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Automated Matlab Solver for Nonlinear Singular BVPs in Membrane
N. Ch. N. Suresh1*, Syamal K. Sen2, J. Vasundhara Devi3, and Ravi P. Agaral!
1*, 3 "VP#Pr$%. V. &a'shmi'an(ham )ns(i(u(e %$r Advaned S(udies and De+ar(men( $%
a(hema(is, "VP C$llege $% -ngineering A/, Visa'ha+a(nam#03!
suresh.ne''an(igv+e.a.in, 4vdevi.liasgv+e.a.in
2 "VP#Pr$%. V. &a'shmi'an(ham )ns(i(u(e %$r Advaned S(udies and als$ De+ar(men( $%
C$m+u(er Siene 5-ngineering, "VP C$llege $% -ngineering A/, Visa'ha+a(nam#03!
s'sen.liasgv+e.a.in
! De+ar(men( $% a(hema(is, 6e7as A5 8niversi(y#Kingsville, 9 8niversi(y :lvd., SC192, Kingsville, 6e7as 93;3#22
Ravi.Agaral(amu'.edu
Abstract A %as( $nverging i(era(ive a(la
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A( (he edge, i.e., a( 1= x , e have (he $ndi(i$n
1/ 3/ y λ = >
here λ is a $ns(an( hih re+resen(s radial s(ress a( (he
here v is (he +$iss$nFs ra(i$.
)% e n$ in(r$due (he hange $% varia
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6he $m+u(a(i$nal +r$edures ($ s$lve (he +r$
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!/ vi@. +r$
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)% ei(her θ 1 $r θ 2 are rela(ively aay %r$m (he $riginal value $% y a( 7Ea $riginal ini(ial
$ndi(i$n/, (hen e en$un(er an in(egra(i$n ($lerane +r$
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n)$- /input!+no. of intervals+#-
feval!3pf14,14v4a4b4n#
else if r))$
disp!+Problem $ data+#
,1)input!+, value )+#-
lmda)input!+lmda value )+#- a).$$$- /!+starting value of 0+#-
b).2- /!+final value of 0+#-
n)$- / !+no. of intervals+#-
feval!3p$f14,14lmda4a4b4n#
else
disp!+invalid number assigned to r+#
end-
end-
end-
Problem P1 C$nsider (he Pr$!160!
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title!+Gonvergence of solns E!0# bet%een lo%er and upper solns+#-
legend!+"o%er soln+4+pper soln+4 +True soln+4 +Iterated solns+#-
/SJGHN KI"J
/subprogram to solve the ode iterativelEfunction'C()p1f!lmda4a4b4h4lb4ub#
0span)a8h8b-
s9)si9e!0span#- l0)s9!14 #- p)-
/ l0 denotes the no of values of D0F at %hich %e are calculating the solution
flag)1- !!ub6lb#7!fub6flb###-
E)'ne%E4(-
'0r4E()ode12s!+Edash1+40span4E#-
si9e0r)si9e!0r#-s9r)si9e0r!141#-
if !s9r))l0#OO
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.!332 .; .;!;
.!;;0 .; .;10
.!II .; .;91.0331 .; .;9!;
.0;;! .; .;99
.0II9 .; .;;;9.;331 .; .;;2!
.;;;! .; .;09I
.;II9 .; .;032
.933 .; .;!2
.9;;3 .; .;!3
.9II; .; .;39;.32I .; .;31I
.;;2 .; .;2;
.II; .; .;1II.I32I .; .;130
.I;;2 .; .;;I
.III0 .; .;1&$er s$ln a( 7E 8++er s$ln a( 7E
.; .9!2
%r E #2.30!e#
a
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6he %$ll$ing gra+h ig.1/ %$r 'E.3 and lmdaE.; gives (he l$er, (he u++er, (he (rue,
and (he i(era(ed s$lu(i$ns.
Kig. 1 ,).$ and ).W for Problem P1 !integration
tolerance Problem not encounterd#
Example 2
>> membrn_bvp(1)Problem 1 datak value =0.7lmda value =0.6
6he s$lu(i$n
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.1 .9I1; .9;I .991I .992
.33! .9I1! .9; .9919 .991
.;;9 .9II .9;1 .9911 .9911
… … …… … .I32I .;0!I .;2I .;203.;20!
.I;;2 .;!30 .;0 .;13 .;131
.III0 .;310 .0I02 .0III .;
Kig. ,).@ and ).W for Problem P1 !integration
tolerance problem not encounterd#
N$(ie (ha( e have n$( en$un(ered any
in(egra(i$n ($lerane +r$ membrn_bvp(1)Problem 1 datak value =0.8lmda value =0.5
6he s$lu(i$n
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reducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19 (n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9
Warning: Failure at t=9.0$0)1)e-001. Unable to meet integrationtolerances withoutreducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19 (n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9Warning: Failure at t=9.62721%e-001. Unable to meet integrationtolerances withoutreducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19
(n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9
6he a
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Kig. $ ,).= and ).2 for Problem P1!obviated
integration tolerance problem automaticallE#
Kig. ,)1.2 and ).W2 for Problem P1
!overcome integration tolerance problem#:
1!
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Kig. 2 ,)$ and ).$@ for Problem P1 !integration
tolerance problem obviated#
>ere hen 'E3 and HE.39 e $me
ar$ss (he in(egra(i$n ($lerane +r$
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Kig. @ ,)1 and )1.2 for Problem P1
!solutionfound onlE in t%o iterations#
Kig. = ,)1 and ) for Problem P1 !final solution
almost coincides %ith upper solution#
Kig. ,)1 and ).2 for Problem P1 !obviated
integration tolerance problem automaticallE#
)% e (a'e 'E2 and vary H, similar +a((erns are $
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Kig. 11 ,) and )1.@2 for Problem P1 !lb and ub
solutions are close and true solution comes verE
close to ub solution#
Kig. 11 ,) and ). for Problem P1 !obviated
integration tolerance problem to get the solution#
Problem P C$nsider (he +r$
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fprintf!+:n Gol 18 04 Gols 4 $4 ..4 last but one8 successive iterated sols E!0#4 Gollast8 true sol E!0# :n+#
C1)'0+ C(4
/fprintf!+/@.f /@.f /@.f /@.f /@.f /@.f /@.f :n+4 0+4 C#
s)si9e!C#- Cf ) s!84 #- ts ) C!84 Cf#-
plot!0+4 lb4 +r5+4 0+4 ub4 +b0+4 0+4 ts4 +,o+4 0+4 C#- 0label!+0+#- Elabel!+Solution E!0# for HJ ?E 5 ,7E 5 $E70)+#-
title!+Gonvergence of solns E!0# bet%een lo%er and upper solns+#-
legend!+"o%er soln+4+pper soln+4 +True soln+4 +Iterated solns+#-
/SJGHN KI"J
/subprogram pf
function'C()pf!v4 a4b4h4 lb4 ub#
0span)a8h8b-
s9)si9e!0span#- l0)s9!14 #- p)-
flag)1- Elb!l041#5Elb!l04#-
fub)!16v#>Eub!l041#5Eub!l04#-
ne%E)lb6!flb>!!ub6lb#7!fub6flb###-
E)'ne%E4(-
'0r4E()ode12s!+Edash1+40span4E#-
1
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if !s9r))l0#OOE!l041#5E!l04#-)soln!84#-
C!84E!#70(-
Numerical results 6he %$reg$ing +r$grams +r$vide (he s$lu(i$n $% +r$ere (he value $% ($ in+u( +arame(ers ' and , are as'ed %$r in (he a(la<
+r$gram and (hen ill
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6he s$lu(i$n
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Q1 E
C$lumns 1 (hr$ugh 0 C$lumns ; (hr$ugh
.1 .99; .9!! .902
.9023 .33! .990 .9!39 .9!II
.9021
.;;9 .9999 .9!3 .9!I3.9012
.1 .99;; .9!2 .9!9I
.901
.133! .9902 .9!; .9!;0.9!0
.1;;9 .993! .93I .9!!0
.9!;9 .2 .9913 .93; .9!22
.9!!!
.2333 .9; .93!3 .93I0.9!1
.2;;; .9;;2 .9310 .93;0
.93I
.2III .9;32 .922 .9333.930
.3332 .90I; .92!0 .92I
.931 .3;;0 .9009 .92! .9203
.9299
.3III .901! .910I .92.9231
.!332 .9!;9 .91I .910
.911 .!;;0 .9!19 .900 .912
.9129
.!II .93;2 .;II0 .9!2
.9; .0331 .93! .;I32 .;I9I
.90
.0;;! .92!1 .;;3 .;I11.;I39
.0II9 .919! .;9I .;3
.;;0 .;331 .913 .;91 .;90I
.;99
.;;;! .92; .;;20 .;;9;
.;90 .;II9 .;I!0 .;03! .;0;
.;;1;
.933 .;0 .;!3 .;!I1.;021
.9;;3 .;9;; .;33! .;3I
.;!2
.9II; .;;; .;22! .;21.;312
.32I .;0;! .;19 .;1;0
.;1I .;;2 .;!03 .0I2 .;!2
.;9;
.II; .;33; .0! .0I1.0I!0
.I32I .;211 .090 .09;I
.0;
.I;;2 .;9 .0003 .0;2.0;0
.III0 .0I39 .03I .0!;
.00 .901! .9010 .9010
.9011 .9012 .9013
.900 .90; .90; .9!I2 .9!I3 .9!I3
.9!99 .9!9 .9!9I
.9!09 .9!0 .9!0 .9!30 .9!3; .9!3;
.9! .9!I .9!I
.939 .939I .939I
.93!; .93!9 .93! .931 .9312 .9312
.92;; .92;9 .92;
.9221 .9222 .9223 .9191 .9192 .9193
.911; .9119 .911
.90; .909 .90 .;II3 .;II! .;II0
.;I20 .;I2; .;I29
22
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.;02 .;03 .;0!
.;99! .;99; .;99;
.;;I1 .;;I2 .;;I3 .;;2 .;;3 .;;!
.;09 .;0 .;0I
.;!; .;!9 .;! .;2I9 .;2II .;2II
.;12 .;1! .;1!
.;; .;;1 .;;2
.0I2 .0I3 .0I31 .09I .09I1 .09I1
.0;! .0;!2 .0;!2
.0!1 .0!3 .0!!
Kig. 1 ,) .@2 and X) .1 for Problem P
e $
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Kig. 1$ ,).@2 and X). for Problem P
!integration tolerance problem not encounterd#
Kig. 1 ,).@2 and X).2 for Problem P !no. of
iterations re;uired to find the true solution is
small#
Kig. 12 ,).@2 and X)Y.W2 for Problem P !no. of
iterations to find the true solution is large#
Kig. 1W ,).@2 and X)6.= for Problem P
!integration tolerance problem is obviated#
Kig. 1@ ,)1.$2 and X).2 for problem P
!integration tolerance problem not encounterd#Kig. 1= ,)1.$2 and X).= for Problem P !the no.
of iterations are small to find true solution#
2!
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Kig. 1 ,)1.$2 and X)Y.@ for Problem P !some of
the iterated solutions crosses ub solution curve but
bounce bac, into lb6ub solutions# Kig. ,)1.$2 and X)Y.= for Problem P
!obviated integration tolerance problem#
Kig. 1 ,) and X).2 for problem P
!integration tolerance problem not encounterd#
Kig. ,) and X).=2 for problem P
!the no. of iterations are small to find true solution#
Kig. $ ,) and X)6.2 for problem P!some of
the iterated solutions crosses ub solution curve but
bounce bac, into lb6ub solutions#
Kig. ,) and X)Y. for problem P
!obviated integration tolerance problem#
20
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Problem P$ B!16!17t!
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/step denotes the length of the interval bet%een lo%er and upper bounds of the
/ solution E!0#. step needs to be appropriatelE divided into subintervals
/%henever the tolerance problem crops up. e have ta,en the
/number of subintervals as 2 and this has %or,ed %ell for most of the problems
/considered. hen this number fails to obviate the tolerance problem bE
/indicating an error message %e increase suitablE this no. of subintervals.
'tlb4Elb()ode12s!+Edash$+4 tspan4 'lb4(#-
'tub4Eub()ode12s!+Edash$+4 tspan4 'ub4(#-
/ the follo%ing part of the program determines %hich of the bounds !lo%er7upper#
/ causes the integration tolerance problem and accordinglE modifE the interval.
/ If the problem occurs at the lo%er bound then %e increase the lo%er bound bE
/ an amount step71 else %e decrease the upper bound bE step71.
ntlb)si9e!tlb+#- mtlb)ntlb!14#-
ntub)si9e!tub+#- mtub)ntub!14#-
if !mtlb))lt mtub))lt#OO!mtlb)lt mtub)lt#
flb)Elb!lt41#6lmda- fub)Eub!lt41#6lmda-ne%t)lb6!flb>!ub6lb#7!fub6flb##-
t)'ne%t4(-
'tr4E()ode12s!+Edash$+4tspan4t#-
/'tr4E()ode$tb!+Edash$+4tspan4t#-
soln)'tr4 E(-
si9etr)si9e!tr#-s9r)si9etr!141#-
n)1
if !s9r))lt#OO!nQ1#
flag)-
else
p)p5$-
lb)lb6!step7p#4
ub)ub6!step7p#4
end-
n)n51
else
p)p5@4
if mtlb)lt
lb)lb5!step7p#-
else ub)ub6!step7p#-
end
/ Rere !17p#th part of the interval !lb4 ub# is deleted from consideration.
/ p value is being modified !incremented bE @# each time the integration
/ tolerance problem is encounterd.
/ e can change this depending on the conte0t and sensitiveness of the solution.
/ e have not come across the condition !m0lb)l0 m0ub)l0# in our
/ e0tensive numerical e0periments4 it is possible to encounter this case though.
end
29
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else
disp!+no solution found+#
brea,-
end
end-
!ub6lb#7!fub6flb##-
t)'ne%t4(-
'tr4E()ode12s!+Edash$+4tspan4t#-
soln)'tr4 E(-
fr)E!lt41#6lmda-)soln!84#-
C!84!17E!1#?#5E!#7t(-
Numerical results for Problem P$
2
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Example 5
>> membrn_bvp(3)Problem 3 datak value =0.5lmda value =0.6
6he s$lu(i$n
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1.22 .92; .;I0 .;I0I
1.9219 .;I1 .;I12 .;I13
1.;313 .;I33 .;;3 .;;0 1.0!I .;2 .;11 .;13
1.!9;2 .;29 .;900 .;909
1.!I2 .;99 .;;I9 .;;II 1.3! .;9I .;;30 .;;39
1.2I1I .;;!0 .;0;I .;091
1.2!3 .;099 .;0 .;02
1.1I29 .;00 .;!29 .;!2I
1.1!0 .;!3 .;30 .;302
1.19; .;30 .;2; .;29 1.;I! .;2;9 .;13 .;10
1.33 .;19 .;I3 .;I0
1.0 .;; .0II .;
Kig. 2 ,) .2 and lmda).W !integration
tolerance problem not encounterd#
e $
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Kig. = ,).2 and ). for problem P$
!integration tolerance problem obviated#
Kig. ,).2 and ).12 for problem P$
!integration tolerance problem that is encounterd
at different stages is obviated automaticallE#
)( is im+$r(an( ($ n$(e (ha( i% 'E.0 and HE.10 (hen e en$un(er in(egra(i$n($lerane +r$
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Kig. $ ,)1.2 and ).2 for problem P$ !the true
solution almost coincides %ith upper solution#
Kig. $$ ,)1.2 and ).2 for problem P$ !obviated
integration tolerance problem to find solution#
Kig. $ ,) and ). for problem P$ !obviated
integration tolerance problem to find solution#
Kig. $2 ,) and ). for problem P$ !true solution
almost coincides %ith upper solution#
32
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. GHNG"SIHNS
Need for lower and upper solutions )n (he suessive linear in(er+$la(i$n +r$edure ($s$lve (he n$nlinear singular :VPs e have made use $% ma(hema(ially derived u++er
and l$er s$lu(i$n
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"h# four signifi!ant digits in the independent variable x N$ measuring devie an
+r$due an auray m$re (han .0T. $r any engineering a++lia(i$n, %$ur signi%ian(digi( in (he %inal resul( ($
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*se of an appropriate method with effi!ient !omputer implementation 6here are manyme(h$ds suh as a sh$$(ing me(h$d and (he ! (h $rder Runge#Ku((a "ill#Runge#Ku((a
me(h$d ($ minimi@e r$unding err$rs/ in li(era(ure ($ s$lve BD-s ass$ia(ed i(h
)VPs?:VPs.hile m$re (han $ne me(h$d may
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+e!imal digit versus signifi!ant digit a!!ura!# )n (his +a+er e have (al'ed a
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I. -.V. Krishnamur(hy and S.K. Sen, Pro4rammin4 in *!56!", A%%ilia(ed -as(#
es( Press, Ne Delhi 23/.
1. S.K. Sen and . Chanda 1I92/G A general sheme %$r s$lving $rdinarydi%%eren(ial e=ua(i$ns under ($#+$in(