Numerical Simulation of One-Dimensional Fractional...

Research ArticleNumerical Simulation of One-DimensionalFractional Nonsteady Heat Transfer Model Based onthe Second Kind Chebyshev Wavelet

Fuqiang Zhao12 Jiaquan Xie12 and Qingxue Huang23

1College of Mechanical Engineering Taiyuan University of Science and Technology Taiyuan Shanxi 030024 China2Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment Taiyuan Shanxi 030024 China3College of Mechanical Engineering Taiyuan University of Technology Taiyuan Shanxi 030024 China

Correspondence should be addressed to Fuqiang Zhao zfqgear163com and Jiaquan Xie xjq371195982163com

Received 23 August 2017 Revised 6 November 2017 Accepted 20 November 2017 Published 11 December 2017

Academic Editor Jorge E Macıas-Dıaz

Copyright copy 2017 Fuqiang Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the current study a numerical technique for solving one-dimensional fractional nonsteady heat transfer model is presentedWe construct the second kind Chebyshev wavelet and then derive the operational matrix of fractional-order integration Theoperationalmatrix of fractional-order integration is utilized to reduce the original problem to a system of linear algebraic equationsand then the numerical solutions obtained by our method are compared with those obtained by CAS wavelet method Lastlyillustrated examples are included to demonstrate the validity and applicability of the technique

1 Introduction

Fractional calculus is a branch of mathematics that deals withgeneralization of the well-known operations of differentia-tions to arbitrary orders Many papers on fractional calculushave been published for the real-world applications in scienceand engineering such as viscoelasticity [1] bioengineering[2] biology [3] and more can be found in [4 5] Moreoverfractional partial differential equations also arewidely used inthe areas of signal processing [6] mechanics [7] economet-rics [8] fluid dynamics [9] and electromagnetics [10] As theanalytical solutions of fractional partial differential equationsare not easy to derive the scholars are committed to obtaintheir numerical solutions of these equations

In recent years various numerical methods have beenproposed for solving fractional diffusion equations thesemethods include wavelets methods [11ndash17] Jacobi Legendreand Chebyshev polynomials methods [18ndash21] spectral meth-ods [22 23] finite element method [24] wavelet Galerkinmethod [25] and finite difference methods [26 27] In [28]a new matrix method is proposed to solve two-dimensionaltime-dependent diffusion equations with Dirichlet boundaryconditions In [29] the authors utilize the second kindChebyshev wavelets to obtain the numerical solutions of the

convection diffusion equations Xie et al use the Cheby-shev operational matrix method to numerically solve one-dimensional fractional convection diffusion equations in[30] In this paper we apply the second kind Chebyshevwavelet method to obtain the numerical solutions of one-dimensional fractional nonsteady heat transfer model Theobtained numerical solutions by our method have beencompared with those obtained by CAS wavelet method

The current paper is organized as follows Section 2 intro-duces the basic definitions of fractional calculus In Section 3the mathematical model of nonsteady heat transfer problemis proposed Section 4 illustrates the second kind Chebyshevwavelets and their properties In Section 5 we apply the sec-ond kind Chebyshev wavelet for solving fractional nonsteadyheat transfer model Numerical examples are presented totest the proposed method in Section 6 Finally a conclusionis drawn in Section 7

2 One-Dimensional Nonsteady HeatTransfer Model

For one infinite plate sample as shown in Figure 1 theheight is 120575 the upper surface and the edge are adiabatic

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 2658124 10 pageshttpsdoiorg10115520172658124

2 Discrete Dynamics in Nature and Society

sample t

x= 0

t

y= 0 t

y= 0

0 y

x

T = tw

Figure 1 Nonsteady heat transfer model with constant temperatureboundary condition

and the lower surface is contacted with the fluid which itstemperature is 119905119908 The heat conductivity coefficient of thesample is120582 the density is120588 and the specific heat capacity is 119888119901The initial temperature is 1199050 taking the origin of coordinateson the sample adiabatic surfaces and the nonsteady heattransfer model with the initial-boundary condition can bedefined as follows [31]

120597119905120597120591 = 12058212059721199051205881198881199011205971199092 120591 = 0119905 = 1199050119909 = 0120597119905120597119909 = 0119909 = 120590119905 = 119905119908

(1)

Obviously when the sample density120588 heat conductivity coef-ficient 120582 specific heat capacity 119888119901 and thickness 120575 are knownwe can obtain the temperature distribution at any position119909 and any time 120591 which is the nonsteady heat conductionmodel with constant temperature boundary condition Basedon the above-mentioned model we give the fractional-ordernonsteady heat transfer model of the following form

120597119879120597119905 = 120582120597120572119879120588119888119901120597119909120572 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 1 lt 120572 le 2

(2)

with the initial condition

119879 (119909 0) = 119891 (119909) 0 le 119909 le 1 (3)

and the boundary conditions

119879 (0 119905) = 1198920 (119905) 119879 (1 119905) = 1198921 (119905)

0 le 119905 le 1(4)

where 119892(119909 119905) denotes source term 119891(119909) is a given functionand 1198920(119905) 1198921(119905) are continuous functions with first-orderderivative

3 Preliminaries of the Fractional Calculus

In this section we give some necessary definitions andmathematical preliminaries on fractional calculus which willbe used further in this paper

Definition 1 TheRiemann-Liouville fractional integral oper-ator 119868120572 (120572 gt 0) of a function 119891(119905) is defined as follows [4]

119868120572119891 (119905) = 1Γ (120572) int119905

0(119905 minus 120591)120572minus1 119891 (120591) d120591

120572 gt 0 120572 isin R+(5)

Some properties of the operator 119868120572 are as follows119868120572119868120573119891 (119905) = 119868120572+120573119891 (119905) (120572 gt 0 120573 gt 0) (6)

119868120572119905120574 = Γ (1 + 120574)Γ (1 + 120574 + 120572)119905120572+120574 (120574 gt minus1) (7)

Definition 2 The Caputo fractional derivative 0119863120572119905 of afunction 119891(119905) is defined as follows [4]

0119863120572119905 119891 (119905) = 1Γ (119899 minus 120572) int119905

0

119891119899 (120591)(119905 minus 120591)119899minus120572+1 d120591

(119899 minus 1 lt 120572 le 119899 119899 isin 119873) (8)

Some properties of the Caputo fractional derivative are asfollows

0119863120572119905 119905120573 = Γ (1 + 120573)Γ (1 + 120573 minus 120572)119905120573minus120572

0 lt 120572 lt 120573 + 1 120573 gt minus1119868120572119863120572119891 (119905) = 119891 (119905) minus 119899minus1sum

119896=0

119891(119896) (0+) 119905119896119896 119899 minus 1 lt 120572 le 119899 119899 isin 119873

(9)

4 The Second Kind ChebyshevWavelet and Its Operational Matrix ofFractional Integration

41 The Second Kind Chebyshev Wavelet and Its PropertiesThe second kind Chebyshev wavelet 120595119899119898(119905) = 120595(119896 119899119898 119905)has four arguments 119899 = 1 2 2119896minus1 119896 isin 119873lowast They aredefined on the interval [0 1) as follows [19]120595119899119898 (119905)=

21198962119898 (2119896119905 minus 2119899 + 1) 119899 minus 12119896minus1 le 119905 lt 1198992119896minus1 0 ow(10)

with

119898 (119905) = radic 2120587119880119898 (119905) 119898 = 0 1 2 119872 minus 1 (11)

Discrete Dynamics in Nature and Society 3

Here 119880119898(119905) are the second kind Chebyshev polynomialswhich are orthogonal with respect to the weight function119908(119905) = radic1 minus 1199052 and satisfy the following recursive formula

1198800 (119905) = 11198801 (119905) = 2119905

119880119898+1 (119905) = 2119905119880119898 (119905) minus 119880119898minus1 (119905) 119898 = 1 2 (12)

A function 119891(119905) defined over [0 1) may be expanded interms of the second kind Chebyshev wavelet as follows

119891 (119905) ≃ 2119896minus1

sum119899=1

119872minus1sum119898=0

119888119899119898120595119899119898 (119905) = 119862119879Ψ (119905) (13)

where

119888119899119898 = (119891 (119905) 120595119899119898 (119905))120596119899 = int1

0120596119899 (119905) 120595119899119898 (119905) d119905 (14)

and the weight function 119908119899(119905) = 119908(2119896119905 minus 2119899 + 1)Moreover119862 and Ψ(119905) are = (2119896minus1119872) column vectors given by

119862 = [11988810 11988811 1198881(119872minus1) 11988820 11988821 1198882(119872minus1) 1198882119896minus10 1198882119896minus1(119872minus1)]119879

Ψ (119905) = [12059510 12059511 1205951(119872minus1) 12059520 12059521 1205952(119872minus1) 1205952119896minus10 1205952119896minus1(119872minus1)]119879

(15)

Take the collocation points as follows

119905119894 = 2119894 minus 12119896119872 119894 = 1 2 2119896minus1119872 = 2119896minus1119872 (16)

We define the second kind Chebyshev wavelet matrix Φtimesas

Φtimes = [Ψ( 12) Ψ ( 32) Ψ (2 minus 12 )] (17)

An arbitrary function of two variables119879(119909 119905)defined over[0 1)times [0 1)may be expanded into Chebyshev wavelets basisas follows

119879 (119909 119905) ≃ sum119894=1

sum119895=1

119889119894119895120595119894 (119909) 120595119895 (119905) = Ψ119879 (119909)119863Ψ (119905) (18)

where119863 = [119889119894119895]times and 119889119894119895 = (120595119894(119909) (119879(119909 119905) 120595119895(119905)))The following theorem discusses the convergence and

accuracy estimation of the proposed method

Theorem 3 Let 119891(119905) be a second-order derivative square-integrable function defined over [0 1) with bounded second-order derivative satisfying |11989110158401015840(119905)| le 119861 for some constants 119861then

(1) 119891(119905) can be expanded as an infinite sum of the secondkind Chebyshev wavelets and the series converge to119891(119905)uniformly that is

119891 (119905) = infinsum119899=0

infinsum119898isin119885

119888119899119898120595119899119898 (119905) (19)

where 119888119899119898 = ⟨119891(119905) 120595119899119898(119905)⟩1198712120596[01)

(2)

120590119891119896119872 lt radic12058711986123 (infinsum119899=2119896minus1+1

11198995infinsum119898=119872

1(119898 minus 1)4)

12 (20)

where 120590119891119896119872 = (int10|119891(119905) minus

sum2119896minus1119899=1 sum119872minus1119898=0 119888119899119898120595119899119898(119905)|2120596119899(119905)119889119905)1242 Operational Matrix of Fractional Integration On theinterval [0 1) we defined a ndash set of block-pulse functions(BPFs) as

119887119894 (119905) = 1 119898 le 119905 lt 119894 + 1 0 ow 119894 = 0 1 2 minus 1 (21)

The functions 119887119894(119905) are disjoint and orthogonal

119887119894 (119905) 119887119895 (119905) = 0 119894 = 119895119887119894 (119905) 119894 = 119895

int10119887119894 (119904) 119887119895 (119904) d119904 =

0 119894 = 1198951119898 119894 = 119895

(22)

Similarly the second kind Chebyshev wavelet may beexpanded into an -term block-pulse functions as

Ψ (119905) = Φtimes119861 (119905) (23)

Kilicman has given the block-pulse functions operationalmatrix of fractional integration 119865120572 of following form

(119868120572119861) (119905) asymp 119865120572119861 (119905) (24)

where

119861 (119905) = [1198870 (119905) 1198871 (119905) 119887minus1 (119905)]119879

119865120572 = 1119898120572 1Γ (120572 + 2)

[[[[[[[[[[[[[

1 1205851 1205852 1205853 sdot sdot sdot 120585minus10 1 1205851 1205852 sdot sdot sdot 120585minus20 0 1 1205851 sdot sdot sdot 120585minus3 d d

0 0 sdot sdot sdot 0 1 12058510 0 0 sdot sdot sdot 0 1

]]]]]]]]]]]]]

(25)

Next we derive the second kind Chebyshev wavelet opera-tional matrix of fractional integration Let

(119868120572Ψ) (119905) = 119875120572timesΨ (119905) (26)

where 119875120572times is called the second kind Chebyshev waveletoperational matrix of fractional integration and it can begiven by

119875120572times = Φtimes119865120572Φminus1times (27)

For More details see [29]


5 Numerical Implementation

In this section we use the second kind Chebyshev waveletsmethod for numerically solving the nonsteady fractional-order heat transfer model with initial-boundary conditionsIn order to solve this problem we assume

12059731198791205971199051205971199092 = Ψ119879 (119909)119863Ψ (119905) (28)

where 119863 = (119889119894119895)times is an unknown matrix which shouldbe determined and Ψ(sdot) is the vector defined in (15) Byintegrating (28) from 0 to 119905 we obtain

12059721198791205971199092 = 12059721198791205971199092100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ

119879 (119909)119863119875timesΨ (119905) (29)

Making use of the initial condition (3) enables one to put (29)in the following form

12059721198791205971199092 = 11989110158401015840 (119909) + Ψ119879 (119909)119863119875timesΨ (119905) (30)

Then we have

120597120572119879120597119909120572 = 1198682minus120572119909 (12059721198791205971199092 )= 1198682minus120572119909 ( 12059721198791205971199092

100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ119879 (119909)119863119875timesΨ (119905))

= 1198682minus120572119909 11989110158401015840 (119909) + Ψ119879 (119909) (1198752minus120572times)119879119863119875timesΨ (119905) (31)

By integrating (30) two times from 0 to 119909 we obtain119879 (119909 119905) = 119879 (0 119905) + 119909 120597119879120597119909

10038161003816100381610038161003816100381610038161003816119909=0 + 119891 (119909) minus 119891 (0)minus 1199091198911015840 (0) + Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905)

(32)

and by putting 119909 = 1 in (32) we get

119879 (119909 119905) = 119879 (0 119905) + 119909119867 (119905) + 119891 (119909) minus 119891 (0) minus 1199091198911015840 (0)+ Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905)

(33)

where

119867(119905) = 119879 (1 119905) minus 119879 (0 119905) + 119891 (0) + 1198911015840 (0) minus 119891 (1)minus Ψ119879 (1) (1198752times)119879119863119875timesΨ (119905)

(34)

By one time differentiation of (33) with respect to 119905 we obtain120597119879120597119905 = 1198791015840 (0 119905) + 1199091198671015840 (119905)

+ Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905) (35)

where

1198671015840 (119905) = 1198791015840 (1 119905) minus 1198791015840 (0 119905)minus Ψ119879 (1) (1198752times)119879119863119875timesΨ (119905)

(36)

10806

x040200

05t

10

002

004

006

008

Ana

lytic

al so

lutio

n

Figure 2 Analytical solution

Now by substituting (31) and (35) into (2) and combining (4)and taking collocation points 119909119894 = (2119894minus1) 119905119895 = (2119895minus1)119894 119895 = 1 2 3 we obtain the following linear system ofalgebraic equations

1198791015840 (0 119905119895) + 119909119894 (1198791015840 (1 119905119895) minus 1198791015840 (0 119905119895)minus Ψ119879 (1) (1198752times)119879119863119875timesΨ(119905119895)) + Ψ119879 (119909119894)sdot (1198752times)119879119863119875timesΨ(119905119895) = 1198861198682minusV119909 11989110158401015840 (119909119894)+ 119886Ψ119879 (119909119894) (P2minusVtimes)119879119863119875timesΨ(119905119895) + 119892 (119909119894 119905119895)

119894 119895 = 1 2 3

(37)

By solving this system to determine 119863 we can get thenumerical solution of this problem by substituting 119863 into(33)

6 Numerical Simulations

In this section we use the proposed method to solve theinitial-boundary problem of nonsteady heat transfer equa-tionsThe followingnumerical examples are given to show theeffectiveness and practicability of the proposed method andthe results have been compared with the analytical solution

Example 4 Consider the following fractional-order non-steady heat transfer model

120597119879120597119905 = 1205821205971511987912058811988811990112059711990915 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (38)

where the parameters 120588 = 7500 119888119901 = 0795 120582 = 800 and119892(119909 119905) = 119909(119909minus1)(2119905minus1)minus030279357104449811990905119905(119905minus1)withinitial-boundary condition 119879(119909 0) = 119879(0 119905) = 119879(1 119905) = 0The analytical solution of this problem is 119879(119909 119905) = 119909119905(119909 minus1)(119905 minus 1) The graph of the analytical solution is shown inFigure 2 The graphs of the numerical solutions when 119896 =119872 = 3 119896 = 119872 = 4 119896 = 119872 = 5 are shown in Figures3ndash5 From Examples 4 6 and 7 it can be concluded that thenumerical solutions approximate to the analytical solution for


10806

x

040200

05t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

3

Figure 3 Numerical solution with 119896 = 119872 = 3

10806x

04020005t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

4


10806

x040200

05t

10

002

004

006

008

Num

eric

al so

lutio

nw

ithk=

M=

5


a given value 119896 as119872 increases or for a given value119872 as 119896increases

Example 5 Consider the following fractional-order non-steady heat transfer equation

120597119879120597119905 = 1205971811987912059711990918 + 21199092119905 minus 2119909

021199052Γ (12) 0 le 119909 le 1 119905 ge 0 (39)

with initial-boundary condition 119879(119909 0) = 119879(0 119905) = 0119879(1 119905) = 1199052 The analytical solution of this problem is119879(119909 119905) = 11990921199052 When 119896 = 119872 = 3 119896 = 119872 = 4 119896 = 119872 = 5the numerical solutions obtained by our method and thoseobtained by CAS wavelet method at some values of 119909 119905 arelisted in Table 1

Example 6 We consider the following second-order non-steady heat transfer model

10806

x

040200

05t

1minus1

minus05

0

05

1

Ana

lytic

al so

lutio

n


10806

x

040200

05t

1

Num

eric

al so

lutio

nminus1

minus050

051

15

with

k=

3

Figure 7 Numerical solution with 119896 = 3

120597119879120597119905 = 212059721198791205971199092 + 3 sin (119909) minus sin (119905) minus 2 cos (119905)

0 le 119909 le 1 119905 gt 0(40)

in such a way that 119879(119909 0) = sin(119909) + 1 119879(0 119905) = cos(119905)119879(1 119905) = sin(1) + cos(119905) The analytical solution of the systemis 119879(119909 119905) = sin(119909) + cos(119905)The absolute errors between thenumerical and analytical solutions obtained by our methodand CAS wavelet method at some values of 119909 119905 when 119896 = 3(119872 = 3119872 = 4119872 = 5) are shown in Table 2 Table 2 showsthat ourmethodhas a better approximation thanCASwaveletmethod

Example 7 Consider the following second-order nonsteadyheat transfer model

120597119879120597119905 = 12058212059721198791205881198881199011205971199092 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (41)

where the parameters 120588 = 7500 119888119901 = 0795120582 = 1000 and 119892(119909 119905) = minus120587 sin(120587119909) sin(120587119905) +01677148846960171205872 sin(120587119909) cos(120587119905) in such a waythat 119879(119909 0) = sin(120587119909) 119879(0 119905) = 119879(1 119905) = 0The analyticalsolution of this problem is 119879(119909 119905) = sin(120587119909) cos(120587119905) Thegraphs of the analytical and numerical solutions when119872 = 3 (119896 = 3 4 5) are shown in Figures 6ndash9

Example 8 Consider (41) with 120572 = 2 19 18 17 thenumerical solutions when 119896 = 119872 = 4 at 119905 = 03 06 095are shown in Figure 10 This example is introduced to verify


Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

Figure 10 The numerical solutions with 120572 = 2 19 18 17 when 119896 = 119872 = 4


the robustness of the proposed method when the fractionalorder gradually approaches to 2 the numerical solutions arein agreement with the analytical solution

7 Conclusions

This paper presents a numerical technique for approximat-ing solutions of one-dimensional fractional nonsteady heattransfer model by combining the second kind Chebyshevwavelet with its operational matrix of fractional-order inte-gration In the proposed method a small number of gridpoints guarantee the necessary accuracyThemain advantageof wavelet method for solving the kinds of equations is thatafter dispersing the coefficients matrix of algebraic equationsis sparse The solution is convenient even though the sizeof increment may be large Several examples are given todemonstrate the powerfulness of the proposed method

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Collaborative InnovationCenter of TaiyuanHeavyMachinery Equipment and the Nat-ural Science Foundation of Shanxi Province (201701D221135)Dr Startup Funds of Taiyuan University of Science andTechnology (20122054) and Postdoctoral Funds of TaiyuanUniversity of Science and Technology (20152034)

References

[1] R L Bagley and P J Torvik ldquoA theoretical basis for theapplication of fractional calilus to visoelasticityrdquo Journal ofRheology vol 27 no 3 pp 201ndash210 1983

[2] R L Magin ldquoFractional calculus in bioengineeringrdquo CriticalReviews in Biomedical Engineering vol 32 no 1 pp 1ndash377 2004

[3] D A Robinson ldquoThe use of control systems analysis in theneurophysiology of eye movementsrdquo Annual Review of Neuro-science vol 4 pp 463ndash503 1981

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

[5] K Diethelm The Analysis of Fractional Differential Equationsvol 2004 of Lecture Notes in Mathematics Springer-VerlagBerlin Germany 2010

[6] Y-M Chen Y-Q Wei D-Y Liu D Boutat and X-K ChenldquoVariable-order fractional numerical differentiation for noisysignals by wavelet denoisingrdquo Journal of Computational Physicsvol 311 pp 338ndash347 2016

[7] A Guerrero and M A Moreles ldquoOn the numerical solutionof the eigenvalue problem in fractional quantum mechanicsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 20 no 2 pp 604ndash613 2015

[8] R T Baillie ldquoLongmemory processes and fractional integrationin econometricsrdquo Journal of Econometrics vol 73 no 1 pp 5ndash59 1996

[9] V V Kulish and J L Lage ldquoApplication of fractional calculus tofluid mechanicsrdquo Journal of Fluids Engineering vol 124 no 3pp 803ndash806 2002

[10] V E Tarasov ldquoFractional integro-differential equations forelectromagnetic waves in dielectric mediardquo Theoretical andMathematical Physics vol 158 no 3 pp 355ndash359 2009

[11] T Liu ldquoA wavelet multiscale method for the inverse problem ofa nonlinear convectionndashdiffusion equationrdquo Journal of Compu-tational and Applied Mathematics vol 330 pp 165ndash176 2018

[12] M H Heydari M R Hooshmandasl and F M M GhainildquoWavelets method for the time fractional diffusion-wave equa-tionrdquo Physics Letters A vol 379 no 3 pp 71ndash76 2015

[13] Y Chen YWu Y Cui ZWang andD Jin ldquoWaveletmethod fora class of fractional convection-diffusion equation with variablecoefficientsrdquo Journal of Computational Science vol 1 no 3 pp146ndash149 2010

[14] G Hariharan and K Kannan ldquoReview of wavelet methodsfor the solution of reaction-diffusion problems in science andengineeringrdquoAppliedMathematicalModelling vol 38 no 3 pp799ndash813 2014

[15] M Yi Y Ma and L Wang ldquoAn efficient method based onthe second kind Chebyshev wavelets for solving variable-orderfractional convection diffusion equationsrdquo International Journalof Computer Mathematics pp 1ndash19 2017

[16] F Zhou and X Xu ldquoThe third kind Chebyshev waveletscollocation method for solving the time-fractional convectiondiffusion equations with variable coefficientsrdquo Applied Mathe-matics and Computation vol 280 pp 11ndash29 2016

[17] MHHeydariMRHooshmandasl C Cattani andGHariha-ran ldquoAn optimization wavelet method for multi variable-orderfractional differential equationsrdquoFundamenta Informaticae vol151 no 1-4 pp 255ndash273 2017

[18] M Behroozifar and A Sazmand ldquoAn approximate solutionbased on Jacobi polynomials for time-fractional convection-diffusion equationrdquoAppliedMathematics and Computation vol296 pp 1ndash17 2017

[19] N H Sweilam A M Nagy and A A El-Sayed ldquoSecondkind shiftedChebyshev polynomials for solving space fractionalorder diffusion equationrdquo Chaos Solitons and Fractals vol 73pp 141ndash147 2015

[20] N H Sweilam A M Nagy and A A El-Sayed ldquoOn thenumerical solution of space fractional order diffusion equationvia shifted Chebyshev polynomials of the third kindrdquo Journal ofKing Saud University - Science vol 28 no 1 pp 41ndash47 2016

[21] S Abbasbandy S Kazem M S Alhuthali and H H Alsu-lami ldquoApplication of the operational matrix of fractional-orderLegendre functions for solving the time-fractional convection-diffusion equationrdquoAppliedMathematics and Computation vol266 pp 31ndash40 2015

[22] Y Yang Y Chen Y Huang and H Wei ldquoSpectral collocationmethod for the time-fractional diffusion-wave equation andconvergence analysisrdquo Computers and Mathematics with Appli-cations vol 73 no 6 pp 1218ndash1232 2017

[23] E Pindza and K M Owolabi ldquoFourier spectral method forhigher order space fractional reaction-diffusion equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 40 pp 112ndash128 2016

[24] F Zeng and C Li ldquoA new Crank-Nicolson finite elementmethod for the time-fractional subdiffusion equationrdquo AppliedNumerical Mathematics vol 121 pp 82ndash95 2017


[25] M H Heydari M R Hooshmandasl G B Loghmani and CCattani ldquoWavelets Galerkin method for solving stochastic heatequationrdquo International Journal of Computer Mathematics vol93 no 9 pp 1579ndash1596 2016

[26] K Burrage A Cardone R DrsquoAmbrosio and B PaternosterldquoNumerical solution of time fractional diffusion systemsrdquoApplied Numerical Mathematics vol 116 pp 82ndash94 2017

[27] V G Pimenov A S Hendy and R H De Staelen ldquoOn aclass of non-linear delay distributed order fractional diffusionequationsrdquo Journal of Computational and Applied Mathematicsvol 318 pp 433ndash443 2017

[28] B Zogheib and E Tohidi ldquoA new matrix method for solv-ing two-dimensional time-dependent diffusion equations withDirichlet boundary conditionsrdquoAppliedMathematics and Com-putation vol 291 pp 1ndash13 2016

[29] F Zhou and X Xu ldquoNumerical solution of the convectiondiffusion equations by the second kind Chebyshev waveletsrdquoApplied Mathematics and Computation vol 247 pp 353ndash3672014

[30] J Xie Q Huang and X Yang ldquoNumerical solution of the one-dimensional fractional convection diffusion equations based onChebyshev operationalmatrixrdquo SpringerPlus vol 5 no 1 articleno 1149 2016

[31] Q Chen Z Dong YMa et al ldquoTest thermo-physical propertiesof solid material based on one dimensional unsteady heattransfer model in constant temperature boundary conditionrdquoJournal of Central South University vol 46 no 12 pp 4686ndash4692 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


sample t

x= 0

t

y= 0 t

y= 0

0 y

x

T = tw

Figure 1 Nonsteady heat transfer model with constant temperatureboundary condition

and the lower surface is contacted with the fluid which itstemperature is 119905119908 The heat conductivity coefficient of thesample is120582 the density is120588 and the specific heat capacity is 119888119901The initial temperature is 1199050 taking the origin of coordinateson the sample adiabatic surfaces and the nonsteady heattransfer model with the initial-boundary condition can bedefined as follows [31]

120597119905120597120591 = 12058212059721199051205881198881199011205971199092 120591 = 0119905 = 1199050119909 = 0120597119905120597119909 = 0119909 = 120590119905 = 119905119908

(1)

Obviously when the sample density120588 heat conductivity coef-ficient 120582 specific heat capacity 119888119901 and thickness 120575 are knownwe can obtain the temperature distribution at any position119909 and any time 120591 which is the nonsteady heat conductionmodel with constant temperature boundary condition Basedon the above-mentioned model we give the fractional-ordernonsteady heat transfer model of the following form

120597119879120597119905 = 120582120597120572119879120588119888119901120597119909120572 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 1 lt 120572 le 2

(2)

with the initial condition

119879 (119909 0) = 119891 (119909) 0 le 119909 le 1 (3)

and the boundary conditions

119879 (0 119905) = 1198920 (119905) 119879 (1 119905) = 1198921 (119905)

0 le 119905 le 1(4)

where 119892(119909 119905) denotes source term 119891(119909) is a given functionand 1198920(119905) 1198921(119905) are continuous functions with first-orderderivative

3 Preliminaries of the Fractional Calculus

In this section we give some necessary definitions andmathematical preliminaries on fractional calculus which willbe used further in this paper

Definition 1 TheRiemann-Liouville fractional integral oper-ator 119868120572 (120572 gt 0) of a function 119891(119905) is defined as follows [4]

119868120572119891 (119905) = 1Γ (120572) int119905

0(119905 minus 120591)120572minus1 119891 (120591) d120591

120572 gt 0 120572 isin R+(5)

Some properties of the operator 119868120572 are as follows119868120572119868120573119891 (119905) = 119868120572+120573119891 (119905) (120572 gt 0 120573 gt 0) (6)

119868120572119905120574 = Γ (1 + 120574)Γ (1 + 120574 + 120572)119905120572+120574 (120574 gt minus1) (7)

Definition 2 The Caputo fractional derivative 0119863120572119905 of afunction 119891(119905) is defined as follows [4]

0119863120572119905 119891 (119905) = 1Γ (119899 minus 120572) int119905

0

119891119899 (120591)(119905 minus 120591)119899minus120572+1 d120591

(119899 minus 1 lt 120572 le 119899 119899 isin 119873) (8)

Some properties of the Caputo fractional derivative are asfollows

0119863120572119905 119905120573 = Γ (1 + 120573)Γ (1 + 120573 minus 120572)119905120573minus120572

0 lt 120572 lt 120573 + 1 120573 gt minus1119868120572119863120572119891 (119905) = 119891 (119905) minus 119899minus1sum

119896=0

119891(119896) (0+) 119905119896119896 119899 minus 1 lt 120572 le 119899 119899 isin 119873

(9)

4 The Second Kind ChebyshevWavelet and Its Operational Matrix ofFractional Integration

41 The Second Kind Chebyshev Wavelet and Its PropertiesThe second kind Chebyshev wavelet 120595119899119898(119905) = 120595(119896 119899119898 119905)has four arguments 119899 = 1 2 2119896minus1 119896 isin 119873lowast They aredefined on the interval [0 1) as follows [19]120595119899119898 (119905)=

21198962119898 (2119896119905 minus 2119899 + 1) 119899 minus 12119896minus1 le 119905 lt 1198992119896minus1 0 ow(10)

with

119898 (119905) = radic 2120587119880119898 (119905) 119898 = 0 1 2 119872 minus 1 (11)



1198800 (119905) = 11198801 (119905) = 2119905

119880119898+1 (119905) = 2119905119880119898 (119905) minus 119880119898minus1 (119905) 119898 = 1 2 (12)


119891 (119905) ≃ 2119896minus1

sum119899=1

119872minus1sum119898=0

119888119899119898120595119899119898 (119905) = 119862119879Ψ (119905) (13)

where

119888119899119898 = (119891 (119905) 120595119899119898 (119905))120596119899 = int1

0120596119899 (119905) 120595119899119898 (119905) d119905 (14)




(15)




Φtimes = [Ψ( 12) Ψ ( 32) Ψ (2 minus 12 )] (17)


119879 (119909 119905) ≃ sum119894=1

sum119895=1

119889119894119895120595119894 (119909) 120595119895 (119905) = Ψ119879 (119909)119863Ψ (119905) (18)





119891 (119905) = infinsum119899=0


119888119899119898120595119899119898 (119905) (19)

where 119888119899119898 = ⟨119891(119905) 120595119899119898(119905)⟩1198712120596[01)

(2)


11198995infinsum119898=119872

1(119898 minus 1)4)

12 (20)

where 120590119891119896119872 = (int10|119891(119905) minus


119887119894 (119905) = 1 119898 le 119905 lt 119894 + 1 0 ow 119894 = 0 1 2 minus 1 (21)


119887119894 (119905) 119887119895 (119905) = 0 119894 = 119895119887119894 (119905) 119894 = 119895

int10119887119894 (119904) 119887119895 (119904) d119904 =

0 119894 = 1198951119898 119894 = 119895

(22)


Ψ (119905) = Φtimes119861 (119905) (23)


(119868120572119861) (119905) asymp 119865120572119861 (119905) (24)

where

119861 (119905) = [1198870 (119905) 1198871 (119905) 119887minus1 (119905)]119879

119865120572 = 1119898120572 1Γ (120572 + 2)

[[[[[[[[[[[[[



]]]]]]]]]]]]]

(25)


(119868120572Ψ) (119905) = 119875120572timesΨ (119905) (26)







12059731198791205971199051205971199092 = Ψ119879 (119909)119863Ψ (119905) (28)


12059721198791205971199092 = 12059721198791205971199092100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ

119879 (119909)119863119875timesΨ (119905) (29)


12059721198791205971199092 = 11989110158401015840 (119909) + Ψ119879 (119909)119863119875timesΨ (119905) (30)

Then we have

120597120572119879120597119909120572 = 1198682minus120572119909 (12059721198791205971199092 )= 1198682minus120572119909 ( 12059721198791205971199092

100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ119879 (119909)119863119875timesΨ (119905))




(32)



(33)

where


(34)


+ Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905) (35)

where


(36)

10806

x040200

05t

10

002

004

006

008

Ana

lytic

al so

lutio

n




119894 119895 = 1 2 3

(37)





120597119879120597119905 = 1205821205971511987912058811988811990112059711990915 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (38)



10806

x

040200

05t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

3


10806x

04020005t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

4


10806

x040200

05t

10

002

004

006

008

Num

eric

al so

lutio

nw

ithk=

M=

5




120597119879120597119905 = 1205971811987912059711990918 + 21199092119905 minus 2119909

021199052Γ (12) 0 le 119909 le 1 119905 ge 0 (39)



10806

x

040200

05t

1minus1

minus05

0

05

1

Ana

lytic

al so

lutio

n


10806

x

040200

05t

1

Num

eric

al so

lutio

nminus1

minus050

051

15

with

k=

3



0 le 119909 le 1 119905 gt 0(40)



120597119879120597119905 = 12058212059721198791205881198881199011205971199092 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (41)




Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






1198800 (119905) = 11198801 (119905) = 2119905

119880119898+1 (119905) = 2119905119880119898 (119905) minus 119880119898minus1 (119905) 119898 = 1 2 (12)


119891 (119905) ≃ 2119896minus1

sum119899=1

119872minus1sum119898=0

119888119899119898120595119899119898 (119905) = 119862119879Ψ (119905) (13)

where

119888119899119898 = (119891 (119905) 120595119899119898 (119905))120596119899 = int1

0120596119899 (119905) 120595119899119898 (119905) d119905 (14)




(15)




Φtimes = [Ψ( 12) Ψ ( 32) Ψ (2 minus 12 )] (17)


119879 (119909 119905) ≃ sum119894=1

sum119895=1

119889119894119895120595119894 (119909) 120595119895 (119905) = Ψ119879 (119909)119863Ψ (119905) (18)





119891 (119905) = infinsum119899=0


119888119899119898120595119899119898 (119905) (19)

where 119888119899119898 = ⟨119891(119905) 120595119899119898(119905)⟩1198712120596[01)

(2)


11198995infinsum119898=119872

1(119898 minus 1)4)

12 (20)

where 120590119891119896119872 = (int10|119891(119905) minus


119887119894 (119905) = 1 119898 le 119905 lt 119894 + 1 0 ow 119894 = 0 1 2 minus 1 (21)


119887119894 (119905) 119887119895 (119905) = 0 119894 = 119895119887119894 (119905) 119894 = 119895

int10119887119894 (119904) 119887119895 (119904) d119904 =

0 119894 = 1198951119898 119894 = 119895

(22)


Ψ (119905) = Φtimes119861 (119905) (23)


(119868120572119861) (119905) asymp 119865120572119861 (119905) (24)

where

119861 (119905) = [1198870 (119905) 1198871 (119905) 119887minus1 (119905)]119879

119865120572 = 1119898120572 1Γ (120572 + 2)

[[[[[[[[[[[[[



]]]]]]]]]]]]]

(25)


(119868120572Ψ) (119905) = 119875120572timesΨ (119905) (26)







12059731198791205971199051205971199092 = Ψ119879 (119909)119863Ψ (119905) (28)


12059721198791205971199092 = 12059721198791205971199092100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ

119879 (119909)119863119875timesΨ (119905) (29)


12059721198791205971199092 = 11989110158401015840 (119909) + Ψ119879 (119909)119863119875timesΨ (119905) (30)

Then we have

120597120572119879120597119909120572 = 1198682minus120572119909 (12059721198791205971199092 )= 1198682minus120572119909 ( 12059721198791205971199092

100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ119879 (119909)119863119875timesΨ (119905))




(32)



(33)

where


(34)


+ Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905) (35)

where


(36)

10806

x040200

05t

10

002

004

006

008

Ana

lytic

al so

lutio

n




119894 119895 = 1 2 3

(37)





120597119879120597119905 = 1205821205971511987912058811988811990112059711990915 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (38)



10806

x

040200

05t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

3


10806x

04020005t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

4


10806

x040200

05t

10

002

004

006

008

Num

eric

al so

lutio

nw

ithk=

M=

5




120597119879120597119905 = 1205971811987912059711990918 + 21199092119905 minus 2119909

021199052Γ (12) 0 le 119909 le 1 119905 ge 0 (39)



10806

x

040200

05t

1minus1

minus05

0

05

1

Ana

lytic

al so

lutio

n


10806

x

040200

05t

1

Num

eric

al so

lutio

nminus1

minus050

051

15

with

k=

3



0 le 119909 le 1 119905 gt 0(40)



120597119879120597119905 = 12058212059721198791205881198881199011205971199092 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (41)




Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







12059731198791205971199051205971199092 = Ψ119879 (119909)119863Ψ (119905) (28)


12059721198791205971199092 = 12059721198791205971199092100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ

119879 (119909)119863119875timesΨ (119905) (29)


12059721198791205971199092 = 11989110158401015840 (119909) + Ψ119879 (119909)119863119875timesΨ (119905) (30)

Then we have

120597120572119879120597119909120572 = 1198682minus120572119909 (12059721198791205971199092 )= 1198682minus120572119909 ( 12059721198791205971199092

100381610038161003816100381610038161003816100381610038161003816119905=0 + Ψ119879 (119909)119863119875timesΨ (119905))




(32)



(33)

where


(34)


+ Ψ119879 (119909) (1198752times)119879119863119875timesΨ (119905) (35)

where


(36)

10806

x040200

05t

10

002

004

006

008

Ana

lytic

al so

lutio

n




119894 119895 = 1 2 3

(37)





120597119879120597119905 = 1205821205971511987912058811988811990112059711990915 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (38)



10806

x

040200

05t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

3


10806x

04020005t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

4


10806

x040200

05t

10

002

004

006

008

Num

eric

al so

lutio

nw

ithk=

M=

5




120597119879120597119905 = 1205971811987912059711990918 + 21199092119905 minus 2119909

021199052Γ (12) 0 le 119909 le 1 119905 ge 0 (39)



10806

x

040200

05t

1minus1

minus05

0

05

1

Ana

lytic

al so

lutio

n


10806

x

040200

05t

1

Num

eric

al so

lutio

nminus1

minus050

051

15

with

k=

3



0 le 119909 le 1 119905 gt 0(40)



120597119879120597119905 = 12058212059721198791205881198881199011205971199092 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (41)




Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





10806

x

040200

05t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

3


10806x

04020005t

1

Num

eric

al so

lutio

n

0

002

004

006

008

with

k=

M=

4


10806

x040200

05t

10

002

004

006

008

Num

eric

al so

lutio

nw

ithk=

M=

5




120597119879120597119905 = 1205971811987912059711990918 + 21199092119905 minus 2119909

021199052Γ (12) 0 le 119909 le 1 119905 ge 0 (39)



10806

x

040200

05t

1minus1

minus05

0

05

1

Ana

lytic

al so

lutio

n


10806

x

040200

05t

1

Num

eric

al so

lutio

nminus1

minus050

051

15

with

k=

3



0 le 119909 le 1 119905 gt 0(40)



120597119879120597119905 = 12058212059721198791205881198881199011205971199092 + 119892 (119909 119905) 0 le 119909 le 1 119905 ge 0 (41)




Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table1Th

enum

ericalsolutio

nsob

tained

byou

rmetho

dandthoseo

btainedby

CASwavele

tmetho

dwhen119896=

119872=3119896

=119872=4

119896=119872=

5119905

119909AnalSol

119896=119872=

3119896=

119872=4

119896=119872=

5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t

02

03

0003600

0000362673

001527126

000360257

000

471281

000360019

000382719

06

00144

000

00144

5390

003638127

00144

0370

001673180

00144

0048

001492319

09

0032400

0003248217

007371928

003240631

003631963

0032400

60003273187

05

03

0022500

0002253176

004

872121

0022504

87002826189

0022500

46002293819

06

00900

000

009061074

012739812

00900

6721

009537428

00900

0059

009072347

09

0202500

0020257431

025873179

020250850

020736183

020250074

020301829

08

03

00576000

005765362

009381981

0057604

89060121872

005760062

005830218

06

02304

000

02304

8904

028237189

02304

0790

023833829

02304

0074

023138192

09

0518400

0051851904

060381038

051841027

052478172

051840112

051953785


Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Table2Th

eabsolutee

rrorso

btainedby

ourm

etho

dandCA

Swavele

tmetho

dwhen119872=

3119872=4

119872=5

(119909119905)

AnalSol

119896=3119872

=3119896=

3119872=4

119896=3119872

=5Our

metho

dCA

Swavele

tOur

metho

dCA

Swavelet

Our

metho

dCA

Swavele

t(00

)10

0000

000

1627162119890minus

42381923119890minus2

8719295119890minus

6873

7819119890minus

4231

9280119890minus

6264

8278119890minus

4(0101)

109483758

1738173119890minus

42731899119890minus2

5371912119890minus

6627

1928119890minus

4284

2802119890minus

6374

8217119890minus

4(0202)

117873590

2371827119890minus4

3759289119890minus2

2361827119890minus

5327

1929119890minus

3483

0209119890minus

6468

4278119890minus

4(0303)

125085669

2731872119890minus4

4542767119890minus2

4731872119890minus

5428

1912119890minus

3537

1982119890minus

6647

2938119890minus

4(0404)

131047933

3261772119890minus4

5251757119890minus2

5219289119890minus

5538

1018119890minus

3738

1928119890minus

7786

3982119890minus

4(0505)

135700810

8271985119890minus5

4378391119890minus2

6319288119890minus

5637

9843119890minus

3623

8299119890minus

6763

5176119890minus

4(0606)

138997808

4268278119890minus4

8373456119890minus3

5738273119890minus

5579

2808119890minus

3830

2930119890minus

6836

8386119890minus

4(0707)

140905987

4791982119890minus4

7371928119890minus

2738

2093119890minus

5772

8732119890minus

3198

3100119890minus

5967

3817119890minus

4(0808)

141406280

5281928119890minus4

6367643119890minus2

8382938119890minus

5873

2763119890minus

3938

1098119890minus

6237

1927119890minus

3(0909)

140493687

6782916119890minus4

7371892119890minus

2938

1982119890minus

5973

8273119890minus

3238

1983119890minus

5428

1988119890minus

3(1010)

138177329

9381928119890minus

48263828119890minus2

9983787119890minus

5942

5146119890minus

3331

3910119890minus

5887

1999119890minus

4


10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





10806

x040200

05t

1N

umer

ical

solu

tion

minus1minus05

005

115

with

k=

4


10806

x

040200

05t

1

Num

eric

al so

lutio

n

minus1minus05

005

115

with

k=

5


minus02

minus01

0

01

02

03

04

05

06

Num

er S

ol

minus045

minus04

minus035

minus03

minus025

minus02

minus015

minus01

minus005

0

Num

er S

ol

minus12

minus1

minus08

minus06

minus04

minus02

0

Num

er S

ol

05 10x

05 10x

05 10x

t = 095t = 06t = 03

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17

= 2

= 19

= 18

= 17




7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






7 Conclusions




Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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