Research ArticleIdentifying an Unknown Coefficient inthe Reaction-Diffusion Equation Using Hersquos VIM

F Parzlivand and A M Shahrezaee

Department of Mathematics Alzahra University Vanak Tehran 19834 Iran

Correspondence should be addressed to A M Shahrezaee ashahrezaeealzahraacir

Received 16 July 2014 Accepted 2 October 2014 Published 19 October 2014

Academic Editor Delin Chu

Copyright copy 2014 F Parzlivand and A M Shahrezaee This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

An inverse heat problem of finding an unknown parameter p(t) in the parabolic initial-boundary value problem is solved withvariational iteration method (VIM) For solving the discussed inverse problem at first we transform it into a nonlinear directproblem and then use the proposed method Also an error analysis is presented for the method and prior and posterior errorbounds of the approximate solution are estimatedThemain property of the method is in its flexibility and ability to solve nonlinearequation accurately and conveniently Some examples are given to illustrate the effectiveness and convenience of the method

1 Introduction

The parameter determination in a parabolic partial differ-ential equation from the overspecified data plays a crucialrole in applied mathematics and physics This technique hasbeen widely used to determine the unknown properties of aregion by measuring data only on its boundary or a specifiedlocation in the domain These unknown properties suchas the conductivity medium are important to the physicalprocess but they usually cannot be measured directly or theprocess of their measurement is very expensive [1] Theseproblems are often ill-posed [2] and a direct inversion of thedata for the unknown function is not possible As a resulta number of researchers have developed various methodsto overcome the ill-posed nature of the inversion problemThese methods include Born approximation [3] neural-networks [4] and Levenberg-Marquardt method [5]

In this paper we solve an inverse problem to a classof reaction-diffusion equation using variational iterationmethod The method is capable of reducing the size of calcu-lations and handles both linear and nonlinear equationshomogeneous or inhomogeneous in a direct manner Themethod gives the solution in the form of a rapidly con-vergent successive approximation that may give the exactsolution if such a solution exists For concrete problemswhere exact solution is not obtainable it was found that a

small number of approximations can be used for numericalpurposes

11 Reaction-Diffusion Systems Reaction-diffusion systemsare mathematical models which explain how the concen-tration of one or more substances distributed in spacechanges under the influence of two processes local chemicalreactions in which the substances are transformed into eachother and diffusion which causes the substances to spreadout over a surface in space This description implies thatreaction-diffusion systems are naturally applied in chemistryHowever the system can also describe dynamical processesof nonchemical nature Examples are found in biology geol-ogy physics and ecology [6 7] Mathematically reaction-diffusion systems take the form of semilinear parabolicpartial differential equations They can be represented in thegeneral form

119906119905= nabla sdot (119863 (119909 119905 119906) nabla119906) + 119891 (119906 nabla119906 119909 119905) (1)

where 119906 = 119906(119909 119905) represents the concentration of onesubstance 119863 is a diffusion coefficient and 119891 is the reactionterm

12 A Class of Reaction-Diffusion Systems In this paperwe consider an inverse problem of simultaneously finding

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2014 Article ID 547973 8 pageshttpdxdoiorg1011552014547973

2 Advances in Numerical Analysis

unknown coefficients 119901(119905) and 119906(119909 119905) satisfying reaction-diffusion equation

119906119905= 119886 (119905) 119906

119909119909+ 119901 (119905) 119906

119909+ 119891 (119909 119905) in 119876

119879 (2)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(3)

where 119906(119909 119905) is the concentration 119886(119905) is a diffusion coef-ficient 119901(119905)119906

119909(119909 119905) + 119891(119909 119905) is the reaction term 119876

119879=

(119909 119905) 0 lt 119909 lt 1 0 lt 119905 lt 119879 119879 gt 0 1198611and 119861

2are boundary

operators (ie 1198611 1198612

= 120597119894120597119909119894 119894 = 0 or 1) and 119886 119891 119906

0 1198920

and 1198921are known functions

An additional boundary condition which can be theintegral overspecification is given in the following form

int

119904(119905)

0

119896 (119909) 119906 (119909 119905) 119889119909 = 119864 (119905) 0 lt 119905 le 119879 0 lt 119904 (119905) lt 1

(4)

where 119864 119904 and 119896 are known functions and for some 120588 gt 0the kernel 119896(119909) satisfies

int

1

0

|119896 (119909)| 119889119909 le 120588 (5)

Without the extrameasurement (4) problem (2)-(3) is under-determined and may have many solutions infinitely On theother hand if too many additional conditions are imposedthe solution may not exist The existence and uniquenessof the solution of this problem and more applications andbackground of the problem are discussed in [8ndash10] Also tointerpret integral equation (4) the reader can refer to [11]

13 A Brief Discussion on Some Inverse Problems Inverseproblems in partial differential equations can be used tomodel many real problems in engineering and other physicalsciences (cf [12 13] for examples) Studying these prob-lems has a great deal of importance both theoretically andpractically We give a quick review of the previous workrelated to our problems For the inverse problem (2)-(3) ofsimultaneously finding unknown coefficients 119886(119905) and 119906(119909 119905)

with 119901(119905) = 119891(119909 119905) = 0 and additional condition

119886 (119905) 119906119909(0 119905) = 119864 (119905) 0 le 119905 le 119879 (6)

Jones [14] obtains the global solvability via an integralequation and Schauder fixed point theorem Moreover in[15] the numerical solution by the finite difference methodis discussed Similar problems in different regions are alsotreated in [16 17] By employing the heat potential andGreenrsquosrepresentation the authors in [18] give the explicit solutionfor inverse problem of finding 119891(119905) in the equation

119906119905= 119906119909119909

+ 119891 (119905) (7)

These examples motivate us to consider the following generalproblem

119906119905= 119886 (119909 119905 119906 119906

119909 119906119909(119909⋆ 119905)) 119906119909119909

+ 119887 (119909 119905 119906 119906119909 119906119909(119909⋆ 119905))

(8)

subject to initial-boundary conditions (3) where 0 lt 119909⋆

lt 1

is a fixed point The global solvability for this problem isestablished in [19]

This paper is organized as follows In Section 2 thevariational iteration method is recapitulated In Section 3an error analysis is presented for the proposed method andprior and posterior error bounds of the approximate solutionare estimated In Section 3 at first we transform the inverseproblem (2)ndash(4) into a nonlinear direct problem and thenthe VIM is used for solving this problem To present a clearoverview of the new method some examples are given inSection 4 A conclusion is presented in the last section

2 Basic Idea of the VariationalIteration Method

The variational iteration method is a powerful tool to searchfor both analytical and approximate solutions of nonlinearequation without requirement of linearization or perturba-tion [20 21] The method was first proposed by He in 1998[22] Also it was successfully applied to various engineeringproblems [23ndash27] In 1978 Inokuti et al [28] proposed ageneral Lagrange multiplier method to solve nonlinear prob-lems which was first proposed to solve problems in quantummechanics (see [28] and the references cited therein) Themain feature of the method is as follows the solution of amathematical problem with linearization assumption is usedas initial approximation or trial-function then a more highlyprecise approximation at some special point can be obtained

Consider the following general nonlinear system

119871119906 (119905) + 119873119906 (119905) = 119891 (119905) (9)

where 119871 and 119873 are linear and nonlinear operators respec-tively and 119891 is source or sink term

Assuming 1199060(119905) is the solution of 119871119906 = 0 according to

[28] we can write down an expression to correct the value ofsome special point for example at 119905 = 1

119906cor (1) = 1199060(1) + int

1

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

(10)

where 120582 is general Lagrange multiplier [28] which canbe identified optimally via the variational theory [29] Thesecond term on the right is called the correction He in 1998[22] modified the above method into an iteration method[20ndash22] in the following way

119906119899+1

(1199050) = 119906119899(1199050) + int

1199050

0

120582 (120591) 119871119906119899(120591) + 119873

119899(120591) minus 119891 (120591) 119889120591

(11)

The subscript 119899 denotes the 119899th order approximation and 119899is

considered as restricted variation [29] which means 120575119899= 0

Advances in Numerical Analysis 3

For arbitrary 1199050 we can rewrite the above equation as

follows119906119899+1

(119905) = 119906119899(119905)

+ int

119905

0

120582 (120591) 119871119906119899(120591) + 119873

119899(120591) minus 119891 (120591) 119889120591 119899 ⩾ 0

(12)

Equation (12) is called a correction functional It is requiredfirst to determine the Lagrange multiplier Employing therestricted variation in correction functional and using inte-gration by part make it easy to compute the Lagrangemultiplier see for instance [30] For linear problems itsexact solution can be obtained by only one iteration stepdue to the fact that no nonlinear operator exists so theLagrange multiplier can be exactly identified Having 120582

determined then several approximations 119906119899+1

(119905) 119899 ⩾ 0 canbe determined Under reasonable choice of 119906

0 the fixed point

of the correction functional is considered as an approximatesolution of the above general differential equation Generallyone iteration leads to high accurate solution by variationaliteration method if the initial solution is carefully chosenwith some unknown parameters Comparison of the methodwith Adomian method was conducted by many authors viaillustrative examples and especially Wazwaz gave a completecomparison between the two methods [31] revealing thevariational iteration method has many merits over Adomianmethod it can completely overcome the difficulty arising inthe calculation of Adomian polynomial

3 Convergence Analysis andError Bound of VIM

This section covers the error analysis of the proposedmethodAlso the sufficient conditions are presented to guarantee theconvergence of VIM when applied to solve the differentialequations

31 Convergence Analysis First we will rewrite (12) in theoperator form as follows

119906119899+1

= 119860 [119906119899] (13)

where the operator 119860 takes the following form

119860 [119906 (119905)] = 119906 (119905) + int

119905

0

120582 (120591) 119871119906 (120591) + 119873119906 (120591) minus 119891 (120591) 119889120591

(14)

Theorem 1 Let (119883 sdot ) be a Banach space and 119860 119883 rarr 119883

is a nonlinear mapping and suppose that

119860 [119906] minus 119860 [] le 120574 119906 minus 119906 isin 119883 (15)

for some constant 120574 lt 1 Then 119860 has a unique fixed pointFurthermore sequence (12) using VIMwith an arbitrary choiceof 1199060isin 119883 converges to the fixed point of 119860 and

1003817100381710038171003817119906119899minus 119906119898

1003817100381710038171003817le

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

119899minus2

sum

119895=119898minus1

120574119895 (16)

Proof (see [32]) According to the above theorem a sufficientcondition for the convergence of the variational iterationmethod is strictly contraction of 119860 Furthermore sequence(12) converges to the fixed method of 119860 which is also thesolution of (9) Also the rate of convergence depends on120574

32 Approximation Error In the following theorem weintroduce an estimation of the error of the approximatesolution of problem (9) and prior and posterior error boundsof the approximate solution are estimated The prior errorbound can be used at the beginning of a calculation forestimating the number of steps necessary to obtain a givenaccuracy and the posterior error bound can be used atintermediate stages or at the end of a calculation

Theorem 2 Under the conditions of Theorem 1 error esti-mates are the prior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(17)

and the posterior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817 (18)

Also if one supposes that 119860[0] = 0 then the error of theapproximate solution 119906

119899to problem (9) can be obtained as

follows

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

1 + 120574

1 minus 120574

120574119899 10038171003817100381710038171199060

1003817100381710038171003817 (19)

Proof Consider

1003817100381710038171003817119906119899+1

minus 119906119899

1003817100381710038171003817=

1003817100381710038171003817119860 [119906119899] minus 119860 [119906

119899minus1]1003817100381710038171003817

le 1205741003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817

le 1205742 1003817100381710038171003817119906119899minus1

minus 119906119899minus2

1003817100381710038171003817

le 120574119899 10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(20)

Therefore for any 119898 gt 119899 we have

119906119898

minus 119906119899 le 119906

119898minus 119906119898minus1

+ 119906119898minus1

minus 119906119898minus2

+ sdot sdot sdot + 119906119899+1

minus 119906119899

le 120574119898minus1

1199061minus 1199060 + 120574119898minus2

1199061minus 1199060

+ sdot sdot sdot + 1205741198991199061minus 1199060

= (120574119898minus1

+ 120574119898minus2

+ sdot sdot sdot + 120574119899) 1199061minus 1199060

= 120574119899 1 minus 120574

119898minus119899

1 minus 120574

1199061minus 1199060

(21)

4 Advances in Numerical Analysis

Since 0 lt 120574 lt 1 in the numerator we have 1 minus 120574119898minus119899

lt 1Consequently

1003817100381710038171003817119906119898

minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817 119898 gt 119899 (22)

The first statement follows from (22) by using Theorem 1 as119898 rarr infin We derive (18) Taking 119899 = 1 and writing V

0for 1199060

and V1for 1199061 we have from (17)

1003817100381710038171003817V1minus 119906

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817V1minus V0

1003817100381710038171003817 (23)

Setting V0= 119906119899minus1

we have V1= 119860[V

0] = 119906119899and obtain (18)

Also we have

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817

int

119905

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

=1003817100381710038171003817119860 [1199060] minus 1199060

1003817100381710038171003817le

1003817100381710038171003817119860 [1199060]1003817100381710038171003817+

10038171003817100381710038171199060

1003817100381710038171003817

(24)

We can write1003817100381710038171003817119860 [1199060]1003817100381710038171003817=

1003817100381710038171003817119860 [1199060] minus 119860 [0]

1003817100381710038171003817le 120574

10038171003817100381710038171199060

1003817100381710038171003817 (25)

Therefore10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le (120574 + 1)

10038171003817100381710038171199060

1003817100381710038171003817 (26)

From Theorem 1 as 119898 rarr infin then 119906119898

rarr 119906 and by using(22)-(26) inequality (19) can be obtained

4 The Application of VIM in Inverse ParabolicProblem (2)ndash(4)

To use the variational iteration method for solving theproblem (2)ndash(4) at first we use the following transformation

41 The Employed Transformation By differentiation withrespect to the variable 119905 in (4) one obtains

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905) + int

119904(119905)

0

119896 (119909) 119906119905(119909 119905) 119889119909

(27)

Substituting (2) into the above equation yields

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

+ int

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119901 (119905) 119906119909(119909 119905) + 119891 (119909 119905)] 119889119909

(28)

and it follows that

119901 (119905) = (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896(119909)119906119909(119909 119905)119889119909)

minus1

(29)

provided that for any 119905 isin [0 119879] int

119904(119905)

0119896(119909)119906

119909(119909 119905)119889119909 and

int

119904(119905)

0119896(119909)[119906

119909119909(119909 119905) + 119891(119909 119905)]119889119909 are exit also int

119904(119905)

0119896(119909)119906

119909

(119909 119905)119889119909 = 0Therefore the inverse parabolic problem (2)ndash(4) is equiv-

alent to the following nonlocal parabolic equation

119906119905= 119906119909119909

+ (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896 (119909) 119906119909(119909 119905) 119889119909)

minus1

119906119909

+ 119891 (119909 119905) in 119876119879

(30)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(31)

Therefore for solving the inverse problem (2)ndash(4) we willinvestigate the direct problem (30)-(31)

42 Application In order to solve problem (30)-(31) usingVIM we can write the following correction functional

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

+ int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(32)

Advances in Numerical Analysis 5

Making the above correction functional stationary note that120575119906119899(0) = 0 we have

120575119906119899+1

(119909 119905)

= 120575119906119899(119909 119905)

+ 120575int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896(119909)119906119899119909

(119909 120591)119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(33)

Thus its stationary condition can be obtained as follows

1205821015840(119905 120591) = 0

1 + 120582(119905 120591)|120591=119905

= 0

(34)

therefore 120582(119905 120591) = minus1Now the following iteration formula can be obtained as

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

minus int

119905

0

119906119899120591

(119909 120591) minus 119906119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

119906119899119909

(119909 120591)

minus 119891 (119909 120591)

119889120591

(35)

Here according to Adomian method we choose its initialapproximate solution as 119906

0(119909 119905) = 119906(119909 0)

Having 119906 = lim119899rarrinfin

119906119899determined then the value of119901(119905)

can be computed by using (29)

5 Illustrative Examples

In this section three examples are presented to demonstratethe applicability and accuracy of the method These tests arechosen such that their analytical solutions are known Butthe method developed in this research can be applied tomore complicated problems The numerical implementationis carried out in Microsoft Maple13

Example 1 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909+ 4119905 (

1

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 0 le 119909 le 1

119906 (0 119905) = 21199052minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119906 (119909 119905) 119889119909 = 21199052minus 2119905 minus

1

12

0 le 119905 le 1

(36)

The true solution is 119906(119909 119905) = (12 minus 119909)119909 + 21199052minus 2119905 while

119901(119905) = 4119905 Let 1199060(119909 119905) = 119906(119909 0) = (12 minus 119909)119909 From (35)

we obtain 1199061(119909 119905) = (12 minus 119909)119909 + 2119905

2minus 2119905 which is the exact

solution Also from (29) we have 119901(119905) = 4119905 which is equal tothe exact 119901(119905) of this example

Example 2 We consider the inverse problem (2)ndash(4) asfollows

119906119905= 119906119909119909

+ 119901 (119905) 119906119909

+ 41198904119905

(

3

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 + 2 0 le 119909 le 1

119906 (0 119905) = 21198904119905

minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119909119906 (119909 119905) 119889119909 = 1198904119905

minus 119905 0 le 119905 le 1

(37)

The exact solution for this example is 119906(119909 119905) = (12 minus 119909)119909 +

21198904119905

minus 2119905 while 119901(119905) = 41198904119905 Let 119906

0(119909 119905) = 119906(119909 0) = (12 minus

119909)119909 + 2Using (35) and (29) we obtain 119906

1(119909 119905) = (12minus119909)119909+2119890

4119905minus

2119905 and 119901(119905) = 41198904119905 respectively which is the exact solution in

this example

Remark 3 From the above two examples it can be seen thatthe exact solution is obtained by using one iteration step only

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Numerical Analysis

unknown coefficients 119901(119905) and 119906(119909 119905) satisfying reaction-diffusion equation

119906119905= 119886 (119905) 119906

119909119909+ 119901 (119905) 119906

119909+ 119891 (119909 119905) in 119876

119879 (2)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(3)

where 119906(119909 119905) is the concentration 119886(119905) is a diffusion coef-ficient 119901(119905)119906

119909(119909 119905) + 119891(119909 119905) is the reaction term 119876

119879=

(119909 119905) 0 lt 119909 lt 1 0 lt 119905 lt 119879 119879 gt 0 1198611and 119861

2are boundary

operators (ie 1198611 1198612

= 120597119894120597119909119894 119894 = 0 or 1) and 119886 119891 119906

0 1198920

and 1198921are known functions

An additional boundary condition which can be theintegral overspecification is given in the following form

int

119904(119905)

0

119896 (119909) 119906 (119909 119905) 119889119909 = 119864 (119905) 0 lt 119905 le 119879 0 lt 119904 (119905) lt 1

(4)

where 119864 119904 and 119896 are known functions and for some 120588 gt 0the kernel 119896(119909) satisfies

int

1

0

|119896 (119909)| 119889119909 le 120588 (5)

Without the extrameasurement (4) problem (2)-(3) is under-determined and may have many solutions infinitely On theother hand if too many additional conditions are imposedthe solution may not exist The existence and uniquenessof the solution of this problem and more applications andbackground of the problem are discussed in [8ndash10] Also tointerpret integral equation (4) the reader can refer to [11]

13 A Brief Discussion on Some Inverse Problems Inverseproblems in partial differential equations can be used tomodel many real problems in engineering and other physicalsciences (cf [12 13] for examples) Studying these prob-lems has a great deal of importance both theoretically andpractically We give a quick review of the previous workrelated to our problems For the inverse problem (2)-(3) ofsimultaneously finding unknown coefficients 119886(119905) and 119906(119909 119905)

with 119901(119905) = 119891(119909 119905) = 0 and additional condition

119886 (119905) 119906119909(0 119905) = 119864 (119905) 0 le 119905 le 119879 (6)

Jones [14] obtains the global solvability via an integralequation and Schauder fixed point theorem Moreover in[15] the numerical solution by the finite difference methodis discussed Similar problems in different regions are alsotreated in [16 17] By employing the heat potential andGreenrsquosrepresentation the authors in [18] give the explicit solutionfor inverse problem of finding 119891(119905) in the equation

119906119905= 119906119909119909

+ 119891 (119905) (7)

These examples motivate us to consider the following generalproblem

119906119905= 119886 (119909 119905 119906 119906

119909 119906119909(119909⋆ 119905)) 119906119909119909

+ 119887 (119909 119905 119906 119906119909 119906119909(119909⋆ 119905))

(8)

subject to initial-boundary conditions (3) where 0 lt 119909⋆

lt 1

is a fixed point The global solvability for this problem isestablished in [19]

This paper is organized as follows In Section 2 thevariational iteration method is recapitulated In Section 3an error analysis is presented for the proposed method andprior and posterior error bounds of the approximate solutionare estimated In Section 3 at first we transform the inverseproblem (2)ndash(4) into a nonlinear direct problem and thenthe VIM is used for solving this problem To present a clearoverview of the new method some examples are given inSection 4 A conclusion is presented in the last section

2 Basic Idea of the VariationalIteration Method

The variational iteration method is a powerful tool to searchfor both analytical and approximate solutions of nonlinearequation without requirement of linearization or perturba-tion [20 21] The method was first proposed by He in 1998[22] Also it was successfully applied to various engineeringproblems [23ndash27] In 1978 Inokuti et al [28] proposed ageneral Lagrange multiplier method to solve nonlinear prob-lems which was first proposed to solve problems in quantummechanics (see [28] and the references cited therein) Themain feature of the method is as follows the solution of amathematical problem with linearization assumption is usedas initial approximation or trial-function then a more highlyprecise approximation at some special point can be obtained

Consider the following general nonlinear system

119871119906 (119905) + 119873119906 (119905) = 119891 (119905) (9)

where 119871 and 119873 are linear and nonlinear operators respec-tively and 119891 is source or sink term

Assuming 1199060(119905) is the solution of 119871119906 = 0 according to

[28] we can write down an expression to correct the value ofsome special point for example at 119905 = 1

119906cor (1) = 1199060(1) + int

1

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

(10)

where 120582 is general Lagrange multiplier [28] which canbe identified optimally via the variational theory [29] Thesecond term on the right is called the correction He in 1998[22] modified the above method into an iteration method[20ndash22] in the following way

119906119899+1

(1199050) = 119906119899(1199050) + int

1199050

0

120582 (120591) 119871119906119899(120591) + 119873

119899(120591) minus 119891 (120591) 119889120591

(11)

The subscript 119899 denotes the 119899th order approximation and 119899is

considered as restricted variation [29] which means 120575119899= 0

Advances in Numerical Analysis 3

For arbitrary 1199050 we can rewrite the above equation as

follows119906119899+1

(119905) = 119906119899(119905)

+ int

119905

0

120582 (120591) 119871119906119899(120591) + 119873

119899(120591) minus 119891 (120591) 119889120591 119899 ⩾ 0

(12)

Equation (12) is called a correction functional It is requiredfirst to determine the Lagrange multiplier Employing therestricted variation in correction functional and using inte-gration by part make it easy to compute the Lagrangemultiplier see for instance [30] For linear problems itsexact solution can be obtained by only one iteration stepdue to the fact that no nonlinear operator exists so theLagrange multiplier can be exactly identified Having 120582

determined then several approximations 119906119899+1

(119905) 119899 ⩾ 0 canbe determined Under reasonable choice of 119906

0 the fixed point

of the correction functional is considered as an approximatesolution of the above general differential equation Generallyone iteration leads to high accurate solution by variationaliteration method if the initial solution is carefully chosenwith some unknown parameters Comparison of the methodwith Adomian method was conducted by many authors viaillustrative examples and especially Wazwaz gave a completecomparison between the two methods [31] revealing thevariational iteration method has many merits over Adomianmethod it can completely overcome the difficulty arising inthe calculation of Adomian polynomial

3 Convergence Analysis andError Bound of VIM

This section covers the error analysis of the proposedmethodAlso the sufficient conditions are presented to guarantee theconvergence of VIM when applied to solve the differentialequations

31 Convergence Analysis First we will rewrite (12) in theoperator form as follows

119906119899+1

= 119860 [119906119899] (13)

where the operator 119860 takes the following form

119860 [119906 (119905)] = 119906 (119905) + int

119905

0

120582 (120591) 119871119906 (120591) + 119873119906 (120591) minus 119891 (120591) 119889120591

(14)

Theorem 1 Let (119883 sdot ) be a Banach space and 119860 119883 rarr 119883

is a nonlinear mapping and suppose that

119860 [119906] minus 119860 [] le 120574 119906 minus 119906 isin 119883 (15)

for some constant 120574 lt 1 Then 119860 has a unique fixed pointFurthermore sequence (12) using VIMwith an arbitrary choiceof 1199060isin 119883 converges to the fixed point of 119860 and

1003817100381710038171003817119906119899minus 119906119898

1003817100381710038171003817le

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

119899minus2

sum

119895=119898minus1

120574119895 (16)

Proof (see [32]) According to the above theorem a sufficientcondition for the convergence of the variational iterationmethod is strictly contraction of 119860 Furthermore sequence(12) converges to the fixed method of 119860 which is also thesolution of (9) Also the rate of convergence depends on120574

32 Approximation Error In the following theorem weintroduce an estimation of the error of the approximatesolution of problem (9) and prior and posterior error boundsof the approximate solution are estimated The prior errorbound can be used at the beginning of a calculation forestimating the number of steps necessary to obtain a givenaccuracy and the posterior error bound can be used atintermediate stages or at the end of a calculation

Theorem 2 Under the conditions of Theorem 1 error esti-mates are the prior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(17)

and the posterior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817 (18)

Also if one supposes that 119860[0] = 0 then the error of theapproximate solution 119906

119899to problem (9) can be obtained as

follows

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

1 + 120574

1 minus 120574

120574119899 10038171003817100381710038171199060

1003817100381710038171003817 (19)

Proof Consider

1003817100381710038171003817119906119899+1

minus 119906119899

1003817100381710038171003817=

1003817100381710038171003817119860 [119906119899] minus 119860 [119906

119899minus1]1003817100381710038171003817

le 1205741003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817

le 1205742 1003817100381710038171003817119906119899minus1

minus 119906119899minus2

1003817100381710038171003817

le 120574119899 10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(20)

Therefore for any 119898 gt 119899 we have

119906119898

minus 119906119899 le 119906

119898minus 119906119898minus1

+ 119906119898minus1

minus 119906119898minus2

+ sdot sdot sdot + 119906119899+1

minus 119906119899

le 120574119898minus1

1199061minus 1199060 + 120574119898minus2

1199061minus 1199060

+ sdot sdot sdot + 1205741198991199061minus 1199060

= (120574119898minus1

+ 120574119898minus2

+ sdot sdot sdot + 120574119899) 1199061minus 1199060

= 120574119899 1 minus 120574

119898minus119899

1 minus 120574

1199061minus 1199060

(21)

4 Advances in Numerical Analysis

Since 0 lt 120574 lt 1 in the numerator we have 1 minus 120574119898minus119899

lt 1Consequently

1003817100381710038171003817119906119898

minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817 119898 gt 119899 (22)

The first statement follows from (22) by using Theorem 1 as119898 rarr infin We derive (18) Taking 119899 = 1 and writing V

0for 1199060

and V1for 1199061 we have from (17)

1003817100381710038171003817V1minus 119906

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817V1minus V0

1003817100381710038171003817 (23)

Setting V0= 119906119899minus1

we have V1= 119860[V

0] = 119906119899and obtain (18)

Also we have

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817

int

119905

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

=1003817100381710038171003817119860 [1199060] minus 1199060

1003817100381710038171003817le

1003817100381710038171003817119860 [1199060]1003817100381710038171003817+

10038171003817100381710038171199060

1003817100381710038171003817

(24)

We can write1003817100381710038171003817119860 [1199060]1003817100381710038171003817=

1003817100381710038171003817119860 [1199060] minus 119860 [0]

1003817100381710038171003817le 120574

10038171003817100381710038171199060

1003817100381710038171003817 (25)

Therefore10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le (120574 + 1)

10038171003817100381710038171199060

1003817100381710038171003817 (26)

From Theorem 1 as 119898 rarr infin then 119906119898

rarr 119906 and by using(22)-(26) inequality (19) can be obtained

4 The Application of VIM in Inverse ParabolicProblem (2)ndash(4)

To use the variational iteration method for solving theproblem (2)ndash(4) at first we use the following transformation

41 The Employed Transformation By differentiation withrespect to the variable 119905 in (4) one obtains

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905) + int

119904(119905)

0

119896 (119909) 119906119905(119909 119905) 119889119909

(27)

Substituting (2) into the above equation yields

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

+ int

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119901 (119905) 119906119909(119909 119905) + 119891 (119909 119905)] 119889119909

(28)

and it follows that

119901 (119905) = (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896(119909)119906119909(119909 119905)119889119909)

minus1

(29)

provided that for any 119905 isin [0 119879] int

119904(119905)

0119896(119909)119906

119909(119909 119905)119889119909 and

int

119904(119905)

0119896(119909)[119906

119909119909(119909 119905) + 119891(119909 119905)]119889119909 are exit also int

119904(119905)

0119896(119909)119906

119909

(119909 119905)119889119909 = 0Therefore the inverse parabolic problem (2)ndash(4) is equiv-

alent to the following nonlocal parabolic equation

119906119905= 119906119909119909

+ (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896 (119909) 119906119909(119909 119905) 119889119909)

minus1

119906119909

+ 119891 (119909 119905) in 119876119879

(30)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(31)

Therefore for solving the inverse problem (2)ndash(4) we willinvestigate the direct problem (30)-(31)

42 Application In order to solve problem (30)-(31) usingVIM we can write the following correction functional

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

+ int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(32)

Advances in Numerical Analysis 5

Making the above correction functional stationary note that120575119906119899(0) = 0 we have

120575119906119899+1

(119909 119905)

= 120575119906119899(119909 119905)

+ 120575int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896(119909)119906119899119909

(119909 120591)119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(33)

Thus its stationary condition can be obtained as follows

1205821015840(119905 120591) = 0

1 + 120582(119905 120591)|120591=119905

= 0

(34)

therefore 120582(119905 120591) = minus1Now the following iteration formula can be obtained as

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

minus int

119905

0

119906119899120591

(119909 120591) minus 119906119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

119906119899119909

(119909 120591)

minus 119891 (119909 120591)

119889120591

(35)

Here according to Adomian method we choose its initialapproximate solution as 119906

0(119909 119905) = 119906(119909 0)

Having 119906 = lim119899rarrinfin

119906119899determined then the value of119901(119905)

can be computed by using (29)

5 Illustrative Examples

In this section three examples are presented to demonstratethe applicability and accuracy of the method These tests arechosen such that their analytical solutions are known Butthe method developed in this research can be applied tomore complicated problems The numerical implementationis carried out in Microsoft Maple13

Example 1 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909+ 4119905 (

1

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 0 le 119909 le 1

119906 (0 119905) = 21199052minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119906 (119909 119905) 119889119909 = 21199052minus 2119905 minus

1

12

0 le 119905 le 1

(36)

The true solution is 119906(119909 119905) = (12 minus 119909)119909 + 21199052minus 2119905 while

119901(119905) = 4119905 Let 1199060(119909 119905) = 119906(119909 0) = (12 minus 119909)119909 From (35)

we obtain 1199061(119909 119905) = (12 minus 119909)119909 + 2119905

2minus 2119905 which is the exact

solution Also from (29) we have 119901(119905) = 4119905 which is equal tothe exact 119901(119905) of this example

Example 2 We consider the inverse problem (2)ndash(4) asfollows

119906119905= 119906119909119909

+ 119901 (119905) 119906119909

+ 41198904119905

(

3

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 + 2 0 le 119909 le 1

119906 (0 119905) = 21198904119905

minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119909119906 (119909 119905) 119889119909 = 1198904119905

minus 119905 0 le 119905 le 1

(37)

The exact solution for this example is 119906(119909 119905) = (12 minus 119909)119909 +

21198904119905

minus 2119905 while 119901(119905) = 41198904119905 Let 119906

0(119909 119905) = 119906(119909 0) = (12 minus

119909)119909 + 2Using (35) and (29) we obtain 119906

1(119909 119905) = (12minus119909)119909+2119890

4119905minus

2119905 and 119901(119905) = 41198904119905 respectively which is the exact solution in

this example

Remark 3 From the above two examples it can be seen thatthe exact solution is obtained by using one iteration step only

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 3

For arbitrary 1199050 we can rewrite the above equation as

follows119906119899+1

(119905) = 119906119899(119905)

+ int

119905

0

120582 (120591) 119871119906119899(120591) + 119873

119899(120591) minus 119891 (120591) 119889120591 119899 ⩾ 0

(12)

Equation (12) is called a correction functional It is requiredfirst to determine the Lagrange multiplier Employing therestricted variation in correction functional and using inte-gration by part make it easy to compute the Lagrangemultiplier see for instance [30] For linear problems itsexact solution can be obtained by only one iteration stepdue to the fact that no nonlinear operator exists so theLagrange multiplier can be exactly identified Having 120582

determined then several approximations 119906119899+1

(119905) 119899 ⩾ 0 canbe determined Under reasonable choice of 119906

0 the fixed point

of the correction functional is considered as an approximatesolution of the above general differential equation Generallyone iteration leads to high accurate solution by variationaliteration method if the initial solution is carefully chosenwith some unknown parameters Comparison of the methodwith Adomian method was conducted by many authors viaillustrative examples and especially Wazwaz gave a completecomparison between the two methods [31] revealing thevariational iteration method has many merits over Adomianmethod it can completely overcome the difficulty arising inthe calculation of Adomian polynomial

3 Convergence Analysis andError Bound of VIM

This section covers the error analysis of the proposedmethodAlso the sufficient conditions are presented to guarantee theconvergence of VIM when applied to solve the differentialequations

31 Convergence Analysis First we will rewrite (12) in theoperator form as follows

119906119899+1

= 119860 [119906119899] (13)

where the operator 119860 takes the following form

119860 [119906 (119905)] = 119906 (119905) + int

119905

0

120582 (120591) 119871119906 (120591) + 119873119906 (120591) minus 119891 (120591) 119889120591

(14)

Theorem 1 Let (119883 sdot ) be a Banach space and 119860 119883 rarr 119883

is a nonlinear mapping and suppose that

119860 [119906] minus 119860 [] le 120574 119906 minus 119906 isin 119883 (15)

for some constant 120574 lt 1 Then 119860 has a unique fixed pointFurthermore sequence (12) using VIMwith an arbitrary choiceof 1199060isin 119883 converges to the fixed point of 119860 and

1003817100381710038171003817119906119899minus 119906119898

1003817100381710038171003817le

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

119899minus2

sum

119895=119898minus1

120574119895 (16)

Proof (see [32]) According to the above theorem a sufficientcondition for the convergence of the variational iterationmethod is strictly contraction of 119860 Furthermore sequence(12) converges to the fixed method of 119860 which is also thesolution of (9) Also the rate of convergence depends on120574

32 Approximation Error In the following theorem weintroduce an estimation of the error of the approximatesolution of problem (9) and prior and posterior error boundsof the approximate solution are estimated The prior errorbound can be used at the beginning of a calculation forestimating the number of steps necessary to obtain a givenaccuracy and the posterior error bound can be used atintermediate stages or at the end of a calculation

Theorem 2 Under the conditions of Theorem 1 error esti-mates are the prior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(17)

and the posterior estimate

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817 (18)

Also if one supposes that 119860[0] = 0 then the error of theapproximate solution 119906

119899to problem (9) can be obtained as

follows

1003817100381710038171003817119906 minus 119906119899

1003817100381710038171003817le

1 + 120574

1 minus 120574

120574119899 10038171003817100381710038171199060

1003817100381710038171003817 (19)

Proof Consider

1003817100381710038171003817119906119899+1

minus 119906119899

1003817100381710038171003817=

1003817100381710038171003817119860 [119906119899] minus 119860 [119906

119899minus1]1003817100381710038171003817

le 1205741003817100381710038171003817119906119899minus 119906119899minus1

1003817100381710038171003817

le 1205742 1003817100381710038171003817119906119899minus1

minus 119906119899minus2

1003817100381710038171003817

le 120574119899 10038171003817100381710038171199061minus 1199060

1003817100381710038171003817

(20)

Therefore for any 119898 gt 119899 we have

119906119898

minus 119906119899 le 119906

119898minus 119906119898minus1

+ 119906119898minus1

minus 119906119898minus2

+ sdot sdot sdot + 119906119899+1

minus 119906119899

le 120574119898minus1

1199061minus 1199060 + 120574119898minus2

1199061minus 1199060

+ sdot sdot sdot + 1205741198991199061minus 1199060

= (120574119898minus1

+ 120574119898minus2

+ sdot sdot sdot + 120574119899) 1199061minus 1199060

= 120574119899 1 minus 120574

119898minus119899

1 minus 120574

1199061minus 1199060

(21)

4 Advances in Numerical Analysis

Since 0 lt 120574 lt 1 in the numerator we have 1 minus 120574119898minus119899

lt 1Consequently

1003817100381710038171003817119906119898

minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817 119898 gt 119899 (22)

The first statement follows from (22) by using Theorem 1 as119898 rarr infin We derive (18) Taking 119899 = 1 and writing V

0for 1199060

and V1for 1199061 we have from (17)

1003817100381710038171003817V1minus 119906

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817V1minus V0

1003817100381710038171003817 (23)

Setting V0= 119906119899minus1

we have V1= 119860[V

0] = 119906119899and obtain (18)

Also we have

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817

int

119905

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

=1003817100381710038171003817119860 [1199060] minus 1199060

1003817100381710038171003817le

1003817100381710038171003817119860 [1199060]1003817100381710038171003817+

10038171003817100381710038171199060

1003817100381710038171003817

(24)

We can write1003817100381710038171003817119860 [1199060]1003817100381710038171003817=

1003817100381710038171003817119860 [1199060] minus 119860 [0]

1003817100381710038171003817le 120574

10038171003817100381710038171199060

1003817100381710038171003817 (25)

Therefore10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le (120574 + 1)

10038171003817100381710038171199060

1003817100381710038171003817 (26)

From Theorem 1 as 119898 rarr infin then 119906119898

rarr 119906 and by using(22)-(26) inequality (19) can be obtained

4 The Application of VIM in Inverse ParabolicProblem (2)ndash(4)

To use the variational iteration method for solving theproblem (2)ndash(4) at first we use the following transformation

41 The Employed Transformation By differentiation withrespect to the variable 119905 in (4) one obtains

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905) + int

119904(119905)

0

119896 (119909) 119906119905(119909 119905) 119889119909

(27)

Substituting (2) into the above equation yields

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

+ int

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119901 (119905) 119906119909(119909 119905) + 119891 (119909 119905)] 119889119909

(28)

and it follows that

119901 (119905) = (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896(119909)119906119909(119909 119905)119889119909)

minus1

(29)

provided that for any 119905 isin [0 119879] int

119904(119905)

0119896(119909)119906

119909(119909 119905)119889119909 and

int

119904(119905)

0119896(119909)[119906

119909119909(119909 119905) + 119891(119909 119905)]119889119909 are exit also int

119904(119905)

0119896(119909)119906

119909

(119909 119905)119889119909 = 0Therefore the inverse parabolic problem (2)ndash(4) is equiv-

alent to the following nonlocal parabolic equation

119906119905= 119906119909119909

+ (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896 (119909) 119906119909(119909 119905) 119889119909)

minus1

119906119909

+ 119891 (119909 119905) in 119876119879

(30)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(31)

Therefore for solving the inverse problem (2)ndash(4) we willinvestigate the direct problem (30)-(31)

42 Application In order to solve problem (30)-(31) usingVIM we can write the following correction functional

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

+ int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(32)

Advances in Numerical Analysis 5

Making the above correction functional stationary note that120575119906119899(0) = 0 we have

120575119906119899+1

(119909 119905)

= 120575119906119899(119909 119905)

+ 120575int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896(119909)119906119899119909

(119909 120591)119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(33)

Thus its stationary condition can be obtained as follows

1205821015840(119905 120591) = 0

1 + 120582(119905 120591)|120591=119905

= 0

(34)

therefore 120582(119905 120591) = minus1Now the following iteration formula can be obtained as

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

minus int

119905

0

119906119899120591

(119909 120591) minus 119906119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

119906119899119909

(119909 120591)

minus 119891 (119909 120591)

119889120591

(35)

Here according to Adomian method we choose its initialapproximate solution as 119906

0(119909 119905) = 119906(119909 0)

Having 119906 = lim119899rarrinfin

119906119899determined then the value of119901(119905)

can be computed by using (29)

5 Illustrative Examples

In this section three examples are presented to demonstratethe applicability and accuracy of the method These tests arechosen such that their analytical solutions are known Butthe method developed in this research can be applied tomore complicated problems The numerical implementationis carried out in Microsoft Maple13

Example 1 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909+ 4119905 (

1

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 0 le 119909 le 1

119906 (0 119905) = 21199052minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119906 (119909 119905) 119889119909 = 21199052minus 2119905 minus

1

12

0 le 119905 le 1

(36)

The true solution is 119906(119909 119905) = (12 minus 119909)119909 + 21199052minus 2119905 while

119901(119905) = 4119905 Let 1199060(119909 119905) = 119906(119909 0) = (12 minus 119909)119909 From (35)

we obtain 1199061(119909 119905) = (12 minus 119909)119909 + 2119905

2minus 2119905 which is the exact

solution Also from (29) we have 119901(119905) = 4119905 which is equal tothe exact 119901(119905) of this example

Example 2 We consider the inverse problem (2)ndash(4) asfollows

119906119905= 119906119909119909

+ 119901 (119905) 119906119909

+ 41198904119905

(

3

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 + 2 0 le 119909 le 1

119906 (0 119905) = 21198904119905

minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119909119906 (119909 119905) 119889119909 = 1198904119905

minus 119905 0 le 119905 le 1

(37)

The exact solution for this example is 119906(119909 119905) = (12 minus 119909)119909 +

21198904119905

minus 2119905 while 119901(119905) = 41198904119905 Let 119906

0(119909 119905) = 119906(119909 0) = (12 minus

119909)119909 + 2Using (35) and (29) we obtain 119906

1(119909 119905) = (12minus119909)119909+2119890

4119905minus

2119905 and 119901(119905) = 41198904119905 respectively which is the exact solution in

this example

Remark 3 From the above two examples it can be seen thatthe exact solution is obtained by using one iteration step only

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Numerical Analysis

Since 0 lt 120574 lt 1 in the numerator we have 1 minus 120574119898minus119899

lt 1Consequently

1003817100381710038171003817119906119898

minus 119906119899

1003817100381710038171003817le

120574119899

1 minus 120574

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817 119898 gt 119899 (22)

The first statement follows from (22) by using Theorem 1 as119898 rarr infin We derive (18) Taking 119899 = 1 and writing V

0for 1199060

and V1for 1199061 we have from (17)

1003817100381710038171003817V1minus 119906

1003817100381710038171003817le

120574

1 minus 120574

1003817100381710038171003817V1minus V0

1003817100381710038171003817 (23)

Setting V0= 119906119899minus1

we have V1= 119860[V

0] = 119906119899and obtain (18)

Also we have

10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le

10038171003817100381710038171003817100381710038171003817

int

119905

0

120582 (120591) 1198711199060(120591) + 119873119906

0(120591) minus 119891 (120591) 119889120591

10038171003817100381710038171003817100381710038171003817

=1003817100381710038171003817119860 [1199060] minus 1199060

1003817100381710038171003817le

1003817100381710038171003817119860 [1199060]1003817100381710038171003817+

10038171003817100381710038171199060

1003817100381710038171003817

(24)

We can write1003817100381710038171003817119860 [1199060]1003817100381710038171003817=

1003817100381710038171003817119860 [1199060] minus 119860 [0]

1003817100381710038171003817le 120574

10038171003817100381710038171199060

1003817100381710038171003817 (25)

Therefore10038171003817100381710038171199061minus 1199060

1003817100381710038171003817le (120574 + 1)

10038171003817100381710038171199060

1003817100381710038171003817 (26)

From Theorem 1 as 119898 rarr infin then 119906119898

rarr 119906 and by using(22)-(26) inequality (19) can be obtained

4 The Application of VIM in Inverse ParabolicProblem (2)ndash(4)

To use the variational iteration method for solving theproblem (2)ndash(4) at first we use the following transformation

41 The Employed Transformation By differentiation withrespect to the variable 119905 in (4) one obtains

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905) + int

119904(119905)

0

119896 (119909) 119906119905(119909 119905) 119889119909

(27)

Substituting (2) into the above equation yields

1198641015840(119905) = 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

+ int

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119901 (119905) 119906119909(119909 119905) + 119891 (119909 119905)] 119889119909

(28)

and it follows that

119901 (119905) = (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896(119909)119906119909(119909 119905)119889119909)

minus1

(29)

provided that for any 119905 isin [0 119879] int

119904(119905)

0119896(119909)119906

119909(119909 119905)119889119909 and

int

119904(119905)

0119896(119909)[119906

119909119909(119909 119905) + 119891(119909 119905)]119889119909 are exit also int

119904(119905)

0119896(119909)119906

119909

(119909 119905)119889119909 = 0Therefore the inverse parabolic problem (2)ndash(4) is equiv-

alent to the following nonlocal parabolic equation

119906119905= 119906119909119909

+ (1198641015840(119905) minus 119904

1015840(119905) 119896 (119904 (119905)) 119906 (119904 (119905) 119905)

minusint

119904(119905)

0

119896 (119909) [119906119909119909

(119909 119905) + 119891 (119909 119905)] 119889119909)

times (int

119904(119905)

0

119896 (119909) 119906119909(119909 119905) 119889119909)

minus1

119906119909

+ 119891 (119909 119905) in 119876119879

(30)

with the initial-boundary conditions

119906 (119909 0) = 1199060(119909) 0 le 119909 le 1

1198611119906 (0 119905) = 119892

0(119905) 0 le 119905 le 119879

1198612119906 (1 119905) = 119892

1(119905) 0 le 119905 le 119879

(31)

Therefore for solving the inverse problem (2)ndash(4) we willinvestigate the direct problem (30)-(31)

42 Application In order to solve problem (30)-(31) usingVIM we can write the following correction functional

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

+ int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(32)

Advances in Numerical Analysis 5

Making the above correction functional stationary note that120575119906119899(0) = 0 we have

120575119906119899+1

(119909 119905)

= 120575119906119899(119909 119905)

+ 120575int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896(119909)119906119899119909

(119909 120591)119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(33)

Thus its stationary condition can be obtained as follows

1205821015840(119905 120591) = 0

1 + 120582(119905 120591)|120591=119905

= 0

(34)

therefore 120582(119905 120591) = minus1Now the following iteration formula can be obtained as

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

minus int

119905

0

119906119899120591

(119909 120591) minus 119906119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

119906119899119909

(119909 120591)

minus 119891 (119909 120591)

119889120591

(35)

Here according to Adomian method we choose its initialapproximate solution as 119906

0(119909 119905) = 119906(119909 0)

Having 119906 = lim119899rarrinfin

119906119899determined then the value of119901(119905)

can be computed by using (29)

5 Illustrative Examples

In this section three examples are presented to demonstratethe applicability and accuracy of the method These tests arechosen such that their analytical solutions are known Butthe method developed in this research can be applied tomore complicated problems The numerical implementationis carried out in Microsoft Maple13

Example 1 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909+ 4119905 (

1

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 0 le 119909 le 1

119906 (0 119905) = 21199052minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119906 (119909 119905) 119889119909 = 21199052minus 2119905 minus

1

12

0 le 119905 le 1

(36)

The true solution is 119906(119909 119905) = (12 minus 119909)119909 + 21199052minus 2119905 while

119901(119905) = 4119905 Let 1199060(119909 119905) = 119906(119909 0) = (12 minus 119909)119909 From (35)

we obtain 1199061(119909 119905) = (12 minus 119909)119909 + 2119905

2minus 2119905 which is the exact

solution Also from (29) we have 119901(119905) = 4119905 which is equal tothe exact 119901(119905) of this example

Example 2 We consider the inverse problem (2)ndash(4) asfollows

119906119905= 119906119909119909

+ 119901 (119905) 119906119909

+ 41198904119905

(

3

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 + 2 0 le 119909 le 1

119906 (0 119905) = 21198904119905

minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119909119906 (119909 119905) 119889119909 = 1198904119905

minus 119905 0 le 119905 le 1

(37)

The exact solution for this example is 119906(119909 119905) = (12 minus 119909)119909 +

21198904119905

minus 2119905 while 119901(119905) = 41198904119905 Let 119906

0(119909 119905) = 119906(119909 0) = (12 minus

119909)119909 + 2Using (35) and (29) we obtain 119906

1(119909 119905) = (12minus119909)119909+2119890

4119905minus

2119905 and 119901(119905) = 41198904119905 respectively which is the exact solution in

this example

Remark 3 From the above two examples it can be seen thatthe exact solution is obtained by using one iteration step only

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 5

Making the above correction functional stationary note that120575119906119899(0) = 0 we have

120575119906119899+1

(119909 119905)

= 120575119906119899(119909 119905)

+ 120575int

119905

0

120582 (119905 120591)

times

119906119899120591

(119909 120591) minus 119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896(119909)119906119899119909

(119909 120591)119889119909)

minus1

times 119899119909

(119909 120591) minus 119891 (119909 120591)

119889120591

(33)

Thus its stationary condition can be obtained as follows

1205821015840(119905 120591) = 0

1 + 120582(119905 120591)|120591=119905

= 0

(34)

therefore 120582(119905 120591) = minus1Now the following iteration formula can be obtained as

119906119899+1

(119909 119905)

= 119906119899(119909 119905)

minus int

119905

0

119906119899120591

(119909 120591) minus 119906119899119909119909

(119909 120591)

minus (1198641015840(120591) minus 119904

1015840(120591) 119896 (119904 (120591)) 119906

119899(119904 (120591) 120591)

minusint

119904(120591)

0

119896 (119909) [119906119899119909119909

(119909 120591) + 119891 (119909 120591)] 119889119909)

times (int

119904(120591)

0

119896 (119909) 119906119899119909

(119909 120591) 119889119909)

minus1

119906119899119909

(119909 120591)

minus 119891 (119909 120591)

119889120591

(35)

Here according to Adomian method we choose its initialapproximate solution as 119906

0(119909 119905) = 119906(119909 0)

Having 119906 = lim119899rarrinfin

119906119899determined then the value of119901(119905)

can be computed by using (29)

5 Illustrative Examples

In this section three examples are presented to demonstratethe applicability and accuracy of the method These tests arechosen such that their analytical solutions are known Butthe method developed in this research can be applied tomore complicated problems The numerical implementationis carried out in Microsoft Maple13

Example 1 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909+ 4119905 (

1

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 0 le 119909 le 1

119906 (0 119905) = 21199052minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119906 (119909 119905) 119889119909 = 21199052minus 2119905 minus

1

12

0 le 119905 le 1

(36)

The true solution is 119906(119909 119905) = (12 minus 119909)119909 + 21199052minus 2119905 while

119901(119905) = 4119905 Let 1199060(119909 119905) = 119906(119909 0) = (12 minus 119909)119909 From (35)

we obtain 1199061(119909 119905) = (12 minus 119909)119909 + 2119905

2minus 2119905 which is the exact

solution Also from (29) we have 119901(119905) = 4119905 which is equal tothe exact 119901(119905) of this example

Example 2 We consider the inverse problem (2)ndash(4) asfollows

119906119905= 119906119909119909

+ 119901 (119905) 119906119909

+ 41198904119905

(

3

2

+ 2119909) 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = (

1

2

minus 119909) 119909 + 2 0 le 119909 le 1

119906 (0 119905) = 21198904119905

minus 2119905 0 le 119905 le 1

119906119909(1 119905) =

minus3

2

0 le 119905 le 1

int

1

0

119909119906 (119909 119905) 119889119909 = 1198904119905

minus 119905 0 le 119905 le 1

(37)

The exact solution for this example is 119906(119909 119905) = (12 minus 119909)119909 +

21198904119905

minus 2119905 while 119901(119905) = 41198904119905 Let 119906

0(119909 119905) = 119906(119909 0) = (12 minus

119909)119909 + 2Using (35) and (29) we obtain 119906

1(119909 119905) = (12minus119909)119909+2119890

4119905minus

2119905 and 119901(119905) = 41198904119905 respectively which is the exact solution in

this example

Remark 3 From the above two examples it can be seen thatthe exact solution is obtained by using one iteration step only

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Numerical Analysis

Table 1 Absolute errors of 119906119899at 119905 = 01 for Example 3

119909 |119906 minus 11990640| |119906 minus 119906

60| |119906 minus 119906

80| |119906 minus 119906

100|

00 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

02 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

04 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

06 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

08 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

10 1005119864 minus 6 3189119864 minus 9 1011119864 minus 11 3207119864 minus 14

Table 2 Absolute errors of 119901119899for Example 3

119905 |119901 minus 11990140| |119901 minus 119901

60| |119901 minus 119901

80| |119901 minus 119901

100|

00 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

02 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

04 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

06 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

08 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

10 7542119864 minus 6 2391119864 minus 8 7585119864 minus 11 2405119864 minus 13

Example 3 We consider the following inverse problem

119906119905= 119906119909119909

+ 119901 (119905) 119906119909 0 lt 119909 lt 1 0 lt 119905 lt 1

119906 (119909 0) = 119909 0 le 119909 le 1

119906 (0 119905) = 119905 0 le 119905 le 1

119906 (1 119905) = 1 + 119905 0 le 119905 le 1

int

radic119905

0

radic119909 119906 (119909 119905) 119889119909 =

2

3

11990574

+

2

5

11990554

0 le 119905 le 1

(38)

The true solution is 119906(119909 119905) = 119909 + 119905 while 119901(119905) = 1 Let1199060(119909 119905) = 119906(119909 0) = 119909 According to (35) one can obtain the

successive approximations 119906119899(119909 119905) of 119906(119909 119905) as follows

1199061(119909 119905) = 119909 +

7

4

119905

1199062(119909 119905) = 119909 +

7

16

119905

1199063(119909 119905) = 119909 +

91

64

119905

1199064(119909 119905) = 119909 +

175

256

119905

(39)

And from (29) one can obtain the successive approximationsof 119901(119905) as follows

1199011(119905) =

7

16

1199012(119905) =

91

64

0 02 04 06 08 1

0

05

10

02

04

06

08

1

x

t

uminusu40

times10minus5

Figure 1 Graph of |119906(119909 119905) minus 11990640(119909 119905)| at 119905 = 01 for Example 3

1199013(119905) =

175

256

1199014(119905) =

1267

1024

(40)

And the rest of the components of iteration formula (35)and (29) are obtained using the Maple PackageThe obtainednumerical results are summarized in Tables 1 and 2 Inaddition the graphs of the error functions |119906 minus 119906

40| and

|119901 minus 11990140| are plotted in Figures 1 and 2

From these results we conclude that the variationaliteration method for this example gives remarkable accuracyin comparison with the exact solution

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 7

0 0201 03 04 05 06 07 08 09 109999

1

10001

10002

t

p(t)

p40(t)

Figure 2 Graph of |119901(119905) minus 11990140(119905)| for Example 3

6 Conclusion

In the present work we have demonstrated the applica-bility of the VIM for solving a class of parabolic inverseproblem in reaction-diffusion equation and introduce anestimation of the absolute error of the approximate solutionfor the proposed method The method needs much lesscomputational work compared with traditional methods anddoes not need discretization The illustrative examples showthe efficiency of the method By this method we obtainremarkable accuracy in comparison with the exact solutionMoreover by using only one iteration step we may getthe exact solution We expect that for more general caseswhere the inhomogeneous term 119891(119909 119910 119911 119905 119906 119906

119909 119906119910 119906119911 119901)

has more complicated structures the present method workswell In conclusion wemention that the VIM can be extendedfor similar two- and three-dimensional inverse parabolicproblems subject to integral overspecification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] MDehghan ldquoAn inverse problemof finding a source parameterin a semilinear parabolic equationrdquo Applied MathematicalModelling vol 25 no 9 pp 743ndash754 2001

[2] J V Beck B Blackwell and C R St Clair Inverse HeatConduction Ill-Posed Problems Wiley-Interscience New YorkNY USA 1985

[3] W C Chew and Y M Wang ldquoReconstruction of two-dime-nsional permittivity distribution using the distorted Born itera-tive methodrdquo IEEE Transactions on Medical Imaging vol 9 no2 pp 218ndash225 1990

[4] C Glorieux J Moulder J Basart and JThoen ldquoDeterminationof electrical conductivity profiles using neural network inver-sion of multi-frequency eddy-current datardquo Journal of PhysicsD Applied Physics vol 32 no 5 pp 616ndash622 1999

[5] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems Kluwer Academic Publishers Boston MassUSA 1996

[6] G Vries T Hillen M Lewis J Muller and B Schonfisch ACourse in Mathematical Biology SIAM Philadelphia Pa USA2006

[7] W Hundsdorfer and J Verwer Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations vol 33Springer Berlin Germany 2003

[8] J R Cannon and S Perez-Esteva ldquoDetermination of the coeffi-cient of u

119909in a linear parabolic equationrdquo Inverse Problems vol

10 no 3 pp 521ndash531 1994[9] J R Cannon and H-M Yin ldquoOn a class of nonlinear parabolic

equations with nonlinear trace type functionalsrdquo Inverse Prob-lems vol 7 no 1 pp 149ndash161 1991

[10] H Azari and F Parzlivand ldquoDetermination of the coefficientin the advection diffusion equation using collocation and radialbasis functionrdquo International Journal of Information and SystemsSciences vol 1 no 1 pp 1ndash11 2009

[11] H Azari ldquoNumerical procedures for the determination ofan unknown coefficient in parabolic differential equationsrdquoDynamics of Continuous Discrete amp Impulsive Systems BApplications amp Algorithms vol 9 no 4 pp 555ndash573 2002

[12] Y P Lin and R J Tait ldquoFinite-difference approximations for aclass of nonlocal parabolic boundary value problemsrdquo Journalof Computational and Applied Mathematics vol 47 no 3 pp335ndash350 1993

[13] J A MacBain ldquoInversion theory for a parameterized diffusionproblemrdquo SIAM Journal on Applied Mathematics vol 47 no 6pp 1386ndash1391 1987

[14] B F Jones ldquoDetermination of a coefficient in a parabolicdifferential equation I Existence and uniquenessrdquo Journal ofMathematics and Mechanics vol 11 pp 907ndash918 1962

[15] J Douglas and B F Jones ldquoThe determination of a coefficient ina parabolic differential equation II Numerical approximationrdquoJournal ofMathematics andMechanics vol 11 pp 919ndash926 1962

[16] J R Cannon ldquoDetermination of certain parameters in heatconduction problemsrdquo Journal of Mathematical Analysis andApplications vol 8 no 2 pp 188ndash201 1964

[17] J Jones ldquoVarious methods for finding unknown coefficients inparabolic differential equationsrdquo Communications on Pure andApplied Mathematics vol 16 pp 33ndash44 1963

[18] J R Cannon and D Zachmann ldquoParameter determinationin parabolic partial differential equations from overspecifiedboundary datardquo International Journal of Engineering Sciencevol 20 no 6 pp 779ndash788 1982

[19] J R Cannon and H-M Yin ldquoA class of nonlinear nonclassicalparabolic equationsrdquo Journal of Differential Equations vol 79no 2 pp 266ndash288 1989

[20] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000

[21] E Yusufoglu ldquoVariational iteration method for constructionof some compact and noncompact structures of Klein-Gordonequationsrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 8 no 2 pp 153ndash158 2007

[22] J H He ldquoApproximate analytical solution for seepage flow withfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Numerical Analysis

[23] L-N Zhang and J-H He ldquoResonance in Sirospun yarnspinning using a variational iteration methodrdquo Computers ampMathematics with Applications vol 54 no 7-8 pp 1064ndash10662007

[24] J-H He Y-P Yu J-Y Yu W-R Li S-Y Wang and N Pan ldquoAnonlinear dynamicmodel for two-strand yarn spinningrdquoTextileResearch Journal vol 75 no 2 pp 181ndash184 2005

[25] M A Abdou and A A Soliman ldquoVariational iteration methodfor solving Burgerrsquos and coupled Burgerrsquos equationsrdquo Journal ofComputational andAppliedMathematics vol 181 no 2 pp 245ndash251 2005

[26] A A Soliman ldquoA numerical simulation and explicit solutions ofKDV-Burgersrsquo and Laxrsquos seventh-order KDVEquationsrdquoChaosSolitons and Fractals vol 29 no 2 pp 294ndash302 2006

[27] Z M Odibat and S Momani ldquoApplication of variationaliteration method to nonlinear differential equations of frac-tional orderrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 7 no 1 pp 27ndash34 2006

[28] M Inokuti H Sekine and T Mura ldquoGeneral use of theLagrange multiplier in nonlinear mathematical physicsrdquo inVariational Method in the Mechanics of Solids S Nemat-NasserEd pp 159ndash162 Pergamon Press Oxford UK 1978

[29] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[30] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[31] A-MWazwaz ldquoA comparison between the variational iterationmethod and Adomian decomposition methodrdquo Journal ofComputational andAppliedMathematics vol 207 no 1 pp 129ndash136 2007

[32] M Tatari and M Dehghan ldquoOn the convergence of Hersquos varia-tional iteration methodrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 121ndash128 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of