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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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