2) is contained in [17]).
Xu [24] proved that the functional p. is uniformly convex on the whole
Banach space E if and only if E is p-uniformly convex, that is, there exists a constantc>0 such that the modulus of convexity (see[8])
forc pE all .20
The normal structure coefficient N(E) of E is defined by Bynum [3] as follows:
KrdiamK
ENK
inf K is a bounded convex subset of E consisting of more than
one point] where
diam K= Kyxyx ,:sup
is the diameter of K and
yxKrKyKx
K supinf
is the Chebyshev radius of K relative to itself.
The space E is said to have uniformly notmal structure if .1EN It is
known that a uniformly convex Banach space has uniformly normal structure
and for a Hilbert space H,N(H)= .2 Recently, Pichugov [18] (cf. Prus [20] calculated
that
.1,2,2min /1/1 pLN pppP
Some estimates for normal structure coefficient in other Banach spaces
may be found in Prus [21]. For a subset K, we denote by Kco the closure of the
convex hull of K.3. Main Result. In this section, we give our main result:
Theorem 3.1. Let E be a LP- space Kp ,21 a nonempty subset of E,G
a left reversible semitopological semigoup and GtTS t : be an admissible class
of asymptotically regular semigroup on K satisfying the condition:
GsxTyyTxcyTyxTxbyxayTxT sssssssss , 22 ...(*)
99
for all ,, kyx where as, bs and cs are non-negative constants satisfying certain
conditions such that
1
2.
12114 2
12
Npp
where
ssa ,suplim
ssc . suplim
Suppose that GtyTt : is bounded for some Ky and there exists a closed
subset C of K such that CstxTco ts : for all .Kx Then there exists a zC
such that zzTs for all sG.
Proof. Let stxTcoxB ts ; and let s s xBxB for Gs and .Kz Define
0: nxn by induction as follows:
,0 yx ,, 11 nnsn xBxBAx for 1n
since KCxB for all nxKx , is well defined. Let
,, mmsm xBxBrr
,, 1 mmsm xBxBrD .1m
Now for each Gts , and ,, Kyx we have
yTxTxTTycyTyxTTxTbyxayTxTT sttssststsssts ..22
and so
(3.0.3) ..22 xTTxTxTycyTyxTTxTbyxayTxTT tsttsststsssts
yTyyxT st .
Then from 11 mttmm xBxBx and a result of Ishihara and Taka-
hashi [11], we have
(3.04) mssmmsm xBdiamN
xBxBrr inf1
,
Now using (3.0.4), we have
sbaxTxTxBdiam mbmas
mss
,:supinfinf
100
mtms
stxTxTsuplimsuplim
mtmst
stxTxTTsuplimsuplim
mtmmstmstmms
stxTxxTTxTbxxTat .suplimsuplim 2
2/1. mstmmtmst xTTxxTxTc
tmtmmstmstmms
stcxTxxTTxTbxxTat .suplimsuplim 2
2/1. mstmsmsmmtmmms xTTxTxTxxTxxxT
Taking the s assuplim and by asymptotic regularity of T, we get
2/12supliminf mmtmmmt
mss
DxTxDDxBdiam .
Again taking the , assuplim t we get
,2inf 2/1mms
sDxBdiam
and hance using (3.0.4), we have
(3.0.5)
,2 2/1
mm DN
r
where
,suplim tt
a tt
csuplim
and N is the normal structure coefficient of E.Again from (2.0.2) and (3.0.3), we have
211
211 111 mtmmsmtm xTxpxTxTx
21
21 1 msmtmsm xTxTxTx
21
21 1 mstmtmsm xTTxTxTx
121 msm xTx mtsmsmtmtmsmt xTTxTxTxbxTxa 11
21
101
1111 mtmmmsmstmsmsmt xTxxxTxTTxTxTxc .
Taking the s assuplim and by asymptotic regularity of T, we have
11222
112 111 mtmmmtmtmmtmm xTxrrcrarxTxpr .
Again taking the , assup t we have
1222
12 111 mmmmmmm DrrarrDpr
or,
.111
122
12
mmmmmm DrrrDpr
Letting ,1 we get
122
12 1 mmmmm DrrDpr
or,
011 22 mm rtrtptF ,
where .1 mDt
It can be easily seen that
0tF for all t .
121142
mrpp
It follows from (3.0.5) that
.
212
114 2/12
1 mm DNp
pD
Hence,
1,.1 mDBD mm
where
,1
212
114 2/12
Np
pB
by the assumption of the theorem. Since
mmsmmsmm xxBrxxBrxx ,, 11
mm Dr
mD2
102
...
...
02 11 DBm as ,m
it follows that mx is a Cauchy sequerce. .lim Let mm xz Then we have
zTxTxTxxzzTz smsmsmms
2/12 .. mssmssmsmsmsmsmm xTzxTxczTzxTxbzxaxTxxz
2/12 . msmmsmssmsmsmsmsmm xTxxzzTzzxczTzxTxbzxaxTxxz
2/12mmsmssmsmsmm DxzzTzzxczTzDbzxaDxz
Taking the limit as m each side, we have
mszTz lim zTzDbzxaDxz smsmsmm
2(
0) 2/1 mmsms DxzzTzzxc
for all .Gs Hence we have zzTs for all .Gs This completes the proof.
As a direct consequence of Theorem 3.1, we have the following results:
Corollary 3.1. Let E be a LP- space Kp ,21 a nonempty subject of E,
and KKT : be a self mapping satisfying the condition:
xTyyTxcyTyxTxbyxayTxT nnn
nnnn
nn ..22
for all Kyx , and ,1n where nn ba , and nc are nonnegative constatnt satisfying
certain conditions. Suppose that 1: nyT n is bounded for some Ky and there
exists a closed subset C of K such that CnkxTco kn : for all .Kx If
,1
212
114 2/12
Npp
where
nna ,suplim
nnc ,suplim
then there exists a point z in C such that Tz=z.By Therorem 3.1 and equation (**), we immediately obtain the following:Theorem 3.2: Let E be a Hillbert space, K a nonempty subset of E, G a left
reversible semitopological semigoup and GtTS t : be an admissible class of
103
asymptotically regular semigroup on K satisfying the condition:
GsxTyyTxcyTyxTxbyxayTxT sssssssss ,22
for all ,, Kyx where ss ba , and sc are non-negative constants satisfying certain
conditions such that
1
2
2.
214 2
12
where
s
sa ,suplim
s
sc ,suplim
Suppose that GtyTt : is bounded for some Ky and there exists a closed
subset C of K such that CstxTco ts : for all .Kx Then there exists a
Cz such that zzTs for all .Gs
If we put 0 ss cb and 2ss ka in inequality (*) of Theorem 3.1, we get the
following results as corollary:Corollary 3.2 (see[ [10], Theorem1): Let K be a nonempty subset of a Hillbert
space H,G a left reversible semitopological semigroup, and GtTS t : a
Lipschitzian semigroup on K with .2suplim ss k Suppose that GtyTt : is
bounded for some Ky and there exists a closed subset C of K such that
CstxTco ts : for all .Kx Then there exists a Cz such that zzTs for
all .GsRemark 1. Theorem 3.2 extends and improves the corresponding results
of Ishihara [10], Ishihara and Takahashi [11] and Downing and Ray [5]. The reasonis that the above authors have considered Lipschitzian of uniformly Lipschitziansemigroups in Hilbert space whereas we consider asymptotically regular semigroupsatisfying the condition (*) which is more general than Lipschitzian or uniformlyLipschitzain semigroups.
Remark 2. Our results also extend several known results given in theliterature.
REFERENCES[1] J. Barros-Neto, An Introduction to the Theory of Distributions, Pure and Applied Mathematics, 14,
Marcel Dekker, New York, 1973.[2] F.E. Browder and V. W. Petryshyn, The solution of nonlinear functional equations in Banach
spaces, Bull. Amer. Math. Soc., 72(1966), 571-576.
104
[3] W.L. Bynum, Normal structure coefficient for Banach space Pacific J. Math., 86 (1980), 427-436.[4] E. Casini and E. Maluta, Fixed points of unifromly lipschitizian mappings in spaces with uniformly
normal structure, Nonlinear Anal. TMA 9 (1985), 103-108.[5] D.J. Dowing and W.O. Ray, Uniformly Lipschitzian semigroups in Hilbert space, Canad. Math. Bull.,
25 No. 2 (1982) 210-214.[6] N. Dunford and J.T. Schwartz, Linear Operators, I. General Theory, Pure and Applied Mathematics,Vol
7 Intersciences Publishers, New York (1958).[7] K. Goebel and W.A. Kirk, A fixed point theorem for transformations whose iterates have unifromly
Lipschitz constant, Studia Math, 47 (1973), 135-140.[8] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math. 28
Cambridge University Press, London, 1990.[9] K. Goebel, W.A. Kirk and R.L. Thele, Uniformly Lipschitzian families of transformations in Banach
Spaces, Canad. J. Math., 26 (1974), 1245-1256.[10] H. Ishihara, Fixed point theorems for Lipschitzian semigroups, Canad. Math. Bull., 32 No.1 (1989),
90-97.[11] H. Ishihara and W. Takahashi, Fixed point theorems for uniformly Lipschitzian semigroups in
Hilbert spaces, J. Math. Anal. Appl. 127 no. 1 (1987) 206-210.[12] H. Ishihara and W. Takahashi, Modulus of convexity, characteristics of convexity and fixed point
theorems, Kodai Math. J. 10 no. 2 (1987) 197-208.[14] E.A. Lifschits, Fixed point theorems for operators in strongly convex spaces (Russian) Voronez. Gos.
Univ. Trudy Mat. Fak. 16 (1975), 23-28.[15] T.C. Lim, On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math., 32 no.
2 (1980) 421-430.[16] T.C. Lim, On some LP inqualities in best approximation theory, J. Math. Anal. Appl., 154 (1991),
523-528.[17] T.C. Lim, H.K. Xu and Z.B. Zu, An Lp inequality and its applications to fixed point theory and
approximation theory in progress in Approximation Theory, P. Neval and A. Pikus (eds.) AcademicPress, New York 1991, 609-624.
[18] S.A. Pichugove, The Jung constant of the space Lp, Mat. Zametki, 43 No.5 (1988), 604-614,Translation in Math. Notes, 43 (1988), 348-354.
[19] B. Prus and R. Smarzewski, Strongly unique best approximations and centers in uniformly convexspaces, J. Math. Anal. Appl. 121 no. 1 (1987), 10-21.
[20] B. Prus, On Bynum's fixed point theorems, Atti. Sem. Mat. Fis. Univ. Modena, 38 No. 2 (1990), 535-
545.[21] B. Prus, Some estimates for the normal structure coefficients in Banach spaces, Rend Circ. Mat.
Palermo (2) 40 1 (1991) 128-135.[22] R. Smarzewski, Strongly unique best approximation in Banach spaces, II J. Approx. Theory 51
No.2 (1987), 202-217.[23] R. Smarzewski. On an inequalityi of Bynum and Drew, J. Math. Anal. Appl. 150 No. 1 (1990), 146-
150.[24] H.K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal. 16 No. 12 (1991), 1127-
1138.
[25] C. Zalinescu, On uniformly convex function, J. Math. Anal. Appl., 95 (1983), 344-374.
105
J~n–an–abha, Vol. 38, 2008
SOME INTEGRAL FORMULAS INVOLVING A GENERAL SEQUENCEOF FUNCTIONS, A GENERAL CLASS OF POLYNOMIALS AND THE
MULTIVARIABLE H-FUNCTIONBy
V.G. GuptaDepartment of Mathematics,
University of Rajasthan, Jaipur-302004, Rajasthan, India and
Suman JainS.S. Jain Subodh Postgraduate College, Jaipur-302004, Rajasthan, Indian
(Received : August 14, 2008)
ABSTRACTIn this paper we establish four integral formulas involving a general
sequence of functions, a general class of polynomials and the multivariable H-function. The results established here are quite general in nature from which onecan derive a large number of (known and new) integrals by specializing theparameters suitably of the various functions involved therein.2000 Mathematics Subject Classification No : Primerey... secondaryKeywords: General sequence of functions, General class of polynomials.Multivariable H-function.
1. Introduction. Srivastava and Panda [c.f.,e.g., [10], [15], [16] and [17]introduced the multivariable H-function in a series of papers defined andrepresented in the following contracted notation ([15], p. 130)
r
r
qrj
rjqjjQ
rjjj
prj
rjpjjp
rjjj
rrr
rrr ddb
cca
z
z
qpqpQpnmnmN
HzzH,1,1
'',1
';
,1,1''
,1'
;1
1,1
111 ,;...;,,...,
,;...;,,...,
;...;:,,;...;,:,0
,...,1
1 (1.1)
to denote the H-function of r complex variables rzz ,...,1 . See Srivastava, Gupta
and Goyal ([18], p.251, Eq.(4.17)) for details of this function.A general class of polynomials occurring in this paper was introduced by
Srivastava [[20], p.158, eq.(1.1)] defined by
mn
k
kkn
mk nxAkn
xnm
S/
0, ,...2,1,0,
! ...(1.2)
where m is arbitrary positive integer and the coefficients 0,, knA kn are arbitary
constants, real or complex.
106
By suitably specializing the coefficients knA , the polynomial xSmn can be
reduced to the well known polynomials. These include among others, the Jacobi,Hermite, Legendre, Tchebycheff, Laguerre polynomials.
Agarwal and Chaubey [1], Srivastava and Manocha [21], Salim [13] andseveral others have studied a general sequence of functions. In the present paperwe shall study the following useful series formula for a general sequence of functions
hetuv
sn xhetuvqpdcbaxR
,,,,
, ,,,,,;,;,,,; ...(1.3)
where
n
v
v
u
n
t
t
e
q
hhetuv
ptvhqknsxdc
0 0 0 0 0,,,,
,1, ...(1.4)
and
!!!!!
,,,, ' hetuvktvdcakb
hetuvn
tcuhvnhvtntn
ve
hveht
kqupe
tnvnn
11
...(1.5)
By suitably specializing the parameters involved in (1.3) a general sequence offunction reduced to generalized polynomial set studied by Raizada [11], a class ofpolynomials introduced by Fujiwara [5], a well known Jacobi polynomials given inRainville [10] and several others.
2. Main Integrals. In this section we establish four main integrals.First Integral
btat
tSbtatt m
n21
0
,;,;,,',',, qpgfdcbtat
tR BA
D
dtzbtat
tz
btatt
H r
r
,...,1
1
s
hewvu
mn
nm bahewvuA
n
21
,,,,
/
0, ,,,,
!
107
rbazbazH r22
11
,, ,...,1
...(2.1)
where
rr
rrr qpqpQp
nmnmNHzzH
;...;:1,1,;...;,:1,0
,...,1,1
111
1,,
r
r
qrj
rjqjjQ
rjjjr
prj
rjpjj
P
rjjjr
rddbs
ccas
z
z
,1,1''
,1'
1
,1,1''
,1
'1
1
,;...;,:,...,;(),,...,;1
,;...;,:,...,;(),,...,;23
1
1
...(2.2)
Provided a,b, , and ri ,...,11 are all positive Re 0 and
r
iiis
1
021
Re
where
0Remin1
ij
ij
ji
di ...(2.3)
The infinite series occurring on the right hand side of (2.1) converages absolutely.
Also hewvu ,,,, is given by (1.5).
Second Integral
,;,;,,',';,/10
1 qpgfdcx
xtR
xxt
nm
Sextx BAD
xt
dxzx
xtz
xxt
H r
r
,...,1
1
hewvu
mn s
nmt t
An
hewvuet,,,,
/
0,
/1
!,,,,
rzt
zt
Hr
,...,12
,,
1
...(2.4)
where
rr
rr
nmnmNqpqpQpr HzzH ,;...;,:1,0
,;...;:1,112
,,11
1,1,...,
108
r
r
qrj
rjqjjQ
rjjj
prj
rjpjjP
rjjjr
rddb
ccas
z
z
,1,1''
,1'
,1,1''
,1'
11
,;...;,:,...,;(
,;...;,:,...,;(),,...,;1
1
1
...(2.5)
r
iiis
1
0Re,0Re,0Re
Third Integral
1
0
1
0 11
111
11
11
xy
y
n
mS
yx
xy
xy
yy
xy
x
,;;,,',';11, pgfdc
xy
yR BA
D
dxdyzxy
y
xy
xz
xy
yy
xy
xH r
rr
11
11,...,
11
11
1
11
rr
hewvu
mn
nm zzzHhewvuA
n ,...,,,,,,! 1
3,,,
,,,,
/
0,
...(2.6)
where
rr
rr
nmnmNqpqpQPr HzzH ,;...;,:2,0
,;...;:1,213
,,,11
1,1,...,
,
,;...;,,,...,;(,...,,1(
,;...;,,,...,;(,,...,.1),,...,;1
,1,1''
,1'
11
,1,1''
,1'
111
1
1
r
r
qrj
rjqjjQ
rjjjrr
prj
rjpjjP
rjjjrr
r
ddbs
ccas
z
z
...(2.7)
where riii ,...,10,,,
and
r
iii
r
iii s
11
0Re,0Re
[ i is given by (2.3)]
Fourth Integral
1
0
1
0 ,11 ,;,;,,',';1111 qpgfdcyRySyxyxyf BA
Dmn
109
dxdyzyxyzyxyH rrrr 11,...,11 1
111
hewvu
mn
rnm rrzzzzHhewvuA
n
,,,,
/
01
1
0 4
,,,, 1,...,1,,,,!
11
dzzzf s 11 ...(2.8)
where
rrzzzzH r
1,...,1 111
4,,,
rr
rr
qpqpQpnmnmN
H;...;:1,2
,;...;,:2,0
1,1
11
,
,;...;,,,...,;(,...,;1(
,;...;,,,...,;(,,...,.1),,...,;1
1
1
,1,1''
,1'
11
,1,1''
,1'
111
1
1
11
r
r
r qrj
rjqjjQ
rjjjrr
prj
rjpjjP
rjjjrr
rddbs
ccas
zz
zz
...(2.9)
Provided that f(z) is so chosen that the integral (2.8) exists, riii ,...,10,,,
and
r
iii
r
iii s
11
0Re,0Re
where i is given by (2.3)
Proof. To establish (2.1) express the general class of polynomial ,xSmn general
sequence of function ,;,;,,,,, qpdcbaxR BAD on the left hand side of (2.1) by its
series (1.2) and (1.3) respectively and the multivariable H-function in terms ofMellin-Barnes type contour integral with the help of (1.1). Change the order ofintegration and summation (which is easily seen to be justifiable due to the absoluteconvergence of the integral and sums involved in the process) under the conditionsmentioned with (2.1) and then evaluate the resulting t-integral with the help of(2.10) a known result given by Gradshteyn and Rayzhik ([7], p.289, 3, 197(7)).
2121
021
badxbxxx
[where 0Re ] ...(2.10)
and then interpreting the result with the help of (1.1) we thus easily arrive at the
110
right-hand side of (2.1).The proof of the formulas (2.4), (2.6) and (2.8) are similar to (2.1) in which
we use the following results ([7],p. 339, eqn. 3.471(3); [3],p. 145-243)
txtetdxextx /1/1
0 1 ...(2.11)
1
0
1
0 111 ,111 Bdydxxyyxy ...(2.12)
1
0
1
0 11 11 dydxyyxxyf
dzzzfB 11
0 1, ...(2.13)
instead of (2.10).3. Special Cases.
(i) On setting g=p=1 and replacing fBB , by and let 1' nk in (2.1), we
get an integral involving general class of polynomials, generalized polynomial setand the multivariavle H-function
btat
tSbtatt m
n1
0 21
kdcq
btatt
S BAD ,,',',,,;,,
rzbtat
tz
btatt
Hr
,...,1
1
dt
hewvu
mn sn
m bahewuvAn
,,,,
/
0
21, ,,,,
!
rbazbazH r22
11
,, ,...,1
...(3.1)
The integral (3.1) is valid under same conditions as given for Integral (2.1) andwhere
!!!!!
1'',,,,1 hewuv
Awvckdhewvu weu
vwwnwD
Dv
DDve
e Dvk
queD
BwADA
1
111
and .wvhqkDs
On applying the same procedure as above in (2.4), (2.6) and (2.8), we can establishthree other integrals.
(ii) If we set 1,0'' qkcdD and 0 in our special case (i)
then the generalized polynomial set reduces to unity and we arrive at the resultobtained by Gupta [8].
(iii) Taking 1,0 0,0 An in special case (ii), we arrive at the results obtained
by Chandel and Jain ([2], p. 1421, eqn. (2.1), (2.4), (2.5) and (2.6)).(iv) Also, the result (2.8) is capable of yielding certain double integrals by anappropriate choice of f(z). Let us take
; 13.2,,, 1211211 inzFzzf
substituting this value of f(z) and evaluating the Integral thus obtained on theright hand side of (2.8), we get the following integral using a result in Erdelyi et al([4], p.399, eqn.(4)):
1
0
1
0 121121111 ;;,1111 xyFyxyx
,;,;,,',';11 , qpgfdcyRyS BAD
mn
dxdyzyxyzyxyH rrr 11,...,11 1
11
hewvu
mn
rnm zzzHhewvuA
n
,,,,
/
021
5,,,,,1, ,...,,,,,,
! 21
where
rr
rr
nmnmNqpqpQPr HzzH ,;...;,:3,0
,;...;:2,315
,,,,,11
1,121,...,
,,...,1(,,...,.1),,...,;1
111
111
rr
rr
rs
s
z
z
r
r
qrj
rjqjjQ
rjjjrr
prj
rjpjjP
rjjjrrr
ddbs
ccas
,1,1''
,1'
1121
,1,1''
,1'
1211
,;...;,),...,;(;,...,;1
,;...;,,,...,;(;,...,;1
1
1
provided that 02Re,0Re 1 s and
02Re 211 s
( i is given by (2.3))
112
Several other new results can be obtained by specializing the parameters involvedin the result (3.1).
REFERENCES[1] B.D. Agrawal and J.P. Chaubey, Certain derivation of generating relations for generalized
polynomials, Indian J. Appl. Math., 10 (1980), 1155-1157 ibid, 11(1981), 357-359.[2] R.S. Chandel and U.C. Jain Certain integral formulas for the multivariable H-functions I, Indian
j. Pure Math., 13 (1982), 1420-1426.[3] J. Edwards, A Treatise on the Integral Calculus, Vol II, Chelsea Publ. Co., New York, (1922).[4] A. Erdélyi et al. Tables of Integral Transforms, Vol. II, McGraw-Hill, New York, 1954.[5] I. Fujiwara, A unified presentation of classical orthogonal polynomials, Math. Japon, 11 (1966),
133-148.[6] S.P. Goyal and T.O. Salim, Multidimensional Fractional integral operators involving a generalized
polynomial set, Kyungpook Math J., 38 (1998), 301-315.[7] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New
York, 1965.[8] V.G. Gupta, Some integral formulas involving the product of the multivariable H-function and a
general class of polynomials, Indian J. Math., 31. (1989), 121-129.[9] S.L. Kalla, S.P. Goyal and R.K. Agrawal, On multiple integral transformations, Math. Notae, 28
(1976-77), 15-27.[10] E. D. Rainville, Special Functions, Chelsea Publ. Co., Bronx, New York (1971).[11] S.K. Raizada, A Study of Unified Representation of Special Functions of Mathematical Physics and
Their Use in Statistical and Boundary Value Problems, Ph.D. Thesis, Bundelkhand Univ., Jhansi,India (1991).
[12] M. Saigo, S.P. Goyal and S. Saxena, A theorem relating a generalized Weyl fractional integral,Laplace and Verma transforms with applications, J. Fractional Calculus, 13 (1998), 43-56.
[13] T.O. Salim, A series formula of generalized class of polynomials associated with Laplace transformand fractional integral operators, J. Rajasthan Acad. Phy. Sci., 1(3) (2002), 167-176.
[14] H. M. Srivastava and R. Panda, A note on certain results involving a general class of poplynomials,Boll. Un. Mat. Ital., A(5), 16(1979), 467-474.
[15] H.M. Srivastava, and R. Panda, Some bilateral generating functions for a class of generalizedhypergeometric polynomials, J. Reine Angew .Math 283/284 (1776), 265-274.
113
J~n–an–abha, Vol. 38, 2008
MAXIMIZING SURVIVABILITY OF ACYCLIC MULTI-STATETRANSMISSION NETWORKS (AMTNs)
ByRaju Singh Gaur and Sanjay Chaudhary
Department of Mathematics, Institute of Basic Sciences, Khandari, Agara,Dr. B.R. Ambedkar University, Agra- 282002, Uttar Pradesh, India
e-mail : [email protected]
(Received : October 5, 2007)
ABSTRACTIn this paper, we evaluate the system survivability of acyclic multi-state
transmission networks (AMTNs) with vulnerable nodes(positions) using theuniversal generating function (UGF) technique. The AMTN survivability is definedas the probability that a signal from root node is transmitted each leaf node. TheAMTNs consist of a number of positions in which multi-state element (MEs) capableof receiving and /or sending a signal are allocated. The MEs located at each non-leaf positions. The two MEs located at first position. The number of MEs is notequal to the number of non-leaf positions. The AMTNs survivability is defined asthe comparison of two networks. All the MEs in the network are assumed to besatistically independent. The signal source is located at each network. The numberof leaf positions that can only receive a signal and a number of intermediatepositions containing MEs copable of transmitting the received signal to some othernodes. The signal transmisson is possible only along links between the nodes. Thenetworks are arranged in such a way that no signal leaving a node can return tothis node through any sequence of nodes otherwise we can say that cycle exists.2000 Mathematics Subject Classification : Primary 62N05; Secondary 90B25.Keywords : Acyclic multi-state transmisson networks, Multi-state, Survivability,Vulnerability, Universal generating function.
1. Introduction. Acyclic multi-state transmission network (AMTN) is ageneralization of tree structured. The example of AMTN is a radio relay station,where the signal source is allocated at root position (node) and receivers allocatedat leaf node. The retransmitter-generating signal situted at each station of radiorelay stations, which can transmit the signal to next stations. The AMTNs consistof a number of positions in which multi-state elements (MEs) capable of receivingand /or sending a signal are allocated. Each network has not position where thesignal source is allocated. The number of leaf nodes that can only receive a signaland the number of non-leaf nodes have retransmitter-generating signal. The event
114
that a ME is in specific state is a random event. The probability of this event isassumed to be known for each ME and for every of its possible state.
This paper presents the comparison of survivability analysis of twonetworks. The MEs located at each non-leaf position, the two MEs located at firstposition. The number of MEs is not equal to the number of non-leaf positions. Thefirst network which has two MEs located at first position and other non leafpositions of this network has only - one multi-state element and in second network,the MEs located at each non-leaf position. If signal transmission is not workingcondition from root position to each leaf node then the networks fail otherwisewhole network is working condition. Malinowski and Preuss [9] discovered thatthe acyclic multi-state transmission network is a gerneralization of the tree-structured multi state systems and Hwang and Yao [4] described the concept ofmulti-state linear consecutively connected network and studied by Kossow andPreuss [5] and Zuo and liang [10]. Gaur and Chaudhary [1,2,3] earlier studied forreliability evaluation of acyclic multi-state network. The algorithm used in thisresearch work is referred by [6,7,8]. In this paper, we consider the case when theMEs allocated at the same mode are subject to a common cause failure. When asystem operates in battle conditions or is affected by acorrosive medium or otherhostile environment. The ability of a system to tolerate both the impact of externalfactors (attack) and internal causes (accidental failures or errors) should beconsidered. The measure of this ability is referred to as system survivability. Thetwo MEs located at the same node in ATMN can be destroyed by a single enternalimpact. An external factor usually causes failures of group of system elementssharing some common resource (allocated) with in the same protective casing,having the same power source gathered geographically, etc.). Therefore addingmore redundant parallel elements will improve system availability but will not beeffective from a vulnerability standpoint without sufficient separation betweenelements.
The paper is organized as follows : Section 2 introduces the model that isthe acyclic multi-state transmission networks model. Section 3 is devoted toSurvivability of AMTNs using universal generating function technique. Section 4presents the Disussion and Conclusion.
2. The Model. In the model considered here, the MEs located at each non-
leaf node Ci MNi 1 where Ci is the non-leaf position can have Ki different
states and each state k has probability ikip . The signal can transmit from non-
leaf node Ci to the nodes belonging to the set i . The ME cannot transmit a signal
to any node : ik : then the condition is total failure and in the case of
115
operational state iikp . There are D available MEs with different characteristics
with probability d
ikip
: A signal can be transmitted by the ME located at Ci only if
it reaches this node, such that
D
k
ai ik
p1
1 where ME d Dd 1 located at Ci can
have Ki different states. The states of all the MEs are independent. The existenceof arc (Ci,Cj) E means that a signal can be transmitted directly from the node i to
node iji CCj :. if ECC ji , where j>i. The number of leaf nodes M: =
{CN-M+1,...,CN} and N is the total number of nodes in AMTN {note that suchnumbering is always possible in acyclic directed graph}.
The MEs located at Ci if some ME n can provide connection from Ci to a set
of nodes ik and ME m can not provide this connection, the state corresponding to
set ik can be defined for both MEs, while 0ni ik
p and 0mi ik
p .
The system survivability S is defined as a probability that a signal generatedat the root node C1 reaches all the M leaf nodes CN–M+1,...,CN.
In general, the resulting polynomial contains 2M-1 terms. The suggestedmethod can depend on the moderate values of M for solving AMTNs. We determine
here UGF technique, obtain zUi~ for each node Ci using operator
zUi~
iik
iik
K
k
Vik
K
k
Vik zzqzq
~
1
0~
~
1
~ ~~. zUi
ˆ (u-functions) is obtained, the values
ivv ikik ˆ,...,1ˆ representing the presence of signal at nodes iCC ,...,1 are not used
further for determining zUmˆ for any m>i. If the signal can not reach any position
from Ci+1 to CN independently of states of MEs located in these positions then in
state k, .0...1 Nviv ikik The only thing one has to know is the sum of
probabilities of states in which these paths exit; it means that we replace all the
values ivv ikik ˆ,...,1ˆ in vector ikv̂ for zUiˆ with zeros and collecting the like terms.
For solving the zUiˆ , the vector ikv̂ , which contain only zero, is removed and
collect the like terms in the resulting polynomial. For calculating the AMTNs
survivability S, first calculate zUzUzU iii 11~
,ˆˆ and at last determine
116
the coefficient of the term of zU NN ˆ in which 1ˆ jv MN for all
.1 NjMN
3. Survivability of AMTNs Using Universal Generating FunctionTechnique. Let us consider AMTN with N=5, M=2, presented in Fig. 1. u-functionof two MEs located at first position C1 and other MEs located at position C2,C3individually and in Fig. 2 individual MEs located at position C1,C2,C3.Case-I. Consider, for example the simplest case in which four identical MEs shouldbe allocated, here the number of MEs is not equal to the number of non-leaf node.When allocated at node C1, the MEs can have four states :* Total failure: ME does not connect node C1 with any other node (Probability
of this state is 121
11 ppp )
* ME connects C1 with C2 (Probability of this state is 212
211
21 ppp )
* ME connects C1 with C3 (Probability of this state is 312
311
31 ppP )
* ME connects C1 with both C2 and C3 (Probability of this state is 1
3,21P
3,212
3,21 pp ).
When allocated at node C2, the MEs can have two states,* Total failure : ME does not connect node C2 with any other node (Probability
of this state is p2).
* ME connects C2 with C3 (Probability of this state is 32p ) and C2 with C5 is
p2{5}, and C2 with C3 and C5 is p2{3,5}).When allocated at node C3, the MEs can have two states,
* Total failure : ME does not connect node C3 with any other node (Probability
of this state is 3p )
* ME connects C3 with C4 is p3{4}.The probability that each node survives during the system operation time
is .
Let us suppose that 21 pp there two possible allocations of the MEs
with in (Fig. 1):(A) Two MEs are located in the first position(B) and other MEs located at second and third position.When both MEs are allocated at position C1 then we have,
117
,0110013,21
00100131
01000121
000001111 zpzpzpzpzu
,0110023,21
00100231
01000221
000002112 zpzpzpzpzu
,001015,32
0000152
0010032
0000022 zpzpzpzpzu
.0001043
0000033 zpzpzu
Following AMTN procedure we obtain,
01000221
0110013,21
00100131
01000121
00000111211 ,, zpzpzpzpzpzuzu
0110023,21
00100231 zpzp
21
131
131
231
11
01000221
121
21
121
221
11
0000021
11 pppppzppppppzpp
2
3,211
312
211
312
3,2,11
212
311
212
3,2111
00100231
131 ppppppppppzpp
011002
3,211
3,212
211
3,2121
13,21 zpppppp
21
131
231
11
01000221
121
21
121
221
11
0000021
11 ppppzppppppzpp
23,21
121
231
121
13,21
131
121
23,21
001000231
131 1 ppppppppzpp
1
3,212
211
3,212
3,212
312
211
3,212
3,211
312
211
31 1 ppppppppppp
,0110023,21
13,21
231 zppp
21
131
231
11
01000221
121
21
121
221
11
0000021
11 ppppzppppppzpp
0110023,21
13,21
13,21
221
131
231
121
23,21
00100231
131 zppppppppzpp ,
01000221
121
21
121
121
221
11
0000021
1112111 ,
~zpppppppzppzuzuzU
2
311
212
3,21001002
311
3111
131
231
11 pppzpppppp
,0110023,21
13,21
13,21
221
131 zppppp
00101
5,3200001
5200100
3200000
222~
zpzpzpzpzuzU ,
.~ 00010
4300000
333 zpzpzuzU
118
Now, ,~ˆ11 zUzU
00100231
131
11
131
231
11
01000221
121
21
121
221
111
ˆ zppppppzppppppzU
.0110023,21
13,21
13,21
221
131
231
121
23,21 zpppppppp
zUzUzU 212ˆ,ˆˆ
00100231
131
11
131
231
11
01000221
121
21
121
221
11 zppppppzpppppp
00000
2011002
3,211
3,211
3,212
211
312
311
212
3,21 , zpzpppppppp
001015,32
0000152
0010032 zpzpzp
11
131
231
11
010002
221
221
121
21
121
221
11 ppppzpppppppp
23,21
0010032
231
131
11
131
231
11
22
231
131 pzpppppppppp
221
131
231
121
23,21
22
23,21
13,21
13,21
221
131
231
121 ppppppppppppp
01100
322
211
2121
121
221
11
232
23,21
13,21
13,21 zppppppppppp
221
131
231
121
23,21
20100152
221
121
21
121
221
11
2 pppppzppppppp
23,21
25,32
221
121
21
121
221
11
252
23,21
13,21
13,21 pppppppppppp
.011015,32
23,21
13,21
13,21
221
131
231
121 zpppppppp
11
131
231
11
010002
221
121
21
121
221
11 ppppzppppppp
00100
3222
311
3111
131
231
11
2231
131 zpppppppppp
231
121
23,21
223,21
13,21
13,21
221
131
231
121
23,21 ppppppppppp
322
211
2121
121
221
11
2322
23,21
13,21
221
131 ppppppppppppp
231
121
23,21
20100152
221
121
21
121
221
11
201100 pppzpppppppz
01101
5,322
211
2121
121
221
11
2
5,32522
3,211
3,211
3,2121
131 2
zppppppp
ppppppp
.
119
13,21
221
131
231
121
23,21
231
131
11
131
231
112
ˆ ppppppppppppzU
221
131
231
121
23,21
231
131
11
131
231
11
223,21
13,21 ppppppppppppp
0010032
221
121
21
121
221
11322
23,21
13,21
13,21 zpppppppppppp
.001015,32
221
121
21
121
221
11
2
5,32522
3,211
3,211
3,212
211
31
231
121
23,21
20000152
221
121
21
121
221
11
2
zppppppp
ppppppp
pppazppppppp
zUzUzU 323~
,ˆ
13,21
221
131
231
121
23,21
231
131
11
131
231
11 pppppppppppp
221
131
231
121
23,21
231
131
11
131
231
11
223,21
13,21 ppppppppppppp
3322
211
2121
121
221
11322
23,21
13,21
13,21 ppppppppppppp
3200001
522
211
2121
121
221
113
200100 pzppppppppz
32
5,32522
3,211
3,211
3,212
211
312
311
212
3,21 ppppppppppp
221
131
231
121
23,2143
3001015,32
221
121
21
121
221
11 ppppppazppppppp
5,322
211
2121
121
221
1143
35,3252
23,21
13,21
13,21 ppppppppppppp
221
131
231
121
23,21
231
131
11
131
231
1143
200111 ppppppppppppz
23,21
231
131
11
131
231
1143
323,21
13,21
13,21 ppppppppppp
21
121
221
11322
23,21
13,21
13,21
221
131
231
121 ppppppppppppp
.0010032
221
121 zppp
1
3,212
211
312
311
212
3,212
311
3111
131
231
11 pppppppppppp
221
131
231
121
23,21
231
131
11
131
231
11
223,21
13,21 ppppppppppppp
322
211
2121
121
221
11322
23,21
13,21
13,21 pppppppppppp
120
2
211
312
311
212
3,212
211
2121
121
221
11
2001003 pppppppppppzp
231
121
23,21
3000015,32523
23,21
13,21
13,21 pppzpppppp
21
121
221
11
3435,3252
23,21
13,21
13,21
221
131 pppppppppppp
23,21
231
131
11
131
231
11
200011435,32
221
121 pppppppzpppp
433222
3,211
3,211
3,212
211
312
311
21 pppppppppp
.000104332
221
121
21
121
221
11
3 zppppppppa
1
3,212
211
312
311
212
3,212
3,211
2121
121
221
11
23
ˆ ppppppppppppzU
2
211
312
311
212
3,21300001
5,325232
3,211
3,21 pppppzppppp
221
121
21
121
221
11
3435,3252
23,21
13,21
13,21 pppppppppppp
231
121
23,21
231
131
11
131
231
11
200011435,32 pppppppppzpp
2
2111
343322
23,21
13,21
13,21
221
131 ppppkpppppp
.000104332
221
121
21
121 zpppppp
5,325232
3,2131213,212
212112
3 222ˆ ppppppppppzU
2113
435,32522
3,2131213,21300001 222 pppppppppz
23,2131213,21
231311
200011435,32
221 222 pppppppzppp
.2 000104332
221211
343322 zpppppppp
The system survivability is equal to the coefficient corresponding to the singleterm of the polynomial:
221211
3435,3252
23,2131213,21
31 222 ppppppppppS
435,32 pp
3,2131213
435,32522
3,2131213,213 1222 pppppppppp
121
435,322
2121 pppp
5,3252432
3,213
5,32215,3252433,2132 pppppppppp
3,2121435,32213
5,32523143213 22 pppppppppp
5,3252432
3,215,32215,3252433,213 2 pppppppppp
.22 3,2121435,32215,3252314321 pppppppppp
Case-II. Let us consider AMTN with N=5 and M=2 presented in Fig. 2 individualMEs located at node C1, C2 and C3 are, here the number of MEs is equal to thenumber of non-leaf positions.
,~ 01100
3,2100100
3101000
2100000
111 zpzpzpzpzuzU
,~ 00101
5,3200001
5200100
3200000
222 zpzpzpzpzuzU
.~ 00010
4300000
333 zpzpzuzU
Following the consecutive procedure, we obtain :
,ˆ~11 zUzU
122
,ˆ 011003,21
0010031
01000211 zpzpzpzU
zUzUzU 212~
,ˆˆ
00100
3200000
201100
3,2100100
3101000
21 , zpzpzpzpzp
001015,32
0000152 zapzp
23,2100100
32312
23101000
221 ppzppppzpp
5231201001
5221201100
32212
323,212 ppzppzpppp
.011015,323,21
25,3221
2523,21
2001015,3231
2 zppppppzpp
3,212
3,2100100
322312
3101000
221 ppzppppzpp
5,32523,21201001
5221201101
32212
322 pppzppzpppp
.011015,3221
2 zpp
3,21322312
3101000
221 pppppzpp
5,32523,21200001
5221200100
32212
3223,212 pppzppzppppp
.001015,3221
2 zpp
32212
3223,212
3,21322312
312ˆ ppppppppppzU
.001015,3221
25,32523,21
2000015221
200100 zpppppzppz
zUzUzU 323~
,ˆˆ ,
32212
3223,212
3,21322312
31 pppppppppp
,001015,3221
25,32523,21
2000015221
200100 zpppppzppz
0001043
000003 zpzp
32212
3223,212
3,21322312
31 pppppppppp
00101
35,32212
5,32523,21200100
3 zppppppzp
123
322313
31200011
435,32213
5,32523,213 ppppzpppppp
.00110433221
33223,21
2 zpppppp
3,21300001
35,32212
5,32523,212
3ˆ pzppppppzU
322313
31200011
435,32213
5,3252 ppppzppppp
.00010433221
33223,21
33,21
2 zppppppp
The system survivability is equal to the coefficient corresponding to the singleterm of the polynomial
435,32213
5,32523,213 ppppppSn (2)
Since 15,3232522 pppp
435,32213
3223,213 1 pppppp . (3)
By comparing eqns. (1) and (2), one can decide which allocation of the elements is
preferable for any given 3121433,21 ,,,, pppp and 5,3252 pp . Condition nSS 1
can be rewritten as,
43213
5,3252432
3,213
5,32215,3252433,213 22 pppppppppppp
5,32523,213
3,2121435,32213
5,325231 2 ppppppppppp
435,32213 ppp (4)
Equation (4) present nSS 1 as function of variables 433,21 pp and 5,3252 pp .
Solution I provides lower system survivability than solution II, when the valuesof located below the curve and the solution II provides lower system survivabilitythan solution I, when the values of located above the curve.
4. Conclusion. This paper is based on universal generating functiontechnique for solving the survivability of AMTNs. The resulting polynomial
contains 12 M term. Therefore, the suggested method can be applied for AMTNs
with moderate values of M where M is the number of leaf nodes. The algorithmwhich is used in this paper for solving AMTNs survivability of MEs in which thenodes vulnerability is taken into account. The networks are arranged in such away that signal transmission from root node to leaf node. The paper suggests asystem survivability method for comparison of two networks.
124
REFERENCES[1] R.S. Gaur and S. Chaudhary, Availability and maximizing survivability of Cable T.V. transmission
system, ,~ bhaananJ 34 (2004), 107-112.
[2] R.S. Gaur and S. Chaudhary, Reliability enhancement and fixed transmission times of acyclictransmission multi-state networks (AMTNs). Proceedings of the Ramunujan Mathematical Society.Agra (2004).
[3] R.S. Gaur and S. Chaudhary, Reliability enhancement and fixed transmission times of acyclictransmission multi-state networks (ATMNs), JMASS, 2 (2006), 46-56.
[4] F. Hwang and Y. Yao, Multi-state consecutively-connected systems, IEEE Transaction onReliability,38(1989), 472-474.
[5] A Kossow and W. Preuss, Reliability of linear consecutively-connected systems with multi-statecomponents, IEEE Transaction on Reliability, 44 (1995), 518-522.
[6] G. Levitin, Optimal allocation of multi-state retransmitters in acyclic transmission networks,Reliability Engineering and System Safety, 75 (2002), 73-82.
[7] G. Levitin, Reliability evaluation for acyclic consecutively-connected networks with multi-stateelements, Reliability Engineering and System Safety, 73 (2001), 137-143.
[8] A. Lisnianski and G. Levitin, Multi-State System Reliability: Assessment, Optimization andApplications, Vol. 6: Series on Quality,Reliability & Engineering, Word Scientific, New Jersey,(2003).
[9] J. Malinowski and W. Preuss, Reliability evaluation for tree-structured systems with multi-statecomponents, Microelectron Reliab, 36 (1996), 9-17.
[10] M. Zuo and M. Liang, Reliability of multi-state consecutively-connected systems, Reliability
Engineering and system Safety, 44 (1994), 173-176.
125
J~n–an–abha, Vol. 38, 2008
THE INTEGRATION OF CERTAIN PRODUCTS INVOLVING H-FUNCTION WITH GENERAL POLYNOMIALS AND INTEGRAL
FUNCTION OF TWO COMPLEX VARIABLESBy
V.G. GuptaDepartment of Mathematics
University of Rajasthan, Jaipur-302004,Rajasthanand
Nawal Kishor JangidDepartment of Mathematics
Global Institute of Technology, Jaipur-302022, Rajasthan(Received : August 14, 2008)
ABSTRACTIn this paper, we wstablish a new integral involving general polynomials of
Srivastava (1985), Laguere polynomials and integral function of two complexvariables of order , given in Dzrbasjan (1957), based on the properties of Fox's H-
function. This Integral is unified in nature and acts as a key formula from whichwe can derive as its specials cases, integrals pertaining to a large number of simplerspecial functions and polynomials. For example, we derive a few special cases ofour integral and main theorem which are also new and of interest by themselves.The results established here are basis in nature and are likely to be of usefulapplications in several field's notably mathematical physics, statistical mechanicsand probability theory.2000 Mathematics Subject Classification : Primary 33C70; Secondary 33C60Keywords : General polynomials, Laguerre polynomial and Fox's H-function.
1. Introduction. Srivastava ([11], P. 185, eq.(7)) has defined and introducedthe general polynomials
R
rr
R
RRr
r
kR
kRR
VU
k R
kVRkVVU
kR
VVUU xxkUkUA
k
U
k
UxxS ...,;...;,
!...
!...,..., 11111
1
1
1 111
/
0 1
1/
01
,...,,...,
(1.1)
where rvv ,...,1 are arbitrary positive integers and the coefficients rr kUkUA ,;...;, 11
are arbitrary constants real or complex.The Fox's H-function is defined in terms of a Mellin-Barnes type integral as
L
s
ppnmqp dszs
bbaa
zH2
1,,...,,,,...,,
11
11,, ...(1.2)
126
Where
p
njjj
q
mjjj
n
jjj
m
jjj
sasb
sasb
s
11
11
1
1
...(1.3)
and .0,1 z An empty product is interpreted as unity. Also m,n,p and q are
integers satisfying ,1 qm ,0 pn qjpj j ,...,1,,...,1 are positive
numbers and qjbpja jj ,...,1,,...,1 are complex numbers. L is the suitable
contour of Barnes type such that the poles of mjsb jj ,...,1 lie to the right
and those of njsa jj ,...,11 lie to the left of L. the Integral converges if
0,0111111
BAq
mjj
m
jj
p
njj
n
jj
q
jj
p
jj
and .2arg Bz These assumptions for the H-function will be adhered to
throughout the paper. ,
,,,
,
ppnmqp b
azH where ppa , stands for
ppaa ,,...,, 11 and qqb , stands for .,,...,, 11 qqbb
The symbol nm, represents for the set of m parameters:
mmn
mn
mn 1
,...,1
,
.
Let
0,
2121
,21
21
2121
!!,
nn
nnnn zznn
azzF ...(1.4)
be an integral function of complex variables z1 and z2. Denote
21,max21
zzFrMrzz
F
the maximum modulus of 21 , zzF .
Dzrabasjan ([4]. P. 257) has given the following definition of order:
The integral function 21, zzF is said to be of order if
0
log
loglog.lim
r
rMSup F
r...(1.5)
2. First Integral. In this section, we evaluate the following integral whichwill be required in our Investigation :
127
dxba
zxHxyxySxLexqq
ppnmqpR
VVUUk
xy R
R
,
,,...,
0,
,1,...,,...,
1
1
R
kk yyL
k,..,
!21 1
2/11
21
1,,,,,1,,1,,2,
,2, kba
zHqq
ppnmqp ...(2.1)
where is a positive integer, 2arg,0,0 BzBA and
R
jjjj kmjb
1
,,...,11/1Re and
. ,;...;,!
...,..., 11
/
0 1
/
01
11
1
RR
VU
k
R
j j
kjkVjVU
kR kUkUA
k
yUyyL
rr
R
j
jj
Proof. To prove (2.1) we first express the general polynomials by (1.1), the H-function by (1.2), then change the order of integration and summation which isjustifiable due to the convergence conditions mentioned with the integral andevaluate the inner integral with the help of ([7], p 76), now using the formula
nn n
11
1
and the Gauss integral multiplication formula ([5], p.4) we arrive at the righthand side of (2.1) after a little simplification.
3. Main Theorem. Let :2,12
arg,0
lll and let
0,
2121
,21
21
2121
!!,
nn
nnnn zznn
azzF be an integral function of two complex variables z1
and z2 of order 0 then for arg 1 =arg 2 , we have
ppnmqp
ttnn b
attzHettP
,,
, 22110 0
,,
1221121,
2211
21
221122111,...,,...,2211 ,...,1
1ttyttySttL R
VVUUk
R
R
2121 , dtdtttF
128
0,
112
11
21
21
2/112/1
2121
21 ,...,
1
!1!2
1nn
Rnnnn
kk yyL
nn
nn
a
k
1,11
,,,
,1,11
,1,11
,
21
2121
2,,2,
knn
bb
annnn
zH
ppnm
qp
...(3.1)Provided (i) is a +ive integer
(ii) 2arg,0,0 BzBA
(iii) mjbnn
jj ,...,1,11
Re 21
(iv) The series in (3.1) is uniformly and absolutely convergent in a suitably choosendomain.
Proof of Theorem. Let us first take 0Re a and consider the integral
0
0 211
21,21
21xxaLexxaI k
xxann
21211,...,,..., ,...,1
1xxayxxayS R
VVUU
R
R
212121
,,
21
,,
dxdxxxba
xxzaH nn
ppnmqp
where , n1 and n2 are positive integers. Changing the variables
tutuxutx 0,10,,1 21
we have
atyatySatLetaI RVVUUk
atnnnn
R
R,...,1
,...,,...,0 0,
1
1
21
21
dtduutxx
uuba
atzH nn
ppnmqp ,
,1
,,
21,,
21
atyatySatLet RVVUUk
atnn R
R,..., 1
,...,,...,
0
1
0 1
1
21
129
dtduuuba
atzH nn
ppnmqp
211,,,
,
.
Evaluating u-integral with the help of the Eulerian integral of the first kind(Copson, 1961, P. 212), we obtain
atyatySatLetnnnn
aI RVVUUk
atnnnn
R
R,...,
!1!!
1,...,,...,
021
21,
1
1
21
21
dtba
atzHqq
ppnmqp
,,,
,
xLexa
nnnn
kx
nnnn
0
111
21
212121
!1
!!
dxba
zxHxyxySqq
ppnmqpR
VVUU
R
R
,,
,..., ,,1
,...,,...,
1
1 . ...(3.2)
Now evaluating x-integral with the help of (2.1) we have
1
211
21
211
121
,21
21
21 !1!!!
,...,21
nn
knn
R
k
nn
annknn
yyLaI
1,11
,,,
,,1, 11
,1;11
,
21
2121
2,,2,
knn
bb
annnn
zH
ppnm
qp
...(3.3)
where 2arg,0,0 BzBA .
mjbnn
jj ,...,1,11
Re 21
and
RR
VU
k
R
j
kj
j
kVjVU
kR kUkUAy
k
UyyL
rr
R
jjj ,;...;,!
...,..., 11
/
0 1
/
01
11
1
Let arg 21 arg and if we denote iea where
,2arg,0cosRe Bza then from (3.1), we obtain
130
0
0 22111
221121,2211
21, ttLettaP k
ttann
221122111,...,,..., ,...,1
1ttayttayS R
VVUU
R
R
21
0,11
21
,2211
,,
21
2121
!!,,
dtdtttnn
aba
ttzaHnn
nnnn
ppnmqp
0
0
12211
0, 21
,1
21
21
!!tt
nn
ae
nn
nni
22112211 ttaLe k
tta
221122111,...,,..., ,...,1
1ttayttayS R
VVUU
R
R
21212211
,,
21
,,
dtdtttba
ttzaH nn
ppnmqp
1
21
10, 21
,1
21
21
21
.
1!!
nn
nn
nni
nn
ae
21
0
0 1
2121 xxaLexx k
xxa
21211,...,,..., ,...,1
1xxayxxayS R
VVUU
R
R
211121
,,
21
,,
dxdxxxba
xxzaH nn
ppnmqp
1
211
0, 21
121
21
21
21
21
!1!21
nn
knn
nn
nnik
ann
ae
k
Rnn yyL ,...,1
112
11
21
1, 11
,,,
,,1, 11
,,1,11
,
21
2121
2,,2,
knn
b
annnn
zH
ppnm
qp
due to relation (3.2) where
RR
VU
k
R
j
kj
kj
VU
kR kUkUAyUyyL
rr
R
j
j
jkjV ,;...;,...,..., 11
/
0 1 !
/
01
11
1
.
131
Thus under conditions ,2,12
arg,argarg,0 121
ll and an appeal to
analytic continuation we have (3.1).The change of order of integration and summations in (3.3) is justified by
the de 1 vallee Poussiu's theorem (Bromwich, 1931, p. 504) under the conditionimposed in the theorem.
4. Special Cases. (i) If take R=1 in (2.1) we get integral involving theproduct of general class of polynomials and Fox's H-function.
dxba
zxHxySxLexqq
ppnmqp
VUk
x
,
,,,
0 11
1
11
1
11
/
0111
1
2/1121
,!
21VU
k
kj
kkk kUAyk
UjkjV
1,,,,,,1,,,1,,
1
112,,2, kkb
akkzH
ppnmqp ...(4.1)
where is a positive integers, 2arg,0,0 BzBA and
1,1/1Re 1 jBbk jj .
(ii) If we take R=1 in (3.1) our result reduces to
022111
02211
1221121,
1
1
2211
21, ttySttLettP V
Uktta
nn
21212211
,, ,
,,
dtdtttFba
ttzHqq
ppnmqp
0,
12
11
21
21
2/1121
2121
21 1!1
21nn
nnnnkk nn
nn
a
11
1
111
/
0111
1
1,
!
VU
k
k kUAyk
UkV
1,11
,,,
,,1,11
,,1,11
,
211
211
211
2,,2,
knn
kb
ann
knn
kzH
ppnm
qp
...(4.2)
132
provided that (i) is a positive integer
(ii) 2arg,0,0 BzBA
(iii) 11
Re 211
jjbnn
k
(iv) The series in (4.2) is uniformly and absolutely convergent in a suitablychoosen domain.
(iii) Letting 1,0 0,01 AU in the equation (4.1) and (4.2) we get the result obtained
by Shah ([9], p715) and Nigam ([7], p.2, eqn (3.1)) respectively as given below.
dxba
HxLexqq
ppnmqpk
x
,,
0 ,
,
1,,, ,,, 1,,, 1,,
21 2,,2,
2/1121
kba
zHqq
ppnmqp
kk...(4.3)
where is a positive integer, 2arg,0,0 BzBA , and 1/1Re jjb
0
0 1
221121,2211
21, tt
nn ettP
21212211
,,2211 ,
,,
dtdtttFba
ttzHttLqq
ppnmqpk
12
11
1
0, 21
211
21
21
21
21
21
!1!21
nn
nn
nn
nnkk
nn
a
k
1,11
,,,
,,1,11
,,1,11
,
21
2121
2,,2,
knn
b
annnn
zH
ppnm
qp
...(4.4)Provided
(i) is a positive integer.
(ii) 2arg,0,0 BzBA
(iii) mjbnn
jj ,...,1,1/1
Re 21
133
(iv) The series in (4.4) is uniformly and absolutely convergent in asuitably chosen domain.(iv) If we make use of the relation. (Sharma; 1965, p. 199)
q
pnmqp
q
pnmqp bb
aazG
bbaa
zH,...,,...,
1,,...,1,1,,...,1,
1
1,,
1
1,,
0
0 22111
22112211 ttLett k
tt
221122111,...,,..., ,...,1
1ttyttyS R
VVUU
R
R
21211
12211
,, ,
,...,,...,
dtdtttFbbaa
ttzGq
nmqp
0,
12
11
21
21
2/112/1
2121
21
1
!1!2
1nn
nnnn
kk
nn
nn
a
k RyyL ,...,1
11
,,,
,...,,11
,,1
,
211
12121
2,,2,
knn
b
annnn
zG
q
pnm
qp (4.5)
where
11
1 0 0 1111 ,;...;,
!...,...,
VU
k
VU
k
R
jRR
kj
j
jR
RR
R
j kUkUAyk
UyyL and
R
jjk
1 (4.6)
(v) If we take R=1 in (4.5) we get the integral involving the product of generalclass of polynomials and G-function
0
0 22111
22112211 ttLett k
tt
21211
12211
,,22111 ,
,...,
,...,1
1dtdtttF
bb
aattzGttyS
q
pnmqp
VU
111
/
0 1
11
21
1
1
0, 21
2/1121
,!!1
21 111
1
11
21
21
21
211 kUAyk
U
nn
a kVU
k
kVnn
nn
nn
nnkkk
134
1
1,,,,
,...,,11
,,11
,
211
121
121
12,
,2,k
nnkb
ann
knn
kzG
pnm
qp
...(4.7)
(vi) Letting 1,0 0,0 AU in equation (4.7) we get the result obtained by Nigam
([7], P. 5, eq.(4.1))
0
0 22111
22112211 ttLett k
tt
21211
12211
,, ,
,...,,...,
dtdtttFbbaa
ttzGq
nmqp
12
11
1
0, 21
2/1k121
k
21
21
21
21
!1!21
nn
nn
nn
nn
nn
a
k
γδ δπ
1
1,,,
,...,,11
,,11
,
21
12121
2,,2,
knn
b
annnn
zG
pnm
qp .
REFERENCES[1] T.J.I.A. Bromwich, An Introduction to the Theory of Infinite Series, London (1931).[2] B.L.J., Baraaksma, Asymptotic expansions and analytic continuations for a class of
Barness-integrals, Compositio Math. 15 (1963), 339-341.[3] E.T. Copson, Theory of Functions of Complex Variables, Oxford, (1961).[4] M.M. Dzrbasjan, Mathematich Cskii Sbornik, 41 (1957) (83), 257 (Russian).[5] A. Erdélyi et.al., Higher Transcendental Functions. Mc-Graw Hill Book Company, Inc.
New york, (1953).[6] C. Fox, The G and H-function as symmetrical Fourier Kernal. Trans. Amer, Math., Soc. 98
(1961), 395-429.[7] A.M. Mathai and R.K. Saxena, Generalized Hypergeometric Functions with Applications
in Statistics and Physical Sciences, Springer-verlag, New York, 1970.[8] H.N. Nigam, Integral involving Fox's H-function and integral function of two complex
variables, 64 (1972), 1-5.[9] M. Shah, Some results on the H-function involving generalized Laguerre polynomail.
Proc, Camb, Phil, Soc. 65 (1969), 713-720.[10] O.P. Sharma, Some finite and infinite integrals involving H-function and Gauss's
hypergeometric functions, Collectanca, Mathematica, 17 (1965), 197-209.[11] H.M. Srivastava, A multilinear generating function for the Konhauser sets of biorthogonal
polynomials suggested by the Laguerre polynomails, Pacific. J. Math. 117 (1985), 183-191.
135
J~n–an–abha, Vol. 38, 2008
STUDY OF VELOCITY AND DISTRIBUTION OF MAGNETIC FIELD INLAMINAR STEADY FLOW BETWEEN PARALLEL PLATES
ByPratap Singh
Department of Mathematics and Statistics, Allahabad, Agricultural Institute-(Deemed University) Allahabad-211007, Uttar Pradesh, India
(Received : October 10, 2008)
ABSTRACTWe studied flow of velocity and distribution of magnetic field in laminar
steady flow between two parallel plates situded at a distance and having relativevelocity between them. For low Hartman number, flow velocity numericallyincreases rapidly in the middle of the plates, then it numerically increases slowlynear the plates. But for large Hartmann number it numerically increases slowlyin the middle of the plates and then it numerically increases rapidly near theplates. It decreases the strengh of the magnetic field decreases. For large Hartmannnumber, the strength of the magnetic field is inversely proportional to the-Hartmann number.2000 Mathematics Subject Classification: Primerey secondaryKeywords: Magnetic permeability/ coefficient of viscocity/electrical conductivity/Hartmann number.
1. Introduction. Contribution of laminar steady and unsteady flow hasbeen made by several authors because of its wide application in Engineering,Physical, Medical Science and Oil refining etc. Problem of heat transfer throughthe annular space when the fluid flow is laminar and there is uniform heatingeither from out side, from in side or from both was investigated [1]. Numeroustheoretical and experimental studies of both laminar and turbulent heat transferin annuli taking various types of wall temperature distributions have been made([2],[4]). An analysis on laminar flow and heat transfer in concentric annuli withmoving core was obtained [5]. The stability of laminar flow of dusty gas was studied[6]. Contribution on laminar flow an electrically conducted liquid in a homogeneousmagnetic field was made [7],[8]. Our problem is to study of velocity and distributionof magnetic field in laminar steady flow between parallel plates situated at adestance haveing relative velocity between them.
2. Formulation of the problem. Let us consider two-dimensional steadylaminar flow of an incompressible and electrically conducting fluid of constantviscosity and enstant electrically conductivity between two parallel insulated plates.The two plates are situated at a distance 2L and relative velocity 2U between
136
them. The velocity of flow is parallel to the plates, which are in the direction of x-axis. An external magnetic field of constant strength H0 in the direction of y-axisis also in consideration. Then conditions of flow are:
. **,*,0,,*
,0,0,**
20
*0
yxpUPHHHyHHH
wvyUuu
zYxx ...(2.1)
where LLyyLxx *,*, and U are the characteristic length and velocity fot the
problem respectively. Using (1) and neglecting stars, the equation of motion
,0. u
,2
. 22
HpHH
DtuD e
e
HVuHHut
HH
2..
takes the form
xp
dydH
Rdy
udR
xH
e
2
21...(2.2)
yp
dydH
HR xxH
...(2.3)
and
01
2
2
dy
HdRdy
du x...(2.4)
where ,,, 2
20 ULR
vH
RUL
R ee
e H
Re= Reynolds number,RH= Magnetic pressure number,
R = Magnetic Reynolds number,
= density of the fluid,= coefficient of viscosity,
e = Magnetic permeability,
137
v = Kinetic coefficient of viscosity,= electrical conductivity.From (2.2) and (2.3), we get
011
2
2
3
3
2 dy
HdRdy
udR
x
h...(2.5)
where hHeh RRRRR , is Hartmann number.
Subtracting (2.4) from (2.5) , we have
023
3
dydu
Rdy
udh ...(2.6)
Consider that the two plates are situated at 1y and there is no pressure
gradient in flow field.Now the boundary conditions are:y=1, u=1; y=0, u=0; y=–1, u=-1. ...(2.7)Using condition (2.7) then solution of (2.6) is
hh
hh
RR
yRyR
eeee
u
. ...(2.8)
Case I. when ,0hR we have
u=y,it is a straight line, whose gradient is 1.
Case II. When hR , we have
u=0 (expect at 1y ).
Plates are insulated. So the boundary conditions are:
1y , Hx =0.
Integrating (2.4) with respect to t and using above boundary conditions, we get
hh
hhhh
RR
yRyRRR
h
x
eeeeee
RRH 1
...(2.9)
It is the expression for magnetic field.3. Results and conclusion. From tabel 1, it is found that for low Hartmann
number, the velocity of flow numerically increases rapidly in the middle of theplates, and then it numerically increases slowly near the plates. But for largeHartmann number, it numerically increases slowly in the middle of the plates, andthen it numerically increases rapidly near the plates. As Hartmann number
138
139
140
increases velocity decreases. From table 2, it is found that on increasing Hartmannnumber strength of the magnetic field decreases. For large Hartmann number, thestrength of magnetic fieldi inversely proportional to Hartmann number i.e.independent of the distance from the plates. Figure 1 and 2 show the pattern of theflow of the velocity and distribution of magnetic field respectively.
REFERENCES[1] M. Jakob and K. Rees, Heat transfer to a fluid in a laminar flow through an annular space, Trans.
AICHE, 37(1941), 619-648.[2] P.A. Mc Cuen, W. M. Kays and W. C. Reynolds, Heat transfer with laminar flow in concentric annuli
with constant and variable wall temperature and heat flux. Report No. AHT-2, Stanford UniversityC A , 1961.
[3] P. A. Mc Cuen, W. M. Kays and W. C. Renolds, Heat transfer with laminar flow and turbulent flowbetween parallel planes with constant and variable wall temperature and heat flux, Report No.AHT-3, Stanford University, CA. 1962.
[4] H. S. Heaton, W. C. Reynolds and W. M. Kays, Heat transfer with laminar flow in concentric annuliwith constant heat flux and simultaneously developing velocity and temperature distributions,Stanford University , C A. 1962.
[5] T. Shigenchi and Y. Lee, An analysis on fully developed laminar flow and heat transfer in concentricannuli with moving cores. Int. J. Heat Mass Transfer. 34(1991) 2593- 2601.
[6] P. G. Safiman , On the stability of laminar flow of dusty gas. J. Fluid Mech., 13 (1962), 120-128.[7] J. Hartman, Hydrodynamics I. Theory of the Laminar Flow of an electrically conductive Liquid in
a homogeneous Magnetic Field, Kgl. Danske Videnskabernes Selskab Math. Fys Med. Copenhagen.No.6 ,15 (1937).
[8] J. Hartmann, and F. Lazarus, Hydrodynamics II-Experimental Investigation on the flow of mercuryin a Homogeneous Magnetic Field, Kgl. Dandke Videnskabernes Selskab Math. Flys Med.l
Copenhagen No. 7 ,14, (1937).
141
J~n–an–abha, Vol. 38, 2008
A NOTE ON THE ERROR BOUND OF A PERIODIC SIGNAL INHOLDER METRIC BY THE DEFERRED CESARO PROCESSOR
ByBhavana Soni
Jawaharlal Institute of Technology Borawan (Khargone)-451228,Madhya Pradesh, India
andPravin Kumar Mahajan
28, Tilak Path Khargone-451001 Madhya Pradesh, IndiaE-mail: pravin_mahajan [email protected]
(Received : November 20, 2007)
ABSTRACTWe determine the error bound of a periodic signal belonging to Hw -space
([6]) by deferred Cesaro-processor ([1]P. 414 and [4], p. 148) and generalize a resultof Zygmund ([7], p. 91).2000 Mathematics Subject Classification : 42A10,42A24.Keywords: Analog signal, Deferred Cesaro Processor, Modulus of continuity,Holder metric.
1. Definitions and Notations. Lets 2,0*Ct be a class of 2 - periodic
analog signals and let the Fourier trigonometric series be given by
ts ~
1 0
0 .sincos21
n nnnn tAntbntaa ...(1.1)
Singh [6] defined the space Hw by
,: 21212 ttKwtstsCtsHw ...(1.2)
and the norm ..w by
,,sup. 21,
.
21
ttsss w
ttcw
...(1.3)
where
, sup20
tsst
c
...(1.4)
and
21
21
21
2021
* ,*
sup, ttttw
tstststts
t
w
...(1.5)
142
and choosing tttts * and ,0, 210 being increasing signals of t. if
cttAtt 2121 ...(1.6)
10,* 2121 ttKtt ...(1.7)
'A' and 'K' being positive constants, then the space
,10,: 21212
tttstsCtsH ...(1.8)
is a Banach spaces (see[5]) and the metric induced by the norm on H is
said to be a Hölder metric.
Let tsn be the thn parital sums of (1.1) and npLet and nq be sequences of
non-negative integers satisfying
nn qP ...(1.9)
and .lim nn
q ...(1.10)
The processor
n
n
q
pkk
nnnn ts
pqsD ,
1...(1.11)
defines the deferred Cesaro-transform nn qpD , ([1], see also [4],p.148). It is known
[1] that nn qpD , is regular under conditions (1.9) and (1.10). Note that -D(0,n) is
the (C,1) transform and let n be a monotone non- decreasing sequence of positive
integers such that 11 and ,11 nn then nnD n , is same as the thngeneralized De la vallee poussian processor [3] generated by the sequence {n}
We shall use following notations:
,222 1111tsttsttstt ..(1.12)
tpqtqpt
tK nnnnn 1sin1sin2sin2
12 . ...(1.13)
2. Main Theorem. Using Fejér operator, Zygmund ([7], p.91) has establishedthe following result.
Theorem A. Let t* be a non-negative and increasing signal defined in a right-
hand neighbourhood of t=0. Suppose that tt* for 10 . Let t be the
143
modulus of continuity for a periodic signal s(t), then if
ottt as,*0 ...(2.1)
we have
,/1*;max 1120
nOtstsnt
...(2.2)
where 1; tsn is the Fejer operator.
The object of the present note is to generalize the above result shall establishfollowing:
Theorem. Let t* be a non-negative and increasing signal defined in the right-
hand neighbourhood of t=0. Suppose that .10for * tt Let t be the
modulus of continuity for a periodic signal s(t), then for 10, Hts and
. as,* ottot ...(2.3)
We have
.1
*1log.1
1
nnnn
nnn pqpq
qOtssD
3. Proof of theorem. Following Zygmund [7], we have
n
n
q
pkn
nnnn tsts
pqtssD 11
1
.1 2
0
dttKt
pq nnn
We write
11 tssDtE nnn
2/
0
1dttKt
pq nnn
and
2121, tEtEttE
2/
0 21
1dttKtt
pq nttnn
144
sayIIpq nn
nn
pq
pq
nn,
121
2/
1
1
0
It is easy to prove
tKtt tt 421
...(3.1)
and .4 2121ttKtt tt ...(3.2)
Now using (3.1), we have
dttpqtqp
t
tpq
OI nnnnpq
nn
nn
1sin1sinsin
12
1
01
dttpqtqp
t
tOpq nnnn
pq
nn
nn
11*1 1
0 2
.1
*1
nnnn
nn
pqpqqp
O
Let 1 nn qp
then
11
1
1lim
n
n
n
n
nn
nn
n
qpqp
pqqp
thus .1
*1
nn pq
OI ...(3.3)
Again using (3.1) we have
dt
t
tpq
OInn pq
nn
2/
1 211
dtt
t
topq
Onn pq
nn
22/
1
*1
1
1
1*
1
nn
nn
nn
nnpq
pq
pq
pqO
145
.1
*
nn pq
O ...(3.4)
Now from (3.2), we have
)(1
0 1 21
1 nn pqntt
nndttktt
pqOI
sayIIpq
O nn
n
n pq
q
q
nn,
11211
1
1
1
0
dttpqqpt
tt
pqoI nnnn
q
nn
n 21
0 221
11 11*1
nq
n dtttqO/1
0 21*
21* ttO ...(3.5)
and
dttpqqpt
tt
pqOI nnnn
pq
qnn
nn
n
1sin1sinsin
*1 /1
1 221
12
dttpqt
tt
pqO nn
pq
qnn
nn
n
1*
1 1
1 221
,log* 21
nn
n
pqq
ttO ...(3.6)
thus
.1log* 211
nn
n
pqq
ttOI ...(3.7)
Again from (3.3)
dt
t
tt
pqOI
nn pqnn
2
1 221
2*1
.* 21 ttO ...(3.8)
Now noting that
,2,1,1 rIII rrr ...(3.9)
we have from (3.3) and (3.7)
146
,1
*1log*1
211
nnnn
n
pqpqq
ttOI ...(3.10)
and from (3.4) and (3.8)
.1
**1
212
nn pqttOI ...(3.11)
Thus from (3.10) and (3.11), we have
211
2121
, *sup,sup
.
21 tt
tEtEttE nn
ntt
.1
*1log1
nnnn
n
pqpqq
O ...(3.12)
It is to be noted that
ssDtE ntcn
201
1
max
nn pq
O1
* . ...(3.13)
Combining (3.12) and (3.13), we get
.1
*1log1
*1
nnnn
nnn pqpq
qOtssD
This completes the proof of theorem 1.REFERENCES
[1] R.P. Agnew, On deferred Cesàro means, Ann. Math .33(1932) 413-421.[2] G. Alexits, Convergence Problems of Orthogonal Series, Pergaman Press, 1961.
[3] P. Chandra, Functions of classes Lp and Lip p, generalized De La Valleepussin means, Math.
Student, 52(1984), 121-125.[4] G. Das , T. Ghose and B.K. Ray, Degree of approximation of functions by their Fourier series in
the generalized Hölder metric. Proc. Indian Acad. Sci (Math. Sci) 106,no (2) (1996), 139-153.[5] S. Prossdorff, Zur, Konvergenz der Fourier reihen Hölder stetiger funktionen, Math. Nachr,
69(1975), 7-14.[6] T. Singh, Degree of approximation of function in a normed spaces, Publ. Math. Debrecen, 40(3-4)
(1992), 261-267.[7] A. Zygmund, Trigonometric Series, Vol. 1, Cambridge University Press, 1959.
J~n–an–abha, Vol. 38, 2008
APPLICATIONS OF SYMMETRY GROUPS IN HEAT EQUATIONBy
V. G. GuptaDepartment of Mathematics
University of Rajasthan Jaipur-302004, RajasthanKapil Pal and Rita Mathur
Department of MathematicsS. S. Jain Subodh Postgraduate College, Jaipur-302004, Rajasthan
E-mail [email protected](Received : December 10, 2008)
ABSTRACTThe main object of present paper is that to obtain the most general solution
for the parital differential equation of one dimensional heat conduction in a finiterod having the thermal diffusivity k0 using the general prolongation formula fortheir symmetry.2000 Mathematics Subject Classification : 17B66, 22Q75, 70G65)Keywords: Scaling, Translation, Linearity, Galilean-Boost.
1. Introduction.1.1 The General Prolongation formula.
Let
q
i
p
i
i
uux
xuxv
11
,, ...(1.1.1)
be a vector field defined on an open subset UXM where X is the space of
independent variables, U is the space of dependent variables, p is the number ofindependent variables and q is the number of dependent variables for the system.
Then thn -prolongation of v is the vector field
q
j J
nJn
uuxvvpr
1
, ...(1.1.2)
defined on the corresponding space nn UXM where X is the space of the
independent variables, nU is the space of the dependent varivables and the
derivatives of the dependent variables up-to n (order of differential equation). The
second summation being over all unordered multi-indices kjjJ ,...,1 with
.1,1 nkpjk The coefficient function J of npr v are given by the following
formula
148
iuuDuxp
ii
ip
ii
iJ
nJ ,,11
...(1.1.3)
where ii xuu / and i
Ji xuiu /, (see Olver [2], eq. 2.38 and 2.39, p.-110).
1.2 Theorem. Suppose ,, nd ux for d=1,...,l is a system of differential equations
of maximal rank defined over UXM . If G is a local group of transformations
acting on M and
0, nd
n uxvpr ...(1.2.1)
for d=1,...,l, whenever 0, nux for every infinitesimal generator v of G, is a
symmetry group of the system, (see Olver[2], eqation 2.25, page 104).2. Mathematical Analysis.
2.1 The Heat conduction equation:The one-dimensional conduction of heat in finite rod, without source, with theassumptions(a)The position of the rod coincides with the x-axis,(b) the rod is homogeneous,(c) It is suffciently thin so that the heat is uniformly distributed over its crosssection at a given time t,(d) The surface of the rod is insulated to prevent any loss of heat through theboundary, is governed by parital differential equation in the standard form
xxt uku 0 ...(2.1.1)
where u(x,t) is the temperature at the point x at time t and k0 be the constantthermal diffusivity, which is the second order differential equation with twoindependent variables and one dependent variable (in our notation given in (1.1)p=2, n=2 and q=1) (see Churchill[1], Simmons[4]).
3 Method. Using the general prolongation formula we obtained the mostgeneral solution for one-dimensional heat conduction equation (2.1.1) (see Olver[2]and Olver [3]).
4 Method of Solution.4.1 Main Theorem. The most general solution of heat conduction equation
xxt uku 0 ...(4.1.1)
is given by tktxxtu 602
52
6536 41/exp.41/1
txttettxef ,41/,41/2 262
16544 ...(4.1.2)
where 61 ,..., are real constants and an arbitrary solution to the heat equation.
149
By using the following lemmas we proved the main theorem (4.1).4.2 Lemma:
Let u
utyxt
utyxx
utyxv
,,,,,,,,, ...(4.2.1)
be a symmery on .UX Then the smooth coefficient functions , and are
given by txutxutxttx ,,,, and ,, where and are arbitary
functions.Proof. Firstly we determine the second prolongation of v by using (1.1.2) to
tt
tt
xt
xt
xx
xx
t
t
x
x
uuuuuvvpr
2...(4.2.2)
and the coefficients present in (4.2.2) can be calculated by using (1.1.3). Using theinfinitesimal criterion (1.2.1) takes the form
xxt k 0 ...(4.2.3)
By substituting the value of t and xx in equation (4.2.3) and replacing tu by
,0 xxuk then equating the coefficient of the terms in the first and second orderparitial derivatives of u, finally we find the determining equations as follows
Table1: The Determining Equations TableMonomial Cofficient
xtxuu uk 020 (a)
xtu xk 020 (b)2
xxu uu kk 20
20 (c)
xxx uu 2uuk 2
00 (d)
xxxuu uxuu kkk 02
00 32 (e)
xxu xuxxtu kkk 202
00 (f)
3xu uuk 00 (g)
2xu xuuuk 20 0 (h)
xu xxxut k 20 (j)
xxt k 0 (k) ...(4.2.4)
The reqirement for (a) and (b) is that be a function of t.(e) shows that does not
depend on u.(h) shows that be linear in txutxutxu ,,,, so, for functions
and .
4.3 Lemma. The most general infinitesimal symmetry of the Heat conduction
150
equation has coefficient functions of the form ,42 6541 xtctcxcc
2642 42 tctcc and txutkxcxckc ,2/1 0
26503 where c1,...,c6
are arbitrary constants.
Proof. Using lemma (4.2), the equation (f) require ttxso txt 2/1,,2
where is only function of t. We infer from xt kj 02 implies be at most a
quadratic function in x given by txkxk ttt 02
0 2/18/1 where is only
function of t. At the end the equation (k) requires that both and be the solution
of the heat conduction equation, i.e., xxt k 0 and .0 xxt k Using the
determining equation of , we find that is quadratic in t, and , is linear in t.
Since all the determining equations are satisfied then the most general infinitesimalsymmetry of the heat conduction equation has coefficient functions of the form
26426541 42,42 tctccxtctcxcc and tkxcxckc 0
26503 2/1
txu , where c1,...,c6 are arbitrary constants and tx, is an arbitrary solution
of the Heat conduction equation.4.4 Lemma. The infinitesimal symmetires of the Heat conduction equation isspanned by the six vector field ,2,,, 4321 txutx txvuvvv
,/12 05 ux xuktv , /244 0022
6 utx uktkxttxv and the infinite-
dimensional sub algebra ., utxv
Proof. The proof is evient by using lemma (4.2) and lemma (4.3).4.5 Lemma. The symmetry Lie algebra of the heat conduction equation is
spanned by the six-vector field v1,...,v6 and the infinite-dimensional sub algebra v .
Proof : The commutation relations between these vector fields as followsTable 2: The Commutation Relation Table
v1 v2 v3 v4 v5 v6 v
v1 0 0 0 v1 30/1 vk 52vx
vv2 0 0 0 2v2 2v1 34 24 vv
tv
v3 0 0 0 0 0 0 v
v4 -v1 -2v2 0 0 v5 2v6 'v
v5 30 2/1 vk -2v1 0 -v5 0 0 ''v
v6 -2v5 2v3-4v40 -2v6 0 0 '''v
vx
vt
v v ' v '' v ''' v 0
151
where 02
0 /24''',/2'',2' ktxtxtkxttx txxtx by the
above table 2 we find that forms the Lie algebra with Lie bracket operation.
5 Result.5.1 Theorem. The one- parameter groups Gi generated by the vi are given asfollows
exp.,,,2(:,,:,,,:,,,:,,,: 52
4321 utytxGutexeGuetxGutxGutxG
tkxtutttxGtxtxGktx 41/exp.41.,41/,41/:,,,:)),/)(( 02
602
where group ,6,...,1iGi is a symmetry group.
Proof. The one parameter groups ,6,...,1iGi are obtained by using the lemma
4.4 and . ~,~,~,,exp utxutxvi
5.2 Theorem. The group invariant solution to the Heat conduction equationcorresponding to its different symmetry groups are given by the function
, ,,,,,, 24321 texefutxfeutxfutxfu
,,,,,2./exp 025 txtxfuttxfktxu
,41/,41/.41/exp.
411
026 tttxftkx
tu
where txfu , be an
given solution to the Heat equation, tx, any other solution and be a real
number.Proof. The group invariant solutions of Heat conduction equation are obtained
by using the relation utxutx ~,~,~,, and putting the values of x,t and u in given
solution txfu , for each symmetry group Gi given in theorem 5.1.
Proof of Main Theorem 4.1. The most general solution txgfu , of the Heat
conduction equation is obtained by group transformations vg exp
1166 exp...exp vv of given solution txfu , and using theorem 5.2 as follows
tktxxtu 602
52
653 41/exp.41/1
txttetltxe ,41/,41/2 262
16544
where 61 ,..., are real constant and an arbitrary solution to the Heat equation.
152
6. Conclusion: In our investigation the groups G3 and G reflect the
linearity of the Heat conduction equation. The G1 and G2 are the time and spaceinvariance of the equation respectively, and reflect the fact that Heat conductionequation has constant coefficients. The group G4 is well known scaling symmetrygroup. The group G5 represent a kind of Galiean boost to a moving corrdinateframe. The group G6 is a genuinely local group of transformations and if u=c bea constant solution then the function
tkxtcu 41/exp.41/ 02
be a solution. The fundamental solution of Heat conduction equation be obtained
by substituting ,/c at the point .4/1,0, 00 tx Now, by translating the
above solution in t using G2, with replaced by ,4/1 we get the fundamental
solution of the problem in the form .4/exp.41/1 02 tkxtu
7. Discussion. The general solution of Heat equation is invarient under
its different symmetry groups ,6,...,1iGi acting on the independent variables.
8. Special Cases:8.1 If take k0=1 then the most general solution to the Heat conduction equation
xxt uu ...(8.1.1)
is given
ttxxtu 62
52
6536 41/exp41/1
txttettxef ,41/,41/2 262
16544 ...(8.1.2)
where 61,..., are real constants and an arbitrary solution (see Olver[2], page
120).ACKNOWLEDGEMENT
The authors are grateful to thank of Ex. Professor M.A. Pathan, Departmentof mathematics, Aligarh Muslim University, Uttar Pradesh, India, for givenvaluable and useful suggestions towards the improvement of this paper.
REFERENCES[1] R.V. Churchill, Operational Mathematics ,3rdedition, McGraw-Hill Kogakusha Ltd., 1972.[2] P. J. Olver, Application of Lie Groups to Differential Equations, Second Edition, Springer-Verlag-
New York, Inc.,1993.[3] P. J. Olver, Symmetry Groups and Group Invariant Solutions of Parital Differential Equation, Diff.
Geom, 14 (1979) 497-542.[4] G. F. Simmons, Differential Equations with Applications and Historical Notes, 2nd., Mc. Graw Hill,
Inc., New York, 1991.
J~n–an–abha, Vol. 38, 2008
A GENERALIZATION OF MULTIVARIABLE POLYNOMIALSBy
R.C. Singh Chandel and K.P. TiwariDepartment of Mathematics
D.V. Postgraduate College, Orai-285001, Uttar Pradesh, IndiaE-mail :[email protected]
(Received : October 7, 2007)
ABSTRACTIn the present paper, we introduce multivariable analogue of Chandel
polynomials [5] and Chandel-Agrawal polynomials [6], Our polynomials are alsogeneralization of multivariable polynomials due to authors Chandel and Tiwari([7],(1.5)).2000 Mathematics subject classification : Primary 33C65; Secondary 33C70.Keywords: Multivariable Analogue of Chandel and Chandel-Agrawal polynomials,Multivariable polynomials due to Chandel and Tiwari.
1.Introduction. Bell polynomials [8] are defined as
(1.1) , ;1,dxd
DeDehgH hgnhgnn
where h is constant and g is some specified function of x.Shrivastava [9] derived from above the polynomials defined by
(1.2) hgn
hgn e
dxd
xeghG
, .
Singh [10] introduced generalized Truesdell polynomials defined byRodrigues’ formula
(1.3) dxd
exexprxTrr pxnpx
n ,,, .
Chandel ([1],[2],[3],[4]) introduced and studied a class of polynomialsdefined by Rodrigues’ formula
(1.4) dxd
xexexprxT kx
pxnx
pxkn
rr
,,,, ,
where 1k .
For 1k , (1.4) reduces to (1.3)Srivastava and Singhal [11] also studied slight variation of (1.4) in
the form
154
(1.5) rn
krknn pxx
dxd
xpxxn
kprxG
exp exp
!1
,,, 1
Further to generalize the polynomials defined by (1.1), (1.2), (1.3)(1.4) and (1.5), Chandel [5] introduced and studied a class of polynomialsdefined by Rodrigues’ formula
(1.6) , ,, hgnx
hgn eekghG
where h is independent of x and g is suitable function of x differentiableany number of times.
Later on Chandel and Agrawal [6] to generalize (1.5) introduced aclass of polynomials defined by
(1.6) ,11,, xhgxhgghS nx
kn
where , h,k are any numbers real or complex independent of x and g(x) isany suitable function of x.
Here in the present paper, we introduce and study multivariableanalogue of Chandel polynomials (1.5) and Chandel-Agrawal polynomials(1.6) defined through Rodrigues’ formula
(1.7) mkkhhb
nn ggG mm
m,...,1
,...,;,...,;,...,
11
1
bmm
nx
nx
bmm ghghghgh m
m ...1 ......1 1111
1
1
where b, h1,...,hm, mkk ,...,1 are real or complex numbers independent of
mxx ,...,1 ; while gi is function of xi differetiable any number of times and
i
kix x
x i
i
, 1ik ; i=1,...,m.
It is clear that
(1.8) m
kkbhbhbnnb
ggG mm
m,...,lim 1
,...,;,...,;,...,
11
1
.,,...,, 1111 mmmnn kghGkghGm
Also in addition for m=1, we can write
(1.9) xghGgG n
kbhbn
b,,lim ,/,
,
where xghGn ,, are polynomials due to Chandel [5] defined by (1.5)Also for m=1, (1.7) reduces to (1.6). That is
(1.10) . ,,;; ghSgG kbn
khbn
where ghS kbn ,, are polynomials due to Chandel and Agrawal [6] defined
through (1.6).
155
Also chossing riiiii pxxgk log,1 (i=1,...,m) and replacing b by
–b, (1.7) reduces to the multivariable polynomials due to authors Chandeland Tiwari ([7],(1.5)] defined by Rodriguies' formula
(1.8) mpprrkkb
nn xxT nmmm
m,...,1
,...,;,...,;,...,;;...;;,...,
1111
1
brmmmm
r mxpxxpx
1111 log...log1 1
brmmm
rnx
nx
mm
mxpxxpx loglog1... 11111
11
1
where ni are positive integers , iiii krkb ,,1,, are arbitrary numbers real
or complex independent of all variables mixi ,....,1; .2. Generating Relation. Starting with Rodrigues’ formula (1.7),
we have
!
...!
,...,1
1
0,...,1
,...,;,...,;,...,
1
1
11
1m
nm
n
nnm
kkhhbnn n
tnt
ggGm
m
mm
m
, ...1...1 11...
1111 b
mmttb
mm ghgheghgh mxmx
Thus making an appeal to the well known result due to Chandel ([1, p.105eq. (2.5)]; also see Srivastava-Singhal [10, p.76 eq. (1.12)])
(2.1)
11
1
11 kk
xt
txk
xfxfe ,
where k1 and f(x) admits Taylor’s series expansion, we finally derivegenerating relation
(2.2)
!...
!,...,
1
1
0,...,1
,...,;,...,;,...,
1
1
11
1m
nm
n
nnm
kkhhbnn n
tnt
ggGm
m
mm
m
11
111
1111
1111
1...1kk
bmm
xtk
xghghgh
b
kkmm
mm
mmxtk
xgh
1
1111
... .
156
3. Applications of Generating Relation. An appeal to generatingrelation (2.2) gives
(3.1) mkkhhbb
nn ggG mm
m,...,1
,...,;,...,;',...,
11
1
mkkhhb
ss
n
s
n
sm
kkhhbsnsn
ns
ns ggGggG mm
n
i
m
m
mm
mm
m
m,...,,...,...... 1
,...,;,...,;',...,
0 01
,...,;,...,;,...,
11
1
111
11
1
1
,
which can be further generalized in the form:
(3.2) m
kkhhbbnn ggG mmq
m,...,1
,...,;,...,;...,...,
111
1
1111 1
111
111
1
... ... 11
,...,;,...,;,..., ,...,......
nss nss
q
jm
kkhhbss
ns
ns
q mmqm
mm
m
m
mjijggG
mkkhhb
ss ggG mmq
mqq,...,... 1
,...,;,...,;,...,
11
1.
4. Recurrence Relations. Starting with generating relation (2.2),we have
!
...!
,...,...11
1
0,...,1
,..,;,...,;1,...,11
1
1
11
1m
nm
n
nnm
kkhhbnnmm n
tnt
ggGghghm
m
mm
m
11
111
111
1
1
0,...,1
,...,;,...,;,...,
11
1
1
11
1
11
1!
...!
,...,kkm
nm
n
nnm
kkhhbnn
xtk
xgh
nt
nt
ggGm
m
mm
m
1
1111
...
mm kk
mm
mmm
xtk
xgh
!...
!,...,
1
1
0,...,1
,...,;,...,;,...,
1
1
11
1m
nm
n
nnm
kkhhbnn n
tnt
ggGm
m
mm
m
0,.., 1
11
,...,;,...,;,...,
1
111
1 !...
!,...,
m
mmm
mnn m
nm
n
mkkhhb
nn nt
nt
ggG
0 0
1 11
111111
1
1
1
11
111...11
s s
ks
kmmm
smsmk
sks
sm
m
m
mm
mxtkxhxtkxh
!
...!
,...,1
1
0,...,1
,...,;,...,;,...,
1
1
11
1m
nm
n
nnm
kkhhbnn n
tnt
ggGm
m
mm
m
157
0,.., 1
11
,...,;,...,;,...,
1
111
1 !...
!,...,
m
mmm
mnn m
nm
n
mkkhhb
nn nt
nt
ggG
0 01
1
111
1
111
1 1
1
11
1
1
1 !1
1s r
rrk
s
ss t
rxk
ks
xh
0 0
1
!1
1...
1
m m
mm
m
m
ms r
rm
m
rkmm
sm
msmsm t
rxk
ks
xh .
Now equating the coefficiants of mnm
n tt ...11 both the sides, we derive
the recurrence relation
(4.1) mkkhhb
nnmm ggGghgh mm
m,...,...1 1
,...,;,...,;1,...,11
11
1
0 0
111
1
1111
,...,;,...,;,...,
1
1
1
111
1
11
11
1,...,
s
n
r
rkssm
kkhhbnn xk
ks
xhggG mm
m
0 0
11
,..,;,...,;,...,, 1
1...,...,11
211
m
m
m
mm
m
m
m
mm
ms
n
r
rkmm
rm
msmsmm
kkhhbnnrn xk
ks
xhggG
mkkhhb
rnnn ggG mm
mmm,...,1
,..,;,...,;,,...,
11
11 .
5. Differential Recurrence Relations. Differentiating (2.2)partially with respect to t1, we have
!...
!!1,...,
2
2
1
11
0,...,1
,..,;,..,;,...,
21
1
11
1m
nm
nn
nnm
kkhhbnn n
tnt
nt
ggGm
m
mm
m
1/111111111
1111 11'...1
kkkkbmm xtkxghghgh
1
1/111/11111
111
11...
111
11
b
kkmmm
mmmkk mmxtk
xgh
xtk
xghb
Therefore,
0,..., 2
2
1
11
1,...,;,...,;
,...,111
2111
1 !...
!!1,...,...1
m
mmm
mnn m
nm
nn
mkkhhb
nnmm nt
nt
nt
ggGghgh
158
0
111
1
1111
1
11
1
1 11
's
sk
s
k xkk
kgxbh
!
...!
,...,1
11
0,...,
,...,;,...,;1,...,
1
1
11
1m
nm
n
mnn
kkhhbnn n
tnt
ggGm
m
mm
m
.
Thus equating the coefficients of mnm
n tt ...11 both the sides, we derive
recurrence relation :
(5.1) mkkhhb
nnnmm ggGghgh mm
m,...,...1 1
,...,;,...,;,...,,111
11
21
1
1
11
211
11
1
1
1
1
01
,...,;,...,;1,...,,
111
1
1111 ,...,1
1'
n
sm
kkhhbnnsn
sk
s
ns
k ggGxkk
kgxbh mm
m ,
which suggests that m-recurrence relations can be written in the followingunified form :
(5.2) mkkhhb
nnnnnmm ggGghgh mm
miii,...,...1 1
,...,;,...,;,...,,1,,...,11
11
111
mkkhhb
nnsnnn
n
s
skii
si
insi
ki ggGxk
kk
gxbh mm
miiii
i
i
ii
i
i
i
i ,...,11
' 1,...,;,...,;1
,...,,,,...,0
11
11
111
,
i=1,...,m.Again differenting (2.2) partially with respect to x1 we have
0,..., 1
11
,...,;,...,;,...,
11
111
1 !...
!,...,
m
mmm
mnn m
nm
n
mkkhhb
nn nt
nt
ggGx
1/11
111
111
11111
11111...1' kk
bmm
xtk
xghghghgbh
bmm
b
kkmmm
mm ghghbxtk
xgh
mm
...1
11... 111/11
1
1
1/111/11111
111
11...
111
11
b
kkmmm
mmmkk mmxtk
xgh
xtk
xgh
12
1111
1111
1/1111111 1
1
1111 11111' kk
kkkk xtkxtkxtkgh .
Therefore,
159
!
...!
,...,...11
11
0,...,
,...,;,...,;,...,
111
1
1
11
1m
nm
n
mnn
kkhhbnnmm n
tnt
ggGx
ghghm
m
mm
m
0,..., 1
11
,...,;,...,;,...,11
1
111
1 !...
!,...,'
m
mmm
mnn m
nm
n
mkkhhb
nn nt
nt
ggGgbh
0,..., 1
11
,...,,,..,;1,...,11
1
111
1 !...
!,...,'
m
mmm
mnn m
nm
n
mkkhhb
nn nt
nt
ggGgbh
0 0 1
1111
1
111
111
1
111
11 1
111
1
11
1
1!
112
1!
11
1
s s
ssk
s
kssk
s st
xkkk
tkxts
xkk
i
.
Thus equating the coefficients of mnm
n tt ...11 , we derive differential
recurrence relation.
(5.3) mkkhhb
nnmm ggGx
ghgh mm
m,...,...1 1
,...,;,...,;,...,
111
11
1
1
1
1
1
1
1
11
10
111
1111
,...,;,...,;1,...,11 1
11
',...,'n
s
sk
s
nsm
kkhhbnn
imm
mxk
kgbhggGgbh
1
0 1
111
11111
,...,;,...,;1,...,,
1
1 1
1
1
111
211 12
1',...,n
s s
ns
km
kkhhbnnsn k
kkxgbhggG mm
m
mkkhhb
nnsn
sk ggGxk mm
m,...,1 1
,...,;,...,;1,...,,
111
11
211
11
,
which further suggests m-differential recurrence relations in the followingunified form :
(5.4) mkkhhb
nni
mm ggGx
ghgh mm
m,...,...1 1
,...,;,...,;,...,11
11
1
i
i
ii
i
i
i
mm
m
n
s
skii
si
nsiim
kkhhbnnii xk
kgbhggGgbh
0
11
,...,;,...,;1,..., 1
11
',...,' 11
1
1
0
111
,...,;,...,;1,...,,,,..., 1
21',...,11
111
i
i i
i
i
imm
miiii
n
s si
insi
kiiim
kkhhbnnsnnn k
kkxgbhggG
mkkhhb
nnsnnn
skii ggGxk mm
miiii
ii ,...,1 1
,...,;,...,;1,...,,1,,...,
1 11
111
, i=1,...,m.
Now eleminating 1'g from (5.2) and (5.4), we further finally derive:
160
(5.5)
i
i
ii
i
i
i
mm
m
i
n
s
skii
si
insm
kkhhbnn
i
ki xk
kk
ggGx
x0
11
,...,;,...,;,..., 1
1,...,11
1
mkkhhb
nnsnnn ggG mm
miiii,...,1
,...,;,...,;1,...,,,,...,
11
111
mkkhhb
nnmkkhhb
nnnnn ggGggG mm
m
mm
miii,...,,..., 1
,..,;,...,;1,...,1
,...,;,...,;,...,,1,,...,
11
1
11
111
mkkhhb
nnsnnn
n
s
skii
ns ggGxk mm
miiii
i
i
iii
i,...,1 1
,...,;,...,;1,...,,,,...,
0
1 11
111
1
0
11 1121
1i
i
ii
i
i
n
s
skii
si
i
i
ii
ki xk
kk
sn
kx
mkkhhb
nnsnnn ggG mm
miiii,...,1
,...,;,...,;1,...,,1,,...,
11
111
, i=1,...,m.
REFERENCES[1] R.C.S. Chandel, Properties of Higher Transcendental Functions and Their Applications,
Ph.D. Thesis, Vikram University, Ujjain, 1970.[2] R.C.S. Chandel, A new class of polynomials, Indian J. Math., 15 (1973), 41-49.
[3] R.C.S. Chandel, A further note on the polynomials prxT kn ,,, , Indian J. Math., 16 (1974),
39-48.[4] R.C.S. Chandel, Generalized Stirling numbers and polynomials, Publ. Del. Istitut
Matematique, 22 (36) (1977), 43-48.
[5] R.C.S. Chandel, A further generalization of the class of polynomials prxT kn ,,, ,
Kyungpook Math. J., 14 (1974), 45-54.[6] R.C.S. Chandel, and S. Agrawal, A generalization of a class of polynomials, bhaananJ~ , 21
(1991) 19-25.[7] R.C.S. Chandel and K.P. Tiwari, Multivariable analogues of generating Truesdell
polynomials, bhaananJ~ , 37, (2007), 167-176.
[8] J. Riordan, An Introduction to the Combinatorial Analysis, 1958.[9] P.N. Shrivastava, On generalized Stirling numbers and polynomials, Riv. Mat. Univ.
Parma (2), 11, (1970), 233-237.[10] R.P. Singh, On generalized Truesdell polynomails, Riv. Mat. Univ. Parma (2). 8 (1967),
345-353.[11] H.M. Srivastava and J.P. Singhal, A class of polynomails defined by generalized Rodrigues’
formula, Ann. Mat. Pura Appl. (4) 90 (1971), 75-85.
161
J~n–an–abha, Vol. 38, 2008
THE DISTRIBUTION OF SUM OF MIXED INDEPENDENT RANDOM
VARIABLES ONE OF THEM ASSOCIATED WITH H -FUNCTIONBy
Mahesh Kumar GuptaDepartment of Mathematics, M.S.J. College,
Bharatpur 321001 , Rajasthan, India(Received : September 15,2007)
ABSTRACTIn the present paper, we shall obtain the distribution of sum of two mixed
independent random variables with different probability density functions. Onewith finite probability density function and the other with infinite probability
density function associated with H -function. The method used is of Laplace
transform and its inverse. The result obtained by us is sufficiently genetral in
nature due to the presence of the H -function in the probability density function.
2000 Mathematics Subject Classification : Primary 60Exx, 62Exx; Secondary62H10.
Keywords: H -function, Laplace Transform, Distribution Function.1. Introduciton. In the study of statistical distributions, there is a vast
literature in the distribution in the linear combination of several independentrandom variables when each random variable follows a particular family ofdistributions. the works of Robins [14], Robins and Pitman [15], Kabe [10], Stacy[20]. Sricastava and Singhal [19], Mathai and Saxena [12], Malik [11], Saxena andDash [16], Goyal and Agrawal [8], Garg and Gupta [6], Garge [5], Garg and Garg[7], is worth mentioning.
It has been observed that the distribution of sum of several independentrandom variables when each random variable is of simply infinite or doubly infiniterange can easily be calculated by means of characteristic function or momentgenetating function. However, when the random variables are distributed overfinite renge, these method are not much useful and the power of integral transformmethod comes sharply into focus.
In this paper, we shall obtain the distribution of two independent randomvariables, X1 and X2, where X1 possess finite uniform probability density function
and X2 follows infinite probability density function involving H -function, given
by the equations (1.1) and (1.2) respectively. Thus
162
f1(x1) =
,0
1 otherwise
ax 00a ...(1.1)
and
otherwise ,0
0, 2,,;,
;,,,2,
,1
222
,1,1
,1,1
2 xzxHeCxxfPNjjNjjj
QMjjjMjj
aAaBbb
yNMQP
x
...(1.2)
where
QMjjjMjj
QNjjNjjjNMQp Bbb
aAazHC
,1,,1
,1,,11,,1
1
;,,,,1;,1;,
...(1.3)
and the following conditions are satisfied :
(i) ,0min,0,01
jjMj
b
(ii) ,01 1 1 1
M
j
N
j
Q
Mj
P
Njjjjjjj BAA ...(1.4)
(iii) The parameters of H -function are real and so restricted that 22 xf remains
positive for 02 x .
The H function occurring in (1.2) is a generalization of well-known FoxH-function [4]. It has been introduced by Inayat Hussain [9] and represented asfollows
QMjjjMjj
PNjjNjjjNMQP
NMQP Bbb
aAazHzH
,1,,1
,1,,1,,
,, ;,,
,;,
,12
1
w
wdz ...(1.5)
Q
Mj
P
Njjj
Bjj
M
j
N
j
Ajjjj
ab
ab
j
j
1 1
1 1
1
1
, ...(1.6)
163
where aj (j=1,...,P) and bj ( j=1,...,Q) are complex parameters, Pjj ,...,10
and Qjj ,...,10 (not all zero simultaneously) and the exponents Aj (j=1,...,N)
and Bj ( j=M+1,...,Q) can take on non-integer values. For the absolute convergence
of the H -function, the sufficient conditions given by (ii) of eq. (1.4) have been
given by Buschman and Srivastava [1].
The behaviour of the H -function for small values of |z| follows easily
from a result recently given by Rathie [13,p.306, eq. (6.9)]. We have
.0,/Remin.01
,,
zbzzH jjMj
NMQP (1.7)
When all the exponents Aj and Bj take on integral values, the H -function
reduces to the well-known Fox H-function [4]. We mention below give few some
interesting special cases of the H -function [9, pp.4126-4127], which are not the
particular cases of Fox H-function.(i) The generalized Riemann Zeta function [2, p.27, eq. (1)] and [9, p.4127,eq.(27)]
0
2,12,2 ;1,,1,0
;1,1,1;1,01,,
n
np p
pzHz
npz . (1.8)
The above function is the generalization of the well-known generalized(Hurwitz's) Zeta function (p,) and Reimann Zeta function (p) [2,p.24, eq.(1),p.32, eq.(1)].(ii) The polylogarithm of order p [2, p.30, eq. 1.11 (14)],
pp
zHpzzfnz
pzFn
p
n
;1,0,1,1;1,1,1;1,1
1,,,1
2,12,2 . (1.9)
For p=2 the above function reduces into Euler's dilogarithm [2,p.31, eq. (22)].(iii) The exact partition function of the Gaussian model in statistical mechanics[9,p. 4127, eq. 28].
dd
HdF dd 1;1,1,1,0;1,2/1,1;1,0,1;1,0
141
41
,' 23,12,32/
2
2/
2
. (1.10)
In the p.d.f defined by (1.2), if we reduce H -function to Fox H-function by
taking Aj=Bj=1, we get the p.d.f. defined by Mathai and Saxena [12, eq. (8), p. 163],which on taking the limit as 0, gives the p.d.f. of studied by Srivastava and
164
Singhal [19]. Further, on specializing the H -function as given above we can obtain
various new p.d.f. s involving the functions defined by equation (1.8), (1.9) and(1.10).
2. Distribution of the Mixed Independent Random VariablesTheorem. Let X1 and X2 be two independent random variables having theprobability density function defined by (1.1) and (1.2) respectively. Then theprobability density function of
21 XXY (2.1)
is given by
,1 ygyg ay 0
,21 ygyg ya (2.2)
where
0 ,1,,1
,1,,11,1,11 1;,,;,
,,1;,1;,
!n QMjjjMjj
pNjjNjjjNMQp
n
nBbbanAa
xyHny
aCy
yg
0y (2.3)
and
0
1,1,12 !n
NMQP
n
ayzHn
aya
ayCyg
aynBbb
anAa
QMjjjMjj
pNjjNjjj
,1;,,;,,,1;,1;,
,1,,1
,1,,1(2.4)
C is given by (1.3) and the following conditions are satisfied :
(i) ,0 0 , 0min1
jjMj
b
(ii)
M
j
N
j
Q
Mj
P
Njjjjjjj BAA
1 1 1 1
0 (2.5)
(iii) The parameters of H - function are real and so restricted that yg1 , 0y
and ,2 yg ay , remains positive.
Proof. To obtain the probability density function of Y=X1+X2, we use the methodof Laplace transform and its inverse. Let the Laplace transform of Y be denoted by
165
sy , then
sxfLsxfLsy ;; 2211 (2.6)
The Laplace transform of 11 xf is a simple integral and to evaluate the
Laplace transform of 22 xf , we express the H -function in terms of Mellin-Barnes
type contour integral (1.5), interchange the order of 2x - and integrals and
evaluate x2 integral as gamma integral to get
QMjjjMjj
pNjjNjjjNMQp
as
Bbb
aAaszH
asC
se
yg,1,1
,1,11,,1 ;,,
,,1;,1,;,1
(2.7)Now, we break above expression in two parts, as follows
QMjjjMjj
pNjjNjjjNMQp Bbb
aAaszH
sasC
yg,1,1
,1,11,,1 ;,,
,,1;,1,;,
QMjjjMjj
pNjjNjjjNMQp
as
Bbb
aAaszH
sasCe
,1,1
,1,11,,1 ;,,
,,1;,1,;, (2.8)
To obtain the inverse Laplace transform of first term of eq. (2.8), we express
the H -function in contour integral, collect the terms involving 's' and take its
inverse Laplace transform and then use the known result [3, p.238, eq.8]. Writing,the confluent hypergeometric function thus obtained in series form and interpretingthe result by definition (1.5), we get the value of g1(y) as given by the eq. (2.3).
The inverse Laplace transform of second term easily follows by the valueof g1(y) and shifting property for Laplace transform.
3. Special Cases
(i) In the theorem obtained in section 2 if we take Ai=Bj=1, the p.d.f. 22 xf
reduces to the p.d.f. defined by Mathai and Saxena [12] as follows
0,
otherwise ,0
,
,
2,1
,12
,,
121
22
2
xb
azxHexCxf
Qjj
PjjNMQP
x
(3.1)
166
where
Qjj
PjjNMQP b
azHC
,1
,11,,1
11 ,
,,,1
and the corresponding p.d.f. of Y as obtained from the eq. (2.2) is given by
,1 yhyh ay 0
,21 yhyh ya (3.3)
where
0 ,1
,11,1,1
11 0,,,,
,,,1
! n Qjj
PjjNMQP
n
ynb
anzyH
ny
ayC
yh (3.4)
and
0 ,1
,11,1,1
12 ,,,
,,,1
! n Qjj
PjjNMQP
n
nb
anayzH
nay
aayC
yh
ay (3.5)
Further, if we take a=1 we get a known result obtained earlier by Garg [5,eq. (2.2), p.79]. This result can be further be reduced to yield the known resultrecorded in book [17].
(ii) In the Theorem, if we reduce the H -function to generalized Riemann Zeta
function as given by relation (1.8), the p.d.f. f2(x2) assumes the following form:
otherwise ,00,,, 22
122
22
2 xpzxexCxfx
(3.6)
where
pp
zHC;1,,1,0
1;,1,;1,1,1;1,03,12,3
12 (3.7)
and the corresponding p.d.f. of Y as obtained from the eq. (2.2) is given by
ayyhyh 0 ,1
,21 yhyh ya (3.8)
where
167
0
3,13,3
21 0,
1;,;1,,1,01;,1,;1,1,1;1,0
!n
n
ynp
npzyH
ny
ayC
yh (3.9)
and
0
3,13,3
22 ,
1;,,;1,,1,01;,1,;1,1,1;1,0
!n
n
aynp
npayzH
nay
aayC
yh
(3.10)ACKNOWLEDGEMENTS
The author is thankful to University Grants Commission, New Delhi, forproviding necessary financial assistance to carry out the present work. The authoris also thankful to Dr. Mridula Garg for her valuable suggestions and guidanceduring the preparation of this paper.
REFERENCES[1] R.G. Buschman and H.M. Srivastava, The H-function associated with a certain class of Feynman
integrals, J. Phys. A : Math. Gen. 23 (1990), 4707-4710.[2] A. Erdélyi et al., Higher Transcendental Functions,Vol.1, McGraw-Hill, New York, 1953.[3] A. Erdélyi et al., Table of Integral Transforms, Vol. I, McGraw-Hill, New York, 1954.[4] C. Fox, The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98 (1961)
395-429.[5] Sheekha Garg, The distribution of the sum of mixed independent random variables, Ganita
Sandesh, 6(2) (1992), 78-82.[6] M. Garg and M.K. Gupta, The distribution of the ratio of two linear functions of independent
random variables associated with Fox H-function, Bull. Cal. Math. Soc., 88 (1996), 125-130.[7] M. Garg and Sheekha Garg; The distribution of a polynomials and the ratio of product of rational
powers of independent random variables, Ganita Sandesh, 9(2), (1995), 79-85.[8] S.P. Goyal and R.K. Agrawal, The distribution of a linear combination and the ratio of products of
random variables associated with the multivariable H-function, bhaananJ~ , 9/10 (1982).[9] A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals,
II, A generalization of the H-function J. Phys. A: Math. Gen., 20 (1987), 4119-4128.[10] D.G. Kabe, On the exact distribution of a class of multivariable test criteria, Annals Math. Soc.,
33 (1962), 1197-1200.[11] H.J. Malik, Distribution of a linear function and the ratio of two independent linear functions of
independent generalized gamma variables, National Research Logistics, 23, (1976), 339-343.[12] A.M. Mathai and R.K. Saxena. On the linear combination of stochastic, variables, Metrika, 20 (3)
(1973), 160-169.[13] A.K. Rathie, A new generalization of generalized hypergeometric functions, Le Math. Fasc. II, 52
(1997), 297-310.[14] H.E. Robbins. The asymptotic distribution of the sum of random number of variables , Bull.
Amer. Math. Soc., 54 (1948) 1151-1161.[15] H.E. Robbins and J.G. Pitman, Application of the method of mixture to variables, Australian J.
Stat., 7 (1949), 110-114.[16] S.P. Saxena and S.P. Dash, The distribution of linear combinations and ratio of product of
168
independent variables associated with H-function, Vijnana Parishad Anusandhan Patrika, 22(1979), 57-65.
[17] M.D. Springer. The Algebra of Random Variables, John Wiley and Sons, New York (1979).[18] H.M. Srivastava, K.C. Gupta and S.P. Goyal. The H-Functions of One and Two Variables with
Applications, South Asian Publishers, New Delhi, Madras. (1982).[19] H.M. Srivastava and J.P. Singhal. On a class of generalized hypergeometric distributions, bhaananJ~
Sect A, 2 (1972), 1-9.[20] E.W. Stacy. A generalization of the gamma distribution, Anal. Math. Statist., 33 (1962), 1187-
1192.
169
J~n–an–abha, Vol. 38, 2008
ON GENERATING RELATIONSHIPS FOR FOX'S H-FUNCTION ANDMULTIVARIABLE H-FUNCTION
ByB.B. Jaimini and Hemlata Saxena
Government Postgraduate College, Kota-324001, Rajasthan, IndiaEmail : [email protected], E-mail :[email protected]
(Received : January 15, 2008)
ABSTRACTIn this paper we have established some new results on bilinear, bilateral
and multilateral generating relationship for Fox's H-function and multivariableH-function. Some known results for the Fox's H-function and multivariable H-function are also obtained as special cases of our main findings.2000 Mathematics Subject Classification: 33C99, 33C90, 33C45Keywords : Bilateral generating functions; Fox's H-function; Multivariable H-function; Generating Function Relationships; Combinatorial identities.
1. Introduction and Results Required. Chen and Shrivastava [1] gavea family of linear, bilateral and multilateral generating functions involving the
sequence 0,
kk z defined by
zkFz vuvuk ;,...,,1;;,..., 11, (1.1)
where for convenience, ; abbreviates the array of parameters
1
,...,1
, 0/0NN
and for its multivariable extension defined by ([1],p.172, equation (5.21)).
0,...,1
111
1
1 ...1
,...,,...,;,...,
r
r
kk
kr
k
k
rrrk zz
kkkA
zzZ (1.2)
rjCNkkkK jjrr ,...,1;,;;... 011 ,
where rkkA ,...,1 is a suitably bounded multiple sequence of complex numbers
and ()k denotes the Pochhammer symbol.
CNkk
kkk ;1...1
0;0,1(1.3)
170
Raina [4] derived the following combinatorial identity as a special case of reductionformula in ([4],p.187, equation (15)).
012
1
;;,111
k
kzzk
kF
kk
kk
kk
1 ,1 zz (1.4)
where
111
knk
nkn
. (1.5)
Recently in an earlier paper Jaimini et al. [3] generalized the results ofabove cited paper [1]. They proved six theorems on the generating functionrelationship in view of the above result (1.4)
The Fox's H-function defined and represented in the following manner([2],p.408), see also ([5], p.265, equation (1.1))
Lqq
ppnmqp dz
iba
zH21
,,,
, (1.6)
where
p
njjj
q
mjjj
n
jjj
m
jjj
ab
ab
11
11
1
1
(1.7)
The multivariable H-function defined and represented in the followingmanner ([6],pp 251-252, equations (C.1)-(C.3)).
rzzH ,...,1
r
rrr
rr
qrj
rjqjjq
rjjj
prj
rjpjjp
rjjj
r
nmnmnoqpqpqp ddb
cca
z
zH
,1,111
,11
,1,111
,111
,,...,,:,,,...,:, ,;...;,:,...,;
,;...;,:,...,;
1
111
11
1
1 ;......,...,......2
111111L L r
sr
srrrr
r
r dsdszzssssi (1.8)
where ; 1i
171
ri
scsd
scsd
si
i
i
i
ii
p
nji
ij
ij
q
mji
ij
ij
n
ji
ij
ij
m
ji
ij
ij
ii ,...,1
1
1
11
11
(1.9)
q
j
r
ji
ijj
p
nj
r
ji
ijj
n
j
r
ji
ijj
r
sbsa
sa
ss
1 11 1
1 11
1
1
,...,. (1.10)
In this paper some generating relations for Fox's H-function andmultivariable H-function defined in (1.6) and (1.8) respectively are established byfollowing the above cited work of Jaimini et al [3]. The importance of these resultslies in the fact that they provide the extensions of the results due to Srivastavaand Raina [7] and also provide a wide range of bilinear, bilateral mixed multilateralgenerating functions for simpler hypergeometric polynomials.
2. Main bilateral generating relationships involving Fox's H-function.
Result-1. Corresponding to an identically nonvanishing function szz ,...,1 of s
complex variables Nszz s ,...,1 and of (complex) order , let
0
11
1,,,, !
,...,;,...,;,
k
kskk
sm mk
tzzatzzy
vv
uusru d
cmkmkyH
,,,11,
1 CNNkak ,;;;0 0 (2.1)
and
mn
kmnksmn tyAzzyM
/
0
,,,,1
,,,,,, ;;,...,;
!!
,...,11 11
mknmk
zza
mknmkn
mknmkn k
skk
(2.2)
where
vv
uusrvu
l l
ll
mnk dclmkn
yHlmkn
ttyA
,,,,1
!; 1,
,10
,,,,, (2.3)
172
then
0
1,,,,,
, ;,...,;n
nsmn tzzyM
m
m
sl
mt
tzz
t
yt 11,,,,
1;,...,;
11 (2.4)
Result-2. Let
0
,,
11
2,, ,
,,...,1;,...,;
k vv
uusrvu
kskk
mk
sm dc
yHtzza
tzzy (2.5)
and
mn
kmnksmn tyUzzyN
/
0
,,,,,1
,,,,,, ;;,...,;
!
,...,111 11
mkn
zza
mknmkn
mknmkn k
skkmk
(2.6)
where
0
1,1,1
,,,,, ,,,
,,,
!;
l vv
uusrvu
l
ll
mnk mkmkdclmkn
yHlmkn
ttyU (2.7)
then
0
1,,,,,
, ;,...,;n
nsmn tzzyN
m
m
smt
tzz
t
yt
112
,,1
1;,...,;
11 ...(2.8)
Result 3. Let tzzy sm ;,..,; 12
,, is defined in 2.5 and
mn
kmnksmn tyVzzyT
/
0
,,,,,,1
,,,,,,, ;;,...,;
!
,...,111 11
mkn
zza
mknmkn
mknmkn k
skmk
k
...(2.9)
173
where
0 1
,,,,,, !
;l
ll
mnk lmknt
tyV
,,,,,.,1,
1,1 kmkdClkmkn
yHvv
uusrvu ...(2.10)
then
0
1,,,,,,
, ;,...,;n
nsmn tzzyT
m
m
smt
tzz
t
yt 11
2,,
1
1;,...,;
11 ...(2.11)
Result -4. Let
011
4,,,,, ,...,1;,...,;
k
kskk
mksm tzzatzzy
,,,,.,11,
1,1 mkkmkdCkmkmk
yHvv
uusrvu ...(2.12)
and
mn
kmnksmn tyWzzy
/
0
,,,,,,1
,,,,,,, ;;,...,;
!
,...,111 11
mkn
zza
mknmkn
mknmkn k
skkmk
...(2.13)
where
0
,,,,,, !
;l l
ll
mnk lmknt
tyW
,,,,.,11,
1,1 mkkmkdClkmkn
yHvv
uusrvu ...(2.14)
then
174
0
1,,,,,,
, 1;,...,;n
smn tzzy
m
m
smt
tzz
t
y11
4,,,,,
1;,...,;
1 ...(2.15)
Proof of Result-1. We denote the left hand side of the assertion 2.4 of Result-1 byH[x,y,t] then we use the definitions in 2.2 and 2.3, we have:
0,
/
0 ! ,,
ln
mn
k l
ll
lmknt
tyxH
!!
,...,11 11
mknmk
zza
mknmkn
mknmkn k
skk
n
vv
uusrvu t
dClmkn
yH ,,
,, ,11,,1
Now using the definition of Fox's H-function from 1.6 and changing theorder of summation and integration and then on making series rearrangementtherein, it takes following form:
0, 0 ! 21
,,ln k l
ll
L lmkmknt
yi
tyxH
dtlmkmknnmk
zza
nmkn
nmkmkn mknkskk
!!
,...,11 11
Now in view of the relation
nn
nn
lni
1
! ...(2.16)
and then interpreting the inner series into Gauss' hypergeometric function 2F1we have
0 0
121
,,k n
L nmkmkn
yi
tyxH
175
nmkmkn
nmkmkn 11 1
nttmkmkn
mkmknF
;;,
12
dmkmkmk
tzza mkkskk .
!
,...,1
Now using the combinatorial identity 1.4 and then on interpreting theresulting contour into H-function with the help of 1.6, we atonce arrive at thedesired result in 2.4
Similarly the proof of Results-2,3,4 would run parallel to that of Result-1,which we have already detailed above fairly adequately.
3. Some Generating Relationships Involving H-Function of SeveralVariables. The Results-5,6,7,8 given below are established for the multivariable H-function defined in 1.8 by following the corresponding results proved in section-2.
Result-5. Let
0
111
5,,,, !
,...,;,...,;,...,
k
kskk
srm mk
tzzatzzyy
,,...,;1
1
1
1
1,;...;,:1,0,;...;,:,1
1
11
11r
r
vuvuvqpqpqp mkmk
ty
tyH
r
rr
rr
r
r
qrj
rjqjjq
rjjj
prj
rjpjjp
rjjj
ddb
cca
,1,111
,11
,1,111
,11
,;...;,:,...,;
,;...;,:,...,;
1
1
...(3.1)
and
mn
ksrmnksrmn tzzyyBzzyyR
/
011
,,,,,11
,,,,,, ;,...,;,...,;,...,;,...,
kskk
mknmk
zza
mknmkn
mknmkn
!!
,...,11 11
...(3.2)
where
176
0
1,,,
,, ! ;,...,
l l
ll
rmnk lmknt
tyyB
,,...,;1 1
1,;...;,:1,0,;...;,:,1
11
11r
r
vuvuvqpqpqp lmkn
y
yH rr
rr
r
r
qrj
rjqjjq
rjjj
prj
rjpjjp
rjjj
ddb
cca
,1,111
,11
,1,111
,11
,;...;,:,...,;
,;...;,:,...,;
1
1
...(3.3)
then
0
11,,,,,
, ;,...,;;,...,n
nsrrmn tzzyyyR
m
m
sr
m t
tzz
t
y
t
yt
r 1115
,,,,1
;,...,;1
,...,1
11 ...(3.4)
Reult-6. Let
0111
6,, . ,...,1;,...,;,...,
k
kskk
mksrm tzzatzzyy
r
rrr
rr
qrj
rjqjjq
rjjj
prj
rjpjjp
rjjj
r
vuvuvqpqpqp ddb
cca
y
yH
,1,111
,11
,1,111
,111
,,...,,:1,0,,...,,:,1 ,;...;,:,...,;
,;...;,:,...,;
1
111
11 ...(3.5)
and
mn
ksrmnksrmn zzyyEzzyyS
/
011
,,,,,11
,,,,,, ;,...,;,...,;,...,;,...,
kskkmk
mkn
zza
mknmkn
mknmkn
!
,...,111 11
...(3.6)
where
0
1,,,
,, ! ;,...,
l l
ll
rmnk lmknt
tyyE rr
rr
vuvuvqpqpqpH ,;...;,:1,0
,;...;,:,111
11
177
r
r
qrj
rjqjjq
rjjjr
prj
rjpjjp
rjjjr
rddbmkmk
ccalmkn
y
y
,1,111
,11
1
,1,111
,11
11
,;...;,:,...,;,...,;
,;...;,:,...,;,...,;
1
1 ...(3.7)
then
0
11,,,,,
, ;,...,;,...,n
nsrmn tzzyyS
m
m
sr
m t
tzz
t
y
t
yt
r 1116
,,1
1;,...,;
1,...,
11
1 ...(3.8)
Result-7. Let tzzyyy srrm ;,..,;;,..., 116
,, is defined in 3.5
and
mn
krmnksrmn tyyFzzyyU
/
01
,,,,,,11
,,,,,,, ;,...,;,...,;,...,
kskkmk
mkn
zza
mknmkn
mknmkn
!
,...,111 11
...(3.9)
where
0
1,,,,
,, ! ;,...,
l l
ll
rmnk lmknt
tyyF rr
rr
vuvuvqpqpqpH ,;...;,:1,0
,;...;,:1,111
11
r
r
qrj
rjqjjq
rjjjr
prj
rjpjjp
rjjjr
rddbmkmkk
ccalkmkn
y
y
,1,111
,11
1
,1,111
,11
11
,;...;,:,...,;, ,...,;
,;...;,:,...,;, ,...,;
1
1
...(3.10)then
0
11,,,,,,
, ;,...,;,...,n
nsrmn tzzyyU
m
m
sr
m t
tzz
t
y
t
yt
r 1116
,,1
1;,...,;
1,...,
11
1 ...(3.11)
178
Result-8. Let
0111
8,,,,, .,...,1;,...,;,...,
k
kskk
mksrm tzzatzzyy
r
r
rr
rr
qrj
rjqjjq
rjjjr
prj
rjpjjp
rjjjr
r
vuvuvqpqpqp
ddbmkkmk
ccamkkmk
y
y
H
,1,111
,11
1
,1,111
,11
11
,;...;,:1,0,;...;,:1,1
,;...;,:,...,;,...,;
,;...;,:,...,;,...,;1
1
1
11
11
...(3.12)
and
mn
krmnksrmn tyyFzzyyV
/
01
,,,,,,11
,,,,,,, ;,...,;,...,;,...,
kskk
mk zzamknmkn
mknmkn
,...,111
1
1
...(3.13)
where
0
1,,,,
,, ! ;,...,
l l
ll
rmnk lmknt
tyyG
rr
rr
vuvuvqpqpqpH ,;...;,:1,0
,;...;,:1,111
11
r
r
qrj
rjqjjq
rjjjr
prj
rjpjjp
rjjjr
rddbmkkmk
ccalmkkn
y
y
,1,111
,11
1
,1,111
,11
11
,;...;,:,...,;,...,;
,;...;,:,...,;,...,;1
1
1 (3.14)
then
0
11,,,,,,
, ;,...,;,...,n
nsrmn tzzyyV
m
m
sr
mt
tzz
t
y
t
yt
r 1116
,,,,,1
;,...,;1
,...,1
11 ...(3.15)
4. Special Cases. If in Results-1 to 5 and in Result-8 we take
0,1,...,1 rk zz and these results reduce to the respective known
results in ([7], pp.37-44, equations (1.10),(1.14),(3.3),(5.3),(6.9),(6.6) at 0 ).
179
If in the results of section -2, 3 we take 0 , and 1,...,1 rk zz then
these results are reduced into certain families of new generating functionsassociated with the Fox's H-function and multivariable H-function, but we skipthe results here.
All the results of section 2 and 3, the product of the essentially arbitarycoefficients
00 Nkak
and the identically nonvanishing function
CNsNkzz sk ;,; ,..., 01
can indeed be notationally into one set of essentially arbitrary (and indenticallynonvanishing) coefficients depending on the order and on one, two or more
variables. In view to applying such results as section 2 above to derive bilateralgenerating relationships involving Fox's H-function and as section 3 to derivemixed multilateral generating relationship involving multivariable H-function.
We find it to be convenient to specialize ak and sk zz ,...,1 individually as well as
separately. Our general results asserted by section 2 and 3 can be shown to yieldvarious families of bilateral and mixed multilateral generating relation for thespecific functions generated in these families but there are not recorded due tolack of space.
REFERENCE[1] M.P. Chen and H.M. Srivastava, Orthogonality relations and generating functions for Jcobi
polynomials and related hypergeometric functions, Appl. Math. Comput., 68 (1995), 153-188.[2] C. Fox, The G-and H-functions as symmetrical Fourier kernesls, Trans. Amer. Math. Soc., 98
(1961) 395-429.[3] B.B. Jaimini, H. Nagar and H.M. Srivastava, Certain classes of generating relations associated
with single and multiple hypergeometric functions, Adv. Stud. Contemp. Math., 12 No.1 (2006),131-142.
[4] R.K. Raina, On a reduction formula involving combinatorial coefficients and hypergeometricfunctions, Boll. Un. Math. Ital. A (Ser. 7) 4 (1990), 183-189.
[5] H.M. Srivastava and R. Panda Some bilateral generating functions for a class of generalizedhypergeometric polynomials, J. Reine Angew., Math., 283/284 (1976), 265-274.
[6] H.M. Srivastava, K.C.Gupta and S.P. Goyal, The H-function of One and Two Variables withApplications (1982), (New Delhi : South Asian Publishers), pp 251-252.
[7] H.M. Srivastava and R.K. Raina, New generating functions for certain polynomial systemsassociated with the H-functions, Hokkaido Math. J., 10 (1981), 34-45.
180
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