Research Article A Meshless Method Based on the...
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Research Article A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem Muhammad Arghand and Majid Amirfakhrian Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran Correspondence should be addressed to Majid Amirfakhrian; [email protected] Received 25 November 2014; Accepted 11 March 2015 Academic Editor: Alkesh Punjabi Copyright © 2015 M. Arghand and M. Amirfakhrian. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a new meshless method to solve a backward inverse heat conduction problem. e numerical scheme, based on the fundamental solution of the heat equation and radial basis functions (RBFs), is used to obtain a numerical solution. Since the coefficients matrix is ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulted system of linear equations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problem demonstrates the stability, accuracy, and efficiency of the proposed method. 1. Introduction Transient heat conduction phenomena are generally described by the parabolic heat conduction equation, and if the initial temperature distribution and the boundary conditions are specified, then this, in general, leads to a well-posed problem which may easily be solved numerically by using various numerical methods. However, in many practical situations when dealing with a heat conducting body it is not always possible to specify the boundary conditions or the initial temperature. Hence, we are faced with an inverse heat conduction problem. Inverse heat conduction problems (IHCPs) occur in many branches of engineering and science. Mechanical and chemical engineers, mathematicians, and specialists in other sciences branches are interested in inverse problems. From another point of view, since the existence, uniqueness, and stability of the solutions of these problems are not usually confirmed, they are generally identified as ill-posed [1–4]. According to the fact that unknown solutions of inverse problems are determined through indirect observable data which contain measurement errors, such problems are naturally unstable. In other words, the main difficulty in the treatment of inverse problems is the unstability of their solution in the presence of noise in the measured data. Hence, several numerical methods have been proposed for solving the various kinds of inverse problems. In addition to, ill-posedness of these kinds of prob- lems, ill-conditioning of the resulting discretized matrix from the traditional methods like the finite differences method (FDM) [5], the finite element method (FEM), and so forth [6, 7], is the main problem making all numerical algorithms for determining the solution of these kinds of problems. Accordingly, within recent years, meshless methods, as the method of fundamental solution (MFS), radial basis func- tions (RBFs) method, and some other methods, have been applied by many scientists in the field of applied sciences and engineering [8–15]. Kupradze and Aleksidze [16] first introduced MFS which defines the solution of the problem as a linear combination of fundamental solutions. Hon et al. [17– 20] applied the MFS to solve some inverse heat conduction problems. In 1990s, Kansa applied RBFs method to solve the different types of partial differential equations [21, 22]. Aſter that, Kansa and many scientists regarded RBFs method to solve different types of mathematical problems from partial or ordinary differential equations to integral equations [23–26]. Following their works, during recent years, many researchers have made some changes in RBFs and MFS methods and have developed advance methods to solve Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 256726, 8 pages http://dx.doi.org/10.1155/2015/256726
Transcript of Research Article A Meshless Method Based on the...
Research ArticleA Meshless Method Based on the FundamentalSolution and Radial Basis Function for Solvingan Inverse Heat Conduction Problem
Muhammad Arghand and Majid Amirfakhrian
Department of Mathematics Central Tehran Branch Islamic Azad University Tehran Iran
Correspondence should be addressed to Majid Amirfakhrian amirfakhrianiauctbacir
Received 25 November 2014 Accepted 11 March 2015
Academic Editor Alkesh Punjabi
Copyright copy 2015 M Arghand and M Amirfakhrian 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
We propose a new meshless method to solve a backward inverse heat conduction problem The numerical scheme based on thefundamental solution of the heat equation and radial basis functions (RBFs) is used to obtain a numerical solution Since thecoefficients matrix is ill-conditioned the Tikhonov regularization (TR) method is employed to solve the resulted system of linearequations Also the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter A test problemdemonstrates the stability accuracy and efficiency of the proposed method
1 Introduction
Transient heat conduction phenomena are generallydescribed by the parabolic heat conduction equation andif the initial temperature distribution and the boundaryconditions are specified then this in general leads to awell-posed problem which may easily be solved numericallyby using various numerical methods However in manypractical situationswhendealingwith a heat conducting bodyit is not always possible to specify the boundary conditions orthe initial temperature Hence we are faced with an inverseheat conduction problem Inverse heat conduction problems(IHCPs) occur in many branches of engineering and scienceMechanical and chemical engineers mathematicians andspecialists in other sciences branches are interested in inverseproblems From another point of view since the existenceuniqueness and stability of the solutions of these problemsare not usually confirmed they are generally identifiedas ill-posed [1ndash4] According to the fact that unknownsolutions of inverse problems are determined throughindirect observable data which contain measurement errorssuch problems are naturally unstable In other words themain difficulty in the treatment of inverse problems is theunstability of their solution in the presence of noise in
the measured data Hence several numerical methods havebeen proposed for solving the various kinds of inverseproblems In addition to ill-posedness of these kinds of prob-lems ill-conditioning of the resulting discretizedmatrix fromthe traditional methods like the finite differences method(FDM) [5] the finite element method (FEM) and so forth[6 7] is the main problem making all numerical algorithmsfor determining the solution of these kinds of problemsAccordingly within recent years meshless methods as themethod of fundamental solution (MFS) radial basis func-tions (RBFs) method and some other methods have beenapplied by many scientists in the field of applied sciencesand engineering [8ndash15] Kupradze and Aleksidze [16] firstintroducedMFS which defines the solution of the problem asa linear combination of fundamental solutionsHon et al [17ndash20] applied the MFS to solve some inverse heat conductionproblems In 1990s Kansa applied RBFs method to solve thedifferent types of partial differential equations [21 22] Afterthat Kansa and many scientists regarded RBFs method tosolve different types of mathematical problems from partialor ordinary differential equations to integral equations[23ndash26] Following their works during recent years manyresearchers have made some changes in RBFs and MFSmethods and have developed advance methods to solve
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 256726 8 pageshttpdxdoiorg1011552015256726
2 Advances in Mathematical Physics
some of these kinds problems [27 28] Consequently in thiswork we will present a meshless numerical scheme basedon combining the radial basis function and the fundamentalsolution of the heat equation in order to approximate thesolution of a backward inverse heat conduction problem(BIHCP) the problem inwhich an unknown initial conditionorand temperature distribution in previous time will bedetermined This kind of problem may emerge in manypractical application areas such as archeology and mantleplumes [29] On the other hand since the system of the linearequations obtained from discretizing the problem in thepresentedmethod is ill-conditioned Tikhonov regularization(TR) method is applied in order to solve it The generalizedcross-validation (GCV) criterion has been assigned to adoptan optimum amount of the regularization parameter Thestructure of the rest of this work is organized as follows InSection 2 we represent the mathematical formulation of theproblem The method of fundamental solutions radial basisfunctions method and method of fundamental solution-radial basis functions (MFSRBF) are described in Section 3Section 4 embraces Tikhonov regularization method with arule for choosing an appropriate regularization parameterIn Section 5 we present the obtained numerical results ofsolving a test problem Section 6 ends in a brief conclusionand some suggestions
2 Mathematical Formulation of the Problem
In this section we consider the following one-dimensionalinverse heat conduction problem
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω = (0 119897) times (0 119905max) (1)
with the following initial and boundary conditions
119906 (119909 0) =
120593 (119909) 119909 isin [0 1199090)
119891 (119909) 119909 isin [1199090 119897]
119906 (0 119905) = 119901 (119905) 119905 isin [0 119905max]
119906 (119897 119905) = 119902 (119905) 119905 isin [0 119905max]
(2)
where 120593(119909) and 119902(119905) are considered as known functions and119905max is a given positive constant while 119891(119909) and 119901(119905) areregarded as unknown functions So in order to estimate119891(119909)and 119901(119905) we consider additional temperature measurementsand heat flux given at a point 119909
0 1199090isin (0 1) as overspecified
conditions119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
(3)
To solve the above problem at first we divide the problem(1)ndash(3) into two separate problems The problem 119860 is asfollows
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω
119860= (0 119909
0) times (0 119905max)
119906 (119909 0) = 120593 (119909) 119909 isin [0 1199090]
119906 (0 119905) = 119901 (119905) 119905 isin [0 119905max]
119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
(4)
and the problem 119861 is considered as follows
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω
119861= (1199090 119897) times (0 119905max)
119906 (119909 0) = 119891 (119909) 119909 isin [1199090 119897]
119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
119906 (119897 119905) = 119902 (119905) 119905 isin [0 119905max]
(5)
Obviously the problems A and B are considered as IHCPwhere 119901(119905) 119906(119909 119905) and 119891(119909) 119906(119909 119905) are unknown functionsin the problems A and B respectively
3 Method of Fundamental Solutions andMethod of Radial Basis Functions
In this section we introduce the numerical scheme forsolving the problem (1)ndash(3) using the fundamental solutionsand radial basis functions
31Method of Fundamental Solutions Thefundamental solu-tion of (1) is presented as below
119896 (119909 119905) =
1
2119886radic120587119905
119890minus1199092(41198862119905)119867(119905) (6)
where 119867(119905) is Heaviside unit function Assuming that 119879 gt
119905max is a constant it can be demonstrated that the time shiftfunction
120601 (119909 119905) = 119896 (119909 119905 + 119879) (7)
is also a nonsingular solution of (1) in the domainΩIn order to solve an IHCP by MFS as the problem 119860
since the basis function 120601 satisfies the heat equation (1)automatically we assume that (119909
119896 119905119896)119873
119896=1is a given set
of scattered points on the boundary 120597Ω119860 An approximate
solution is defined as a linear combination of 120601 as follows
Φ (119909 119905) =
119873
sum
119896=1
120582119896120601 (119909 minus 119909
119896 119905 minus 119905119896) (8)
where120601(119909 119905) is given by (7) and 120582119896rsquos are unknown coefficients
which can be determined by solving the following matrixequation
AΛ = b (9)
where Λ = [1205821 120582
119873]119879 and b is a 119873 times 1 known vector
Also as the fundamental functions are the solution of theheat equation only the initial and boundary conditions arepracticed to make the system of linear equations that is A is
Advances in Mathematical Physics 3
Table 1 Some well-known radial basis functions
Infinitely smooth RBFs 120595(119903)
Gaussian (GA) 119890(minus1198881199032)
Multiquadrics (MQ) radic1199032+ 1198882
Inverse multiquadrics (IMQ) (radic1199032+ 1198882)
minus1
Inverse quadric (IQ) (1199032+ 1198882)minus1
a119873 times119873 square matrix which is defined using the initial andthe boundary conditions as follows
A = [119886119894119895] 119886
119894119895= Φ (119909 119905) (119909 119905) isin 120597Ω
119860 119894 119895 = 1 119873
(10)
For more details see [13 18]
32 Method of Radial Basis Functions In this section weconsider RBF method for interpolation of scattered dataSuppose that xlowast and x are a fixed point and an arbitrary pointin R119889 respectively A radial function 120595lowast is defined via 120595lowast =120595lowast(119903) where 119903 = x minus xlowast
2 That is the radial function 120595lowast
depends only on the distance between x and xlowastThis propertyimplies that the RBFs 120595lowast are radially symmetric about xlowastSomewell-known infinitely smooth RBFs are given in Table 1As it is observed these functions depend on a free parameter119888 known as the shape parameter which has an important rolein approximation theory using RBFs
Let x119896119873
119896=1be a given set of distinct points in domain Ω
in R119889 The main idea of using the RBFs is interpolation witha linear combination of RBFs of the same types as follows
Ψ (x) =119873
sum
119896=1
120582119896120595119896(x) (11)
where 120595119896(x) = 120595(x minus x
119896) and 120582
119896rsquos are unknown coefficients
for 119896 = 1 2 119873 Assume that we want to interpolate thegiven values 119891
119896= 119891(x
119896) 119896 = 1 2 119873 The unknown
coefficients 120582119896rsquos are obtained so thatΨ(x
119896) = 119891119896 119896 = 1 2
119873 which results in the following system of linear equations
AΛ = b (12)
where Λ = [1205821 120582
119873]119879 b = [119891
1 119891
119873]119879 and A = [119886
119894119895]
for 119894 119895 = 1 2 119873 with entries 119886119894119895= 120595119895(x119894) Generally
the matrix A has been shown to be positive definite (andtherefore nonsingular) for distinct interpolation points forinfinitely smooth RBFs by Schoenbergrsquos theorem [30] Alsousing Micchellirsquos theorem [31] it is shown that A is aninvertible matrix for a distinct set of scattered nodes in thecase of MQ-RBF
4 Method of Fundamental Solutions-RadialBasis Functions
This section is specified to introduce the numerical schemefor solving the problem (1)ndash(3) using the fundamental solu-tions and radial basis functions At first we are going to
figure out how radial basis functions and the fundamentalsolutions are applied to approximate the solution of theproblem 119860 2 Similarly this method will be used for solvingthe problem 119861 on domain Ω
119861 Because the one-dimensional
heat conduction equation depends on both parameters of 119909and 119905119873 scattered nodes are considered in the domain Ω
119860=
[0 1199090] times [0 119905max + 119879] We assume that
Ξ119868= (119909119896 119905119896)119873119868
119896=1 Ξ
119887= (119909119896 119905119896)119873119861
119896=1(13)
are two sets of scattered points in the domainΩ119860 where119873 =
119873119868+ 119873119861and Ξ
119868and Ξ
119887are the interior and the boundary
points in domains Ω119860= [0 119909
0] times [0 119905max + 119879] and Ω119887 =
[0 1199090] times [0 119905max] respectively Also we assume that
Ξ119887= Γ1cup Γ2cup Γ3 (14)
whereΓ1= (119909119894 119905119894) 119905119894= 0 0 ⩽ 119909
119894⩽ 1199090 119894 = 1 119899
Γ2= (119909
119895 119905119895) 119909119895= 1199090 0 lt 119905
119895⩽ 119905max 119895 = 1 119898
Γ3= (119909119903 119905119903) 119909119903= 1199090 0 lt 119905
119903⩽ 119905max 119903 = 1 119904
(15)
so that119873119861= 119899 + 119898 + 119904
We suppose that the solution of the problem 2 in Ω119860can
be expressed as follows
(119909 119905) =
119873119861
sum
119896=1
120582119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582119896120595119896(119909 119905) (16)
where 120601119896(119909 119905) = 119896(119909 minus 119909
119896 119905 minus 119905
119896+ 119879) and 119896(119909 119905) is the
fundamental solution of the heat equation which is describedin Section 32 Also 120595
119896= 120595((119909 119905) minus (119909
119896 119905119896)2) and 120595 is a
radial function which is defined in Section 31 To determinethe coefficients in (16) we impose the approximate solution to satisfy the given partial differential equation with the otherconditions at any point (119909 119905) isin Ξ
119868cup Ξ119861 so we achieve the
following system of linear equations
AΛ = b (17)
whereΛ = [1205821 120582
119873]119879 b = [0 120593(119909
119894) 119892(119905119895) ℎ(119905119903)]119879 and 0 is
119873119868times1 zeromatrix Also the119873times119873matrixA can be subdivided
into two submatrices as follows
A = [A119868
A119861
] (18)
where A119868= [119886(1)
119897119901] is the 119873
119868times 119873 obtained submatrix of
applying the interior points whose entries are defined asfollows
119886(1)
119897119901= 0 119897 = 1 119873
119868 119901 = 1 119873
119861 (19)
because as already mentioned above the fundamental func-tions 120601
119896satisfy the heat equation and also
119886(1)
119897119901= (
120597
120597119905
minus 1198862 1205972
1205971199092)120595119896(119909 119905)
119897 = 1 119873119868 119901 119896 = 119873
119861+ 1 119873 (119909 119905) isin Ξ
119868
(20)
4 Advances in Mathematical Physics
Using the boundary points of domain Ω119887 namely Ξ
119887 the
submatrix A119861= [119886(2)
119897119901] results where
119886(2)
119897119901= 120601119896(119909 119905)
119897 = 1 119899 119901 119896 = 1 119873119861 (119909 119905) isin Γ
1
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 1 119899 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
1
119886(2)
119897119901=
120597
120597119909
120601119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 1 119873119861 (119909 119905) isin Γ
2
119886(2)
119897119901=
120597
120597119909
120595119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
2
119886(2)
119897119901= 120601119896(119909 119905)
119897=119899 + 119898 + 1 119899 + 119898 + 119904 119901 119896=1 119873119861 (119909 119905)isinΓ
3
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 119899 + 119898 + 1 119899 + 119898 + 119904
119901 119896 = 119873119861+ 1 119873
(119909 119905) isin Γ3
(21)
It is essential to keep in mind that inherent errors inmeasurement data are inevitable On the other hand as itwas mentioned a radial basis function depends on the shapeparameter 119888 which has an effect on the condition number ofA Therefore the obtained system of linear equations (17) isill-conditioned So it cannot be solved directly Since smallperturbation in initial data may produce a large amount ofperturbation in the solution we use the rand function inmatlab in the numerical example presented in Section 6 andwe produce noisy data as the follows
119887119894= 119887119894(1 + 120575 sdot rand (119894)) 119894 = 1 119873 (22)
where 119887119894is the exact data rand(119894) is a random number
uniformly distributed in [minus1 1] and themagnitude120575displaysthe noise level of the measurement data
5 Regularization Method
Solving the system of linear equations (17) usually does notlead to accurate results by most numerical methods becausecondition number of matrix A is large It means that theill-conditioning of matrix A makes the numerical solutionunstable Now in methods as the proposed method in thisstudy which is based on RBFs the condition number of Adepends on some factors such as the shape parameter 119888 aswell On the other hand for fixed values of the shape param-eter 119888 the condition number increases with the number of
scattered nodes 119873 In practice the shape parameter mustbe adjusted with the number of interpolating points Alsothe accuracy of radial basis functions relies on the shapeparameter So in case a suitable amount of it is chosenthe accuracy of the approximate solution will be increasedDespite various research works which are done findingthe optimal choice of the shape parameter is still an openproblem [32ndash35] Accordingly some regularization methodsare presented to solve such ill-conditioned systems Tikhonovregularization (TR) method is mostly used by researchers[36] In this method the regularized solution Λ120572 for thesystem of linear equations (17) is explained as the solution ofthe following minimization problem
minΛ
10038171003817100381710038171003817AΛ minus b1003817100381710038171003817
1003817
2
+ 1205722Λ2 120572 gt 0 (23)
where sdot denotes the Euclidean norm and 120572 is called theregularization parameter Some methods such as 119871-curve[37] cross-validation (CV) and generalized cross-validation(GCV) [38] are carried out to determine the regularizationparameter 120572 for the TR method In this work we apply(GCV) to obtain regularization parameter In this methodregularization parameter 120572 minimizes the following (GCV)function
119866 (120572) =
10038171003817100381710038171003817AΛ120572 minus b1003817100381710038171003817
1003817
2
(trace (I119873minus AA119868))2
120572 gt 0 (24)
where A119868 = (AtrA + 1205722I119873)minus1Atr
The regularized solution (17) is shown by Λ120572lowast
= [120582120572lowast
1
120582120572lowast
119873]119879 in which 120572lowast is a minimizer of119866Then the approximate
solution for the problem 2 is written as follows
lowast
120572(119909 119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595119896(119909 119905)
119901 (119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601 (0 minus 119909
119896 119905 minus 119905119896) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595 (0 minus 119909
119896 119905 minus 119905119896)
(25)
6 Numerical Experiment
In this section we investigate the performance and the abilityof the present method by giving a test problem Thereforein order to illustrate the efficiency and the accuracy of theproposed method along with the TR method initially wedefine the root mean square (RMS) error the relative rootmean square (RES) error and the maximum absolute error119864infin(119906) as follows
RMS (119906) = radic 1
119873119905
119873119905
sum
119896=1
(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
RES (119906) =radicsum119873119905
119896=1(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
radicsum119873119905
119896=1(119906 (119909119896 119905119896))2
119864infin(119906) = max
1⩽119896⩽119873119905
1003816100381610038161003816119906 (119909119896 119905119896) minus lowast(119909119896 119905119896)1003816100381610038161003816
(26)
Advances in Mathematical Physics 5
Table 2 The obtained values of RMS(119906)RES(119906)RMS(119901) and RES(119901) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem A
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119901) RES(119901) 119864
infin(119901) cond(119860)
11 0017391 0012603 0044511 0014968 0012701 0024588 16111119890 + 09
25 0005081 0003682 0019141 0007528 0006388 0019141 35475119890 + 10
5 0005061 0003668 0014484 0005682 0004821 0014485 53298119890 + 11
10 0002907 0002107 0008510 0003587 0003043 0008510 23557119890 + 13
20 0003463 0002510 0013148 0004864 0004128 0013147 79497119890 + 13
30 0004472 0003241 0021264 0007954 0006749 0021264 11148119890 + 11
50 0003613 0002618 0012343 0005781 0004905 0012343 14673119890 + 11
where119873119905is the total number of testing points in the domain
Ω119887 119906(119909119896 119905119896) and lowast(119909
119896 119905119896) are the exact and the approxi-
mated values at these points respectively RMS RES and 119864infin
errors of the functions 119901(119905) and 119891(119909) are similarly definedas well In our computation the Matlab code developed byHansen [39] is used for solving the discrete ill-conditionedsystem of linear equations (17)
Example For simplicity we assume that 1199090= 05 and 119886 =
119897 = 119905max = 1 By these assumptions the exact solution of theproblem (1)ndash(3) is given as follows [40]
119906 (119909 119905) = 119890minus119905 sin (119909) + 1199092 + 2119905
119901 (119905) = 2119905
119891 (119909) = sin (119909) + 1199092
(27)
Since for a large fixed number of scattered points 119873the matrix A will be more ill-conditioned and also smallervalues of the shape parameter 119888 generate more accurateapproximations we suppose that 119888 = 01 and the numberof interpolating points is 119873 = 20 The obtained values ofRMS(119906) RES(119906) 119864
infin(119906) RMS(119901) RES(119901) 119864
infin(119901) RMS(119891)
RES(119891) and 119864infin(119891) as well as condition number of A for
120575 = 001 and 119873119905= 208 and various values of 119879 are given
in Tables 2 and 4 Numerical results indicate that this methodis not depended on parameter 119879 By the assumptions119873 = 20and 119879 = 41 and with various levels of noise added into thedata Tables 3 and 5 illustrate the relative root mean squareerror of 119901(119905) and 119891(119909) at three points 119909
0= 01 05 and
1199090= 09 of the interval [0 1] respectively Figures 1 and 3
feature out a comparison between the exact solutions and theapproximate solutions for 119909
0= 05 and 119879 = 3 and various
levels of noise added into the data It is observed that as thenoise level increases the approximated functions will haveacceptable accuracy Figures 2 and 4 display the relationshipbetween the accuracy and the parameter 119879with noise of level120575 = 001 added into the dataThey elucidate not only that thenumerical results are stable with respect to parameter 119879 butalso that they retain at the same level of accuracy for a widerange of values 119879 Therefore the accuracy of the numericalsolutions is not relatively dependent on the parameter 119879In addition the study of the presented numerical resultsvia MFS in [40] indicates that with noise of level 120575 = 0namely with the noiseless data and for 119909
0= 05 of the
interval [0 1] RMS(119901) = 17 times 10minus3 with 119879 = 19 and
Table 3 The resulted values of RES(119901) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 3491119890 minus 02 2949119890 minus 03 5095119890 minus 04 4129119890 minus 04
05 2703119890 minus 02 3306119890 minus 03 7707119890 minus 04 3069119890 minus 04
09 6242119890 minus 02 9011119890 minus 03 3292119890 minus 03 1126119890 minus 03
0 01 02 03 04 05 06 07 08 09 10
05
1
15
2
25
Exact
p(t) and its approximation for x0 = 05 T = 3p(t)
andplowast(t)
120575 = 001
120575 = 003
120575 = 005
t
Figure 1 The exact 119901 and its approximation 119901lowast obtained with119873 =
20 119879 = 3 120575 = 001 and 1199090= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
RMS(119891) = 189 times 10minus2 for 119879 = 3 Also when the data areconsidered with noise RMS(119901) = 284 times 10minus2 and RMS(119891) =353 times 10
minus2 while by the proposed method in this work andwith the same assumptions we achieve RMS(119901) = 30479 times10minus5 and RMS(119891) = 35691 times 10minus6 for 120575 = 0 Also with noise
of level 120575 = 10minus4 we obtain RMS(119901) = 11021 times 10minus4 andRMS(119891) = 19204 times 10minus4 Accordingly the MFSRBF whichis based on the fundamental solutions of the heat equationand radial basis functions is more accurate in comparison toMFS
Table 3 indicates the values of the obtained RES(119901) forvarious levels of noise different values of 119909
0and 119879 = 41 by
MFSRBF
6 Advances in Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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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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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Mathematical Physics
some of these kinds problems [27 28] Consequently in thiswork we will present a meshless numerical scheme basedon combining the radial basis function and the fundamentalsolution of the heat equation in order to approximate thesolution of a backward inverse heat conduction problem(BIHCP) the problem inwhich an unknown initial conditionorand temperature distribution in previous time will bedetermined This kind of problem may emerge in manypractical application areas such as archeology and mantleplumes [29] On the other hand since the system of the linearequations obtained from discretizing the problem in thepresentedmethod is ill-conditioned Tikhonov regularization(TR) method is applied in order to solve it The generalizedcross-validation (GCV) criterion has been assigned to adoptan optimum amount of the regularization parameter Thestructure of the rest of this work is organized as follows InSection 2 we represent the mathematical formulation of theproblem The method of fundamental solutions radial basisfunctions method and method of fundamental solution-radial basis functions (MFSRBF) are described in Section 3Section 4 embraces Tikhonov regularization method with arule for choosing an appropriate regularization parameterIn Section 5 we present the obtained numerical results ofsolving a test problem Section 6 ends in a brief conclusionand some suggestions
2 Mathematical Formulation of the Problem
In this section we consider the following one-dimensionalinverse heat conduction problem
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω = (0 119897) times (0 119905max) (1)
with the following initial and boundary conditions
119906 (119909 0) =
120593 (119909) 119909 isin [0 1199090)
119891 (119909) 119909 isin [1199090 119897]
119906 (0 119905) = 119901 (119905) 119905 isin [0 119905max]
119906 (119897 119905) = 119902 (119905) 119905 isin [0 119905max]
(2)
where 120593(119909) and 119902(119905) are considered as known functions and119905max is a given positive constant while 119891(119909) and 119901(119905) areregarded as unknown functions So in order to estimate119891(119909)and 119901(119905) we consider additional temperature measurementsand heat flux given at a point 119909
0 1199090isin (0 1) as overspecified
conditions119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
(3)
To solve the above problem at first we divide the problem(1)ndash(3) into two separate problems The problem 119860 is asfollows
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω
119860= (0 119909
0) times (0 119905max)
119906 (119909 0) = 120593 (119909) 119909 isin [0 1199090]
119906 (0 119905) = 119901 (119905) 119905 isin [0 119905max]
119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
(4)
and the problem 119861 is considered as follows
119906119905minus 1198862119906119909119909= 0 (119909 119905) isin Ω
119861= (1199090 119897) times (0 119905max)
119906 (119909 0) = 119891 (119909) 119909 isin [1199090 119897]
119906119909(1199090 119905) = 119892 (119905) 119905 isin [0 119905max]
119906 (1199090 119905) = ℎ (119905) 119905 isin [0 119905max]
119906 (119897 119905) = 119902 (119905) 119905 isin [0 119905max]
(5)
Obviously the problems A and B are considered as IHCPwhere 119901(119905) 119906(119909 119905) and 119891(119909) 119906(119909 119905) are unknown functionsin the problems A and B respectively
3 Method of Fundamental Solutions andMethod of Radial Basis Functions
In this section we introduce the numerical scheme forsolving the problem (1)ndash(3) using the fundamental solutionsand radial basis functions
31Method of Fundamental Solutions Thefundamental solu-tion of (1) is presented as below
119896 (119909 119905) =
1
2119886radic120587119905
119890minus1199092(41198862119905)119867(119905) (6)
where 119867(119905) is Heaviside unit function Assuming that 119879 gt
119905max is a constant it can be demonstrated that the time shiftfunction
120601 (119909 119905) = 119896 (119909 119905 + 119879) (7)
is also a nonsingular solution of (1) in the domainΩIn order to solve an IHCP by MFS as the problem 119860
since the basis function 120601 satisfies the heat equation (1)automatically we assume that (119909
119896 119905119896)119873
119896=1is a given set
of scattered points on the boundary 120597Ω119860 An approximate
solution is defined as a linear combination of 120601 as follows
Φ (119909 119905) =
119873
sum
119896=1
120582119896120601 (119909 minus 119909
119896 119905 minus 119905119896) (8)
where120601(119909 119905) is given by (7) and 120582119896rsquos are unknown coefficients
which can be determined by solving the following matrixequation
AΛ = b (9)
where Λ = [1205821 120582
119873]119879 and b is a 119873 times 1 known vector
Also as the fundamental functions are the solution of theheat equation only the initial and boundary conditions arepracticed to make the system of linear equations that is A is
Advances in Mathematical Physics 3
Table 1 Some well-known radial basis functions
Infinitely smooth RBFs 120595(119903)
Gaussian (GA) 119890(minus1198881199032)
Multiquadrics (MQ) radic1199032+ 1198882
Inverse multiquadrics (IMQ) (radic1199032+ 1198882)
minus1
Inverse quadric (IQ) (1199032+ 1198882)minus1
a119873 times119873 square matrix which is defined using the initial andthe boundary conditions as follows
A = [119886119894119895] 119886
119894119895= Φ (119909 119905) (119909 119905) isin 120597Ω
119860 119894 119895 = 1 119873
(10)
For more details see [13 18]
32 Method of Radial Basis Functions In this section weconsider RBF method for interpolation of scattered dataSuppose that xlowast and x are a fixed point and an arbitrary pointin R119889 respectively A radial function 120595lowast is defined via 120595lowast =120595lowast(119903) where 119903 = x minus xlowast
2 That is the radial function 120595lowast
depends only on the distance between x and xlowastThis propertyimplies that the RBFs 120595lowast are radially symmetric about xlowastSomewell-known infinitely smooth RBFs are given in Table 1As it is observed these functions depend on a free parameter119888 known as the shape parameter which has an important rolein approximation theory using RBFs
Let x119896119873
119896=1be a given set of distinct points in domain Ω
in R119889 The main idea of using the RBFs is interpolation witha linear combination of RBFs of the same types as follows
Ψ (x) =119873
sum
119896=1
120582119896120595119896(x) (11)
where 120595119896(x) = 120595(x minus x
119896) and 120582
119896rsquos are unknown coefficients
for 119896 = 1 2 119873 Assume that we want to interpolate thegiven values 119891
119896= 119891(x
119896) 119896 = 1 2 119873 The unknown
coefficients 120582119896rsquos are obtained so thatΨ(x
119896) = 119891119896 119896 = 1 2
119873 which results in the following system of linear equations
AΛ = b (12)
where Λ = [1205821 120582
119873]119879 b = [119891
1 119891
119873]119879 and A = [119886
119894119895]
for 119894 119895 = 1 2 119873 with entries 119886119894119895= 120595119895(x119894) Generally
the matrix A has been shown to be positive definite (andtherefore nonsingular) for distinct interpolation points forinfinitely smooth RBFs by Schoenbergrsquos theorem [30] Alsousing Micchellirsquos theorem [31] it is shown that A is aninvertible matrix for a distinct set of scattered nodes in thecase of MQ-RBF
4 Method of Fundamental Solutions-RadialBasis Functions
This section is specified to introduce the numerical schemefor solving the problem (1)ndash(3) using the fundamental solu-tions and radial basis functions At first we are going to
figure out how radial basis functions and the fundamentalsolutions are applied to approximate the solution of theproblem 119860 2 Similarly this method will be used for solvingthe problem 119861 on domain Ω
119861 Because the one-dimensional
heat conduction equation depends on both parameters of 119909and 119905119873 scattered nodes are considered in the domain Ω
119860=
[0 1199090] times [0 119905max + 119879] We assume that
Ξ119868= (119909119896 119905119896)119873119868
119896=1 Ξ
119887= (119909119896 119905119896)119873119861
119896=1(13)
are two sets of scattered points in the domainΩ119860 where119873 =
119873119868+ 119873119861and Ξ
119868and Ξ
119887are the interior and the boundary
points in domains Ω119860= [0 119909
0] times [0 119905max + 119879] and Ω119887 =
[0 1199090] times [0 119905max] respectively Also we assume that
Ξ119887= Γ1cup Γ2cup Γ3 (14)
whereΓ1= (119909119894 119905119894) 119905119894= 0 0 ⩽ 119909
119894⩽ 1199090 119894 = 1 119899
Γ2= (119909
119895 119905119895) 119909119895= 1199090 0 lt 119905
119895⩽ 119905max 119895 = 1 119898
Γ3= (119909119903 119905119903) 119909119903= 1199090 0 lt 119905
119903⩽ 119905max 119903 = 1 119904
(15)
so that119873119861= 119899 + 119898 + 119904
We suppose that the solution of the problem 2 in Ω119860can
be expressed as follows
(119909 119905) =
119873119861
sum
119896=1
120582119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582119896120595119896(119909 119905) (16)
where 120601119896(119909 119905) = 119896(119909 minus 119909
119896 119905 minus 119905
119896+ 119879) and 119896(119909 119905) is the
fundamental solution of the heat equation which is describedin Section 32 Also 120595
119896= 120595((119909 119905) minus (119909
119896 119905119896)2) and 120595 is a
radial function which is defined in Section 31 To determinethe coefficients in (16) we impose the approximate solution to satisfy the given partial differential equation with the otherconditions at any point (119909 119905) isin Ξ
119868cup Ξ119861 so we achieve the
following system of linear equations
AΛ = b (17)
whereΛ = [1205821 120582
119873]119879 b = [0 120593(119909
119894) 119892(119905119895) ℎ(119905119903)]119879 and 0 is
119873119868times1 zeromatrix Also the119873times119873matrixA can be subdivided
into two submatrices as follows
A = [A119868
A119861
] (18)
where A119868= [119886(1)
119897119901] is the 119873
119868times 119873 obtained submatrix of
applying the interior points whose entries are defined asfollows
119886(1)
119897119901= 0 119897 = 1 119873
119868 119901 = 1 119873
119861 (19)
because as already mentioned above the fundamental func-tions 120601
119896satisfy the heat equation and also
119886(1)
119897119901= (
120597
120597119905
minus 1198862 1205972
1205971199092)120595119896(119909 119905)
119897 = 1 119873119868 119901 119896 = 119873
119861+ 1 119873 (119909 119905) isin Ξ
119868
(20)
4 Advances in Mathematical Physics
Using the boundary points of domain Ω119887 namely Ξ
119887 the
submatrix A119861= [119886(2)
119897119901] results where
119886(2)
119897119901= 120601119896(119909 119905)
119897 = 1 119899 119901 119896 = 1 119873119861 (119909 119905) isin Γ
1
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 1 119899 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
1
119886(2)
119897119901=
120597
120597119909
120601119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 1 119873119861 (119909 119905) isin Γ
2
119886(2)
119897119901=
120597
120597119909
120595119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
2
119886(2)
119897119901= 120601119896(119909 119905)
119897=119899 + 119898 + 1 119899 + 119898 + 119904 119901 119896=1 119873119861 (119909 119905)isinΓ
3
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 119899 + 119898 + 1 119899 + 119898 + 119904
119901 119896 = 119873119861+ 1 119873
(119909 119905) isin Γ3
(21)
It is essential to keep in mind that inherent errors inmeasurement data are inevitable On the other hand as itwas mentioned a radial basis function depends on the shapeparameter 119888 which has an effect on the condition number ofA Therefore the obtained system of linear equations (17) isill-conditioned So it cannot be solved directly Since smallperturbation in initial data may produce a large amount ofperturbation in the solution we use the rand function inmatlab in the numerical example presented in Section 6 andwe produce noisy data as the follows
119887119894= 119887119894(1 + 120575 sdot rand (119894)) 119894 = 1 119873 (22)
where 119887119894is the exact data rand(119894) is a random number
uniformly distributed in [minus1 1] and themagnitude120575displaysthe noise level of the measurement data
5 Regularization Method
Solving the system of linear equations (17) usually does notlead to accurate results by most numerical methods becausecondition number of matrix A is large It means that theill-conditioning of matrix A makes the numerical solutionunstable Now in methods as the proposed method in thisstudy which is based on RBFs the condition number of Adepends on some factors such as the shape parameter 119888 aswell On the other hand for fixed values of the shape param-eter 119888 the condition number increases with the number of
scattered nodes 119873 In practice the shape parameter mustbe adjusted with the number of interpolating points Alsothe accuracy of radial basis functions relies on the shapeparameter So in case a suitable amount of it is chosenthe accuracy of the approximate solution will be increasedDespite various research works which are done findingthe optimal choice of the shape parameter is still an openproblem [32ndash35] Accordingly some regularization methodsare presented to solve such ill-conditioned systems Tikhonovregularization (TR) method is mostly used by researchers[36] In this method the regularized solution Λ120572 for thesystem of linear equations (17) is explained as the solution ofthe following minimization problem
minΛ
10038171003817100381710038171003817AΛ minus b1003817100381710038171003817
1003817
2
+ 1205722Λ2 120572 gt 0 (23)
where sdot denotes the Euclidean norm and 120572 is called theregularization parameter Some methods such as 119871-curve[37] cross-validation (CV) and generalized cross-validation(GCV) [38] are carried out to determine the regularizationparameter 120572 for the TR method In this work we apply(GCV) to obtain regularization parameter In this methodregularization parameter 120572 minimizes the following (GCV)function
119866 (120572) =
10038171003817100381710038171003817AΛ120572 minus b1003817100381710038171003817
1003817
2
(trace (I119873minus AA119868))2
120572 gt 0 (24)
where A119868 = (AtrA + 1205722I119873)minus1Atr
The regularized solution (17) is shown by Λ120572lowast
= [120582120572lowast
1
120582120572lowast
119873]119879 in which 120572lowast is a minimizer of119866Then the approximate
solution for the problem 2 is written as follows
lowast
120572(119909 119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595119896(119909 119905)
119901 (119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601 (0 minus 119909
119896 119905 minus 119905119896) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595 (0 minus 119909
119896 119905 minus 119905119896)
(25)
6 Numerical Experiment
In this section we investigate the performance and the abilityof the present method by giving a test problem Thereforein order to illustrate the efficiency and the accuracy of theproposed method along with the TR method initially wedefine the root mean square (RMS) error the relative rootmean square (RES) error and the maximum absolute error119864infin(119906) as follows
RMS (119906) = radic 1
119873119905
119873119905
sum
119896=1
(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
RES (119906) =radicsum119873119905
119896=1(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
radicsum119873119905
119896=1(119906 (119909119896 119905119896))2
119864infin(119906) = max
1⩽119896⩽119873119905
1003816100381610038161003816119906 (119909119896 119905119896) minus lowast(119909119896 119905119896)1003816100381610038161003816
(26)
Advances in Mathematical Physics 5
Table 2 The obtained values of RMS(119906)RES(119906)RMS(119901) and RES(119901) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem A
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119901) RES(119901) 119864
infin(119901) cond(119860)
11 0017391 0012603 0044511 0014968 0012701 0024588 16111119890 + 09
25 0005081 0003682 0019141 0007528 0006388 0019141 35475119890 + 10
5 0005061 0003668 0014484 0005682 0004821 0014485 53298119890 + 11
10 0002907 0002107 0008510 0003587 0003043 0008510 23557119890 + 13
20 0003463 0002510 0013148 0004864 0004128 0013147 79497119890 + 13
30 0004472 0003241 0021264 0007954 0006749 0021264 11148119890 + 11
50 0003613 0002618 0012343 0005781 0004905 0012343 14673119890 + 11
where119873119905is the total number of testing points in the domain
Ω119887 119906(119909119896 119905119896) and lowast(119909
119896 119905119896) are the exact and the approxi-
mated values at these points respectively RMS RES and 119864infin
errors of the functions 119901(119905) and 119891(119909) are similarly definedas well In our computation the Matlab code developed byHansen [39] is used for solving the discrete ill-conditionedsystem of linear equations (17)
Example For simplicity we assume that 1199090= 05 and 119886 =
119897 = 119905max = 1 By these assumptions the exact solution of theproblem (1)ndash(3) is given as follows [40]
119906 (119909 119905) = 119890minus119905 sin (119909) + 1199092 + 2119905
119901 (119905) = 2119905
119891 (119909) = sin (119909) + 1199092
(27)
Since for a large fixed number of scattered points 119873the matrix A will be more ill-conditioned and also smallervalues of the shape parameter 119888 generate more accurateapproximations we suppose that 119888 = 01 and the numberof interpolating points is 119873 = 20 The obtained values ofRMS(119906) RES(119906) 119864
infin(119906) RMS(119901) RES(119901) 119864
infin(119901) RMS(119891)
RES(119891) and 119864infin(119891) as well as condition number of A for
120575 = 001 and 119873119905= 208 and various values of 119879 are given
in Tables 2 and 4 Numerical results indicate that this methodis not depended on parameter 119879 By the assumptions119873 = 20and 119879 = 41 and with various levels of noise added into thedata Tables 3 and 5 illustrate the relative root mean squareerror of 119901(119905) and 119891(119909) at three points 119909
0= 01 05 and
1199090= 09 of the interval [0 1] respectively Figures 1 and 3
feature out a comparison between the exact solutions and theapproximate solutions for 119909
0= 05 and 119879 = 3 and various
levels of noise added into the data It is observed that as thenoise level increases the approximated functions will haveacceptable accuracy Figures 2 and 4 display the relationshipbetween the accuracy and the parameter 119879with noise of level120575 = 001 added into the dataThey elucidate not only that thenumerical results are stable with respect to parameter 119879 butalso that they retain at the same level of accuracy for a widerange of values 119879 Therefore the accuracy of the numericalsolutions is not relatively dependent on the parameter 119879In addition the study of the presented numerical resultsvia MFS in [40] indicates that with noise of level 120575 = 0namely with the noiseless data and for 119909
0= 05 of the
interval [0 1] RMS(119901) = 17 times 10minus3 with 119879 = 19 and
Table 3 The resulted values of RES(119901) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 3491119890 minus 02 2949119890 minus 03 5095119890 minus 04 4129119890 minus 04
05 2703119890 minus 02 3306119890 minus 03 7707119890 minus 04 3069119890 minus 04
09 6242119890 minus 02 9011119890 minus 03 3292119890 minus 03 1126119890 minus 03
0 01 02 03 04 05 06 07 08 09 10
05
1
15
2
25
Exact
p(t) and its approximation for x0 = 05 T = 3p(t)
andplowast(t)
120575 = 001
120575 = 003
120575 = 005
t
Figure 1 The exact 119901 and its approximation 119901lowast obtained with119873 =
20 119879 = 3 120575 = 001 and 1199090= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
RMS(119891) = 189 times 10minus2 for 119879 = 3 Also when the data areconsidered with noise RMS(119901) = 284 times 10minus2 and RMS(119891) =353 times 10
minus2 while by the proposed method in this work andwith the same assumptions we achieve RMS(119901) = 30479 times10minus5 and RMS(119891) = 35691 times 10minus6 for 120575 = 0 Also with noise
of level 120575 = 10minus4 we obtain RMS(119901) = 11021 times 10minus4 andRMS(119891) = 19204 times 10minus4 Accordingly the MFSRBF whichis based on the fundamental solutions of the heat equationand radial basis functions is more accurate in comparison toMFS
Table 3 indicates the values of the obtained RES(119901) forvarious levels of noise different values of 119909
0and 119879 = 41 by
MFSRBF
6 Advances in Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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 Mathematical Physics 3
Table 1 Some well-known radial basis functions
Infinitely smooth RBFs 120595(119903)
Gaussian (GA) 119890(minus1198881199032)
Multiquadrics (MQ) radic1199032+ 1198882
Inverse multiquadrics (IMQ) (radic1199032+ 1198882)
minus1
Inverse quadric (IQ) (1199032+ 1198882)minus1
a119873 times119873 square matrix which is defined using the initial andthe boundary conditions as follows
A = [119886119894119895] 119886
119894119895= Φ (119909 119905) (119909 119905) isin 120597Ω
119860 119894 119895 = 1 119873
(10)
For more details see [13 18]
32 Method of Radial Basis Functions In this section weconsider RBF method for interpolation of scattered dataSuppose that xlowast and x are a fixed point and an arbitrary pointin R119889 respectively A radial function 120595lowast is defined via 120595lowast =120595lowast(119903) where 119903 = x minus xlowast
2 That is the radial function 120595lowast
depends only on the distance between x and xlowastThis propertyimplies that the RBFs 120595lowast are radially symmetric about xlowastSomewell-known infinitely smooth RBFs are given in Table 1As it is observed these functions depend on a free parameter119888 known as the shape parameter which has an important rolein approximation theory using RBFs
Let x119896119873
119896=1be a given set of distinct points in domain Ω
in R119889 The main idea of using the RBFs is interpolation witha linear combination of RBFs of the same types as follows
Ψ (x) =119873
sum
119896=1
120582119896120595119896(x) (11)
where 120595119896(x) = 120595(x minus x
119896) and 120582
119896rsquos are unknown coefficients
for 119896 = 1 2 119873 Assume that we want to interpolate thegiven values 119891
119896= 119891(x
119896) 119896 = 1 2 119873 The unknown
coefficients 120582119896rsquos are obtained so thatΨ(x
119896) = 119891119896 119896 = 1 2
119873 which results in the following system of linear equations
AΛ = b (12)
where Λ = [1205821 120582
119873]119879 b = [119891
1 119891
119873]119879 and A = [119886
119894119895]
for 119894 119895 = 1 2 119873 with entries 119886119894119895= 120595119895(x119894) Generally
the matrix A has been shown to be positive definite (andtherefore nonsingular) for distinct interpolation points forinfinitely smooth RBFs by Schoenbergrsquos theorem [30] Alsousing Micchellirsquos theorem [31] it is shown that A is aninvertible matrix for a distinct set of scattered nodes in thecase of MQ-RBF
4 Method of Fundamental Solutions-RadialBasis Functions
This section is specified to introduce the numerical schemefor solving the problem (1)ndash(3) using the fundamental solu-tions and radial basis functions At first we are going to
figure out how radial basis functions and the fundamentalsolutions are applied to approximate the solution of theproblem 119860 2 Similarly this method will be used for solvingthe problem 119861 on domain Ω
119861 Because the one-dimensional
heat conduction equation depends on both parameters of 119909and 119905119873 scattered nodes are considered in the domain Ω
119860=
[0 1199090] times [0 119905max + 119879] We assume that
Ξ119868= (119909119896 119905119896)119873119868
119896=1 Ξ
119887= (119909119896 119905119896)119873119861
119896=1(13)
are two sets of scattered points in the domainΩ119860 where119873 =
119873119868+ 119873119861and Ξ
119868and Ξ
119887are the interior and the boundary
points in domains Ω119860= [0 119909
0] times [0 119905max + 119879] and Ω119887 =
[0 1199090] times [0 119905max] respectively Also we assume that
Ξ119887= Γ1cup Γ2cup Γ3 (14)
whereΓ1= (119909119894 119905119894) 119905119894= 0 0 ⩽ 119909
119894⩽ 1199090 119894 = 1 119899
Γ2= (119909
119895 119905119895) 119909119895= 1199090 0 lt 119905
119895⩽ 119905max 119895 = 1 119898
Γ3= (119909119903 119905119903) 119909119903= 1199090 0 lt 119905
119903⩽ 119905max 119903 = 1 119904
(15)
so that119873119861= 119899 + 119898 + 119904
We suppose that the solution of the problem 2 in Ω119860can
be expressed as follows
(119909 119905) =
119873119861
sum
119896=1
120582119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582119896120595119896(119909 119905) (16)
where 120601119896(119909 119905) = 119896(119909 minus 119909
119896 119905 minus 119905
119896+ 119879) and 119896(119909 119905) is the
fundamental solution of the heat equation which is describedin Section 32 Also 120595
119896= 120595((119909 119905) minus (119909
119896 119905119896)2) and 120595 is a
radial function which is defined in Section 31 To determinethe coefficients in (16) we impose the approximate solution to satisfy the given partial differential equation with the otherconditions at any point (119909 119905) isin Ξ
119868cup Ξ119861 so we achieve the
following system of linear equations
AΛ = b (17)
whereΛ = [1205821 120582
119873]119879 b = [0 120593(119909
119894) 119892(119905119895) ℎ(119905119903)]119879 and 0 is
119873119868times1 zeromatrix Also the119873times119873matrixA can be subdivided
into two submatrices as follows
A = [A119868
A119861
] (18)
where A119868= [119886(1)
119897119901] is the 119873
119868times 119873 obtained submatrix of
applying the interior points whose entries are defined asfollows
119886(1)
119897119901= 0 119897 = 1 119873
119868 119901 = 1 119873
119861 (19)
because as already mentioned above the fundamental func-tions 120601
119896satisfy the heat equation and also
119886(1)
119897119901= (
120597
120597119905
minus 1198862 1205972
1205971199092)120595119896(119909 119905)
119897 = 1 119873119868 119901 119896 = 119873
119861+ 1 119873 (119909 119905) isin Ξ
119868
(20)
4 Advances in Mathematical Physics
Using the boundary points of domain Ω119887 namely Ξ
119887 the
submatrix A119861= [119886(2)
119897119901] results where
119886(2)
119897119901= 120601119896(119909 119905)
119897 = 1 119899 119901 119896 = 1 119873119861 (119909 119905) isin Γ
1
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 1 119899 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
1
119886(2)
119897119901=
120597
120597119909
120601119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 1 119873119861 (119909 119905) isin Γ
2
119886(2)
119897119901=
120597
120597119909
120595119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
2
119886(2)
119897119901= 120601119896(119909 119905)
119897=119899 + 119898 + 1 119899 + 119898 + 119904 119901 119896=1 119873119861 (119909 119905)isinΓ
3
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 119899 + 119898 + 1 119899 + 119898 + 119904
119901 119896 = 119873119861+ 1 119873
(119909 119905) isin Γ3
(21)
It is essential to keep in mind that inherent errors inmeasurement data are inevitable On the other hand as itwas mentioned a radial basis function depends on the shapeparameter 119888 which has an effect on the condition number ofA Therefore the obtained system of linear equations (17) isill-conditioned So it cannot be solved directly Since smallperturbation in initial data may produce a large amount ofperturbation in the solution we use the rand function inmatlab in the numerical example presented in Section 6 andwe produce noisy data as the follows
119887119894= 119887119894(1 + 120575 sdot rand (119894)) 119894 = 1 119873 (22)
where 119887119894is the exact data rand(119894) is a random number
uniformly distributed in [minus1 1] and themagnitude120575displaysthe noise level of the measurement data
5 Regularization Method
Solving the system of linear equations (17) usually does notlead to accurate results by most numerical methods becausecondition number of matrix A is large It means that theill-conditioning of matrix A makes the numerical solutionunstable Now in methods as the proposed method in thisstudy which is based on RBFs the condition number of Adepends on some factors such as the shape parameter 119888 aswell On the other hand for fixed values of the shape param-eter 119888 the condition number increases with the number of
scattered nodes 119873 In practice the shape parameter mustbe adjusted with the number of interpolating points Alsothe accuracy of radial basis functions relies on the shapeparameter So in case a suitable amount of it is chosenthe accuracy of the approximate solution will be increasedDespite various research works which are done findingthe optimal choice of the shape parameter is still an openproblem [32ndash35] Accordingly some regularization methodsare presented to solve such ill-conditioned systems Tikhonovregularization (TR) method is mostly used by researchers[36] In this method the regularized solution Λ120572 for thesystem of linear equations (17) is explained as the solution ofthe following minimization problem
minΛ
10038171003817100381710038171003817AΛ minus b1003817100381710038171003817
1003817
2
+ 1205722Λ2 120572 gt 0 (23)
where sdot denotes the Euclidean norm and 120572 is called theregularization parameter Some methods such as 119871-curve[37] cross-validation (CV) and generalized cross-validation(GCV) [38] are carried out to determine the regularizationparameter 120572 for the TR method In this work we apply(GCV) to obtain regularization parameter In this methodregularization parameter 120572 minimizes the following (GCV)function
119866 (120572) =
10038171003817100381710038171003817AΛ120572 minus b1003817100381710038171003817
1003817
2
(trace (I119873minus AA119868))2
120572 gt 0 (24)
where A119868 = (AtrA + 1205722I119873)minus1Atr
The regularized solution (17) is shown by Λ120572lowast
= [120582120572lowast
1
120582120572lowast
119873]119879 in which 120572lowast is a minimizer of119866Then the approximate
solution for the problem 2 is written as follows
lowast
120572(119909 119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595119896(119909 119905)
119901 (119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601 (0 minus 119909
119896 119905 minus 119905119896) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595 (0 minus 119909
119896 119905 minus 119905119896)
(25)
6 Numerical Experiment
In this section we investigate the performance and the abilityof the present method by giving a test problem Thereforein order to illustrate the efficiency and the accuracy of theproposed method along with the TR method initially wedefine the root mean square (RMS) error the relative rootmean square (RES) error and the maximum absolute error119864infin(119906) as follows
RMS (119906) = radic 1
119873119905
119873119905
sum
119896=1
(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
RES (119906) =radicsum119873119905
119896=1(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
radicsum119873119905
119896=1(119906 (119909119896 119905119896))2
119864infin(119906) = max
1⩽119896⩽119873119905
1003816100381610038161003816119906 (119909119896 119905119896) minus lowast(119909119896 119905119896)1003816100381610038161003816
(26)
Advances in Mathematical Physics 5
Table 2 The obtained values of RMS(119906)RES(119906)RMS(119901) and RES(119901) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem A
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119901) RES(119901) 119864
infin(119901) cond(119860)
11 0017391 0012603 0044511 0014968 0012701 0024588 16111119890 + 09
25 0005081 0003682 0019141 0007528 0006388 0019141 35475119890 + 10
5 0005061 0003668 0014484 0005682 0004821 0014485 53298119890 + 11
10 0002907 0002107 0008510 0003587 0003043 0008510 23557119890 + 13
20 0003463 0002510 0013148 0004864 0004128 0013147 79497119890 + 13
30 0004472 0003241 0021264 0007954 0006749 0021264 11148119890 + 11
50 0003613 0002618 0012343 0005781 0004905 0012343 14673119890 + 11
where119873119905is the total number of testing points in the domain
Ω119887 119906(119909119896 119905119896) and lowast(119909
119896 119905119896) are the exact and the approxi-
mated values at these points respectively RMS RES and 119864infin
errors of the functions 119901(119905) and 119891(119909) are similarly definedas well In our computation the Matlab code developed byHansen [39] is used for solving the discrete ill-conditionedsystem of linear equations (17)
Example For simplicity we assume that 1199090= 05 and 119886 =
119897 = 119905max = 1 By these assumptions the exact solution of theproblem (1)ndash(3) is given as follows [40]
119906 (119909 119905) = 119890minus119905 sin (119909) + 1199092 + 2119905
119901 (119905) = 2119905
119891 (119909) = sin (119909) + 1199092
(27)
Since for a large fixed number of scattered points 119873the matrix A will be more ill-conditioned and also smallervalues of the shape parameter 119888 generate more accurateapproximations we suppose that 119888 = 01 and the numberof interpolating points is 119873 = 20 The obtained values ofRMS(119906) RES(119906) 119864
infin(119906) RMS(119901) RES(119901) 119864
infin(119901) RMS(119891)
RES(119891) and 119864infin(119891) as well as condition number of A for
120575 = 001 and 119873119905= 208 and various values of 119879 are given
in Tables 2 and 4 Numerical results indicate that this methodis not depended on parameter 119879 By the assumptions119873 = 20and 119879 = 41 and with various levels of noise added into thedata Tables 3 and 5 illustrate the relative root mean squareerror of 119901(119905) and 119891(119909) at three points 119909
0= 01 05 and
1199090= 09 of the interval [0 1] respectively Figures 1 and 3
feature out a comparison between the exact solutions and theapproximate solutions for 119909
0= 05 and 119879 = 3 and various
levels of noise added into the data It is observed that as thenoise level increases the approximated functions will haveacceptable accuracy Figures 2 and 4 display the relationshipbetween the accuracy and the parameter 119879with noise of level120575 = 001 added into the dataThey elucidate not only that thenumerical results are stable with respect to parameter 119879 butalso that they retain at the same level of accuracy for a widerange of values 119879 Therefore the accuracy of the numericalsolutions is not relatively dependent on the parameter 119879In addition the study of the presented numerical resultsvia MFS in [40] indicates that with noise of level 120575 = 0namely with the noiseless data and for 119909
0= 05 of the
interval [0 1] RMS(119901) = 17 times 10minus3 with 119879 = 19 and
Table 3 The resulted values of RES(119901) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 3491119890 minus 02 2949119890 minus 03 5095119890 minus 04 4129119890 minus 04
05 2703119890 minus 02 3306119890 minus 03 7707119890 minus 04 3069119890 minus 04
09 6242119890 minus 02 9011119890 minus 03 3292119890 minus 03 1126119890 minus 03
0 01 02 03 04 05 06 07 08 09 10
05
1
15
2
25
Exact
p(t) and its approximation for x0 = 05 T = 3p(t)
andplowast(t)
120575 = 001
120575 = 003
120575 = 005
t
Figure 1 The exact 119901 and its approximation 119901lowast obtained with119873 =
20 119879 = 3 120575 = 001 and 1199090= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
RMS(119891) = 189 times 10minus2 for 119879 = 3 Also when the data areconsidered with noise RMS(119901) = 284 times 10minus2 and RMS(119891) =353 times 10
minus2 while by the proposed method in this work andwith the same assumptions we achieve RMS(119901) = 30479 times10minus5 and RMS(119891) = 35691 times 10minus6 for 120575 = 0 Also with noise
of level 120575 = 10minus4 we obtain RMS(119901) = 11021 times 10minus4 andRMS(119891) = 19204 times 10minus4 Accordingly the MFSRBF whichis based on the fundamental solutions of the heat equationand radial basis functions is more accurate in comparison toMFS
Table 3 indicates the values of the obtained RES(119901) forvarious levels of noise different values of 119909
0and 119879 = 41 by
MFSRBF
6 Advances in Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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 Mathematical Physics
Using the boundary points of domain Ω119887 namely Ξ
119887 the
submatrix A119861= [119886(2)
119897119901] results where
119886(2)
119897119901= 120601119896(119909 119905)
119897 = 1 119899 119901 119896 = 1 119873119861 (119909 119905) isin Γ
1
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 1 119899 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
1
119886(2)
119897119901=
120597
120597119909
120601119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 1 119873119861 (119909 119905) isin Γ
2
119886(2)
119897119901=
120597
120597119909
120595119896(119909 119905)
119897 = 119899 + 1 119899 + 119898 119901 119896 = 119873119861+ 1 119873 (119909 119905) isin Γ
2
119886(2)
119897119901= 120601119896(119909 119905)
119897=119899 + 119898 + 1 119899 + 119898 + 119904 119901 119896=1 119873119861 (119909 119905)isinΓ
3
119886(2)
119897119901= 120595119896(119909 119905)
119897 = 119899 + 119898 + 1 119899 + 119898 + 119904
119901 119896 = 119873119861+ 1 119873
(119909 119905) isin Γ3
(21)
It is essential to keep in mind that inherent errors inmeasurement data are inevitable On the other hand as itwas mentioned a radial basis function depends on the shapeparameter 119888 which has an effect on the condition number ofA Therefore the obtained system of linear equations (17) isill-conditioned So it cannot be solved directly Since smallperturbation in initial data may produce a large amount ofperturbation in the solution we use the rand function inmatlab in the numerical example presented in Section 6 andwe produce noisy data as the follows
119887119894= 119887119894(1 + 120575 sdot rand (119894)) 119894 = 1 119873 (22)
where 119887119894is the exact data rand(119894) is a random number
uniformly distributed in [minus1 1] and themagnitude120575displaysthe noise level of the measurement data
5 Regularization Method
Solving the system of linear equations (17) usually does notlead to accurate results by most numerical methods becausecondition number of matrix A is large It means that theill-conditioning of matrix A makes the numerical solutionunstable Now in methods as the proposed method in thisstudy which is based on RBFs the condition number of Adepends on some factors such as the shape parameter 119888 aswell On the other hand for fixed values of the shape param-eter 119888 the condition number increases with the number of
scattered nodes 119873 In practice the shape parameter mustbe adjusted with the number of interpolating points Alsothe accuracy of radial basis functions relies on the shapeparameter So in case a suitable amount of it is chosenthe accuracy of the approximate solution will be increasedDespite various research works which are done findingthe optimal choice of the shape parameter is still an openproblem [32ndash35] Accordingly some regularization methodsare presented to solve such ill-conditioned systems Tikhonovregularization (TR) method is mostly used by researchers[36] In this method the regularized solution Λ120572 for thesystem of linear equations (17) is explained as the solution ofthe following minimization problem
minΛ
10038171003817100381710038171003817AΛ minus b1003817100381710038171003817
1003817
2
+ 1205722Λ2 120572 gt 0 (23)
where sdot denotes the Euclidean norm and 120572 is called theregularization parameter Some methods such as 119871-curve[37] cross-validation (CV) and generalized cross-validation(GCV) [38] are carried out to determine the regularizationparameter 120572 for the TR method In this work we apply(GCV) to obtain regularization parameter In this methodregularization parameter 120572 minimizes the following (GCV)function
119866 (120572) =
10038171003817100381710038171003817AΛ120572 minus b1003817100381710038171003817
1003817
2
(trace (I119873minus AA119868))2
120572 gt 0 (24)
where A119868 = (AtrA + 1205722I119873)minus1Atr
The regularized solution (17) is shown by Λ120572lowast
= [120582120572lowast
1
120582120572lowast
119873]119879 in which 120572lowast is a minimizer of119866Then the approximate
solution for the problem 2 is written as follows
lowast
120572(119909 119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601119896(119909 119905) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595119896(119909 119905)
119901 (119905) =
119873119861
sum
119896=1
120582120572lowast
119896120601 (0 minus 119909
119896 119905 minus 119905119896) +
119873
sum
119896=119873119861+1
120582120572lowast
119896120595 (0 minus 119909
119896 119905 minus 119905119896)
(25)
6 Numerical Experiment
In this section we investigate the performance and the abilityof the present method by giving a test problem Thereforein order to illustrate the efficiency and the accuracy of theproposed method along with the TR method initially wedefine the root mean square (RMS) error the relative rootmean square (RES) error and the maximum absolute error119864infin(119906) as follows
RMS (119906) = radic 1
119873119905
119873119905
sum
119896=1
(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
RES (119906) =radicsum119873119905
119896=1(119906 (119909119896 119905119896) minus lowast(119909119896 119905119896))2
radicsum119873119905
119896=1(119906 (119909119896 119905119896))2
119864infin(119906) = max
1⩽119896⩽119873119905
1003816100381610038161003816119906 (119909119896 119905119896) minus lowast(119909119896 119905119896)1003816100381610038161003816
(26)
Advances in Mathematical Physics 5
Table 2 The obtained values of RMS(119906)RES(119906)RMS(119901) and RES(119901) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem A
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119901) RES(119901) 119864
infin(119901) cond(119860)
11 0017391 0012603 0044511 0014968 0012701 0024588 16111119890 + 09
25 0005081 0003682 0019141 0007528 0006388 0019141 35475119890 + 10
5 0005061 0003668 0014484 0005682 0004821 0014485 53298119890 + 11
10 0002907 0002107 0008510 0003587 0003043 0008510 23557119890 + 13
20 0003463 0002510 0013148 0004864 0004128 0013147 79497119890 + 13
30 0004472 0003241 0021264 0007954 0006749 0021264 11148119890 + 11
50 0003613 0002618 0012343 0005781 0004905 0012343 14673119890 + 11
where119873119905is the total number of testing points in the domain
Ω119887 119906(119909119896 119905119896) and lowast(119909
119896 119905119896) are the exact and the approxi-
mated values at these points respectively RMS RES and 119864infin
errors of the functions 119901(119905) and 119891(119909) are similarly definedas well In our computation the Matlab code developed byHansen [39] is used for solving the discrete ill-conditionedsystem of linear equations (17)
Example For simplicity we assume that 1199090= 05 and 119886 =
119897 = 119905max = 1 By these assumptions the exact solution of theproblem (1)ndash(3) is given as follows [40]
119906 (119909 119905) = 119890minus119905 sin (119909) + 1199092 + 2119905
119901 (119905) = 2119905
119891 (119909) = sin (119909) + 1199092
(27)
Since for a large fixed number of scattered points 119873the matrix A will be more ill-conditioned and also smallervalues of the shape parameter 119888 generate more accurateapproximations we suppose that 119888 = 01 and the numberof interpolating points is 119873 = 20 The obtained values ofRMS(119906) RES(119906) 119864
infin(119906) RMS(119901) RES(119901) 119864
infin(119901) RMS(119891)
RES(119891) and 119864infin(119891) as well as condition number of A for
120575 = 001 and 119873119905= 208 and various values of 119879 are given
in Tables 2 and 4 Numerical results indicate that this methodis not depended on parameter 119879 By the assumptions119873 = 20and 119879 = 41 and with various levels of noise added into thedata Tables 3 and 5 illustrate the relative root mean squareerror of 119901(119905) and 119891(119909) at three points 119909
0= 01 05 and
1199090= 09 of the interval [0 1] respectively Figures 1 and 3
feature out a comparison between the exact solutions and theapproximate solutions for 119909
0= 05 and 119879 = 3 and various
levels of noise added into the data It is observed that as thenoise level increases the approximated functions will haveacceptable accuracy Figures 2 and 4 display the relationshipbetween the accuracy and the parameter 119879with noise of level120575 = 001 added into the dataThey elucidate not only that thenumerical results are stable with respect to parameter 119879 butalso that they retain at the same level of accuracy for a widerange of values 119879 Therefore the accuracy of the numericalsolutions is not relatively dependent on the parameter 119879In addition the study of the presented numerical resultsvia MFS in [40] indicates that with noise of level 120575 = 0namely with the noiseless data and for 119909
0= 05 of the
interval [0 1] RMS(119901) = 17 times 10minus3 with 119879 = 19 and
Table 3 The resulted values of RES(119901) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 3491119890 minus 02 2949119890 minus 03 5095119890 minus 04 4129119890 minus 04
05 2703119890 minus 02 3306119890 minus 03 7707119890 minus 04 3069119890 minus 04
09 6242119890 minus 02 9011119890 minus 03 3292119890 minus 03 1126119890 minus 03
0 01 02 03 04 05 06 07 08 09 10
05
1
15
2
25
Exact
p(t) and its approximation for x0 = 05 T = 3p(t)
andplowast(t)
120575 = 001
120575 = 003
120575 = 005
t
Figure 1 The exact 119901 and its approximation 119901lowast obtained with119873 =
20 119879 = 3 120575 = 001 and 1199090= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
RMS(119891) = 189 times 10minus2 for 119879 = 3 Also when the data areconsidered with noise RMS(119901) = 284 times 10minus2 and RMS(119891) =353 times 10
minus2 while by the proposed method in this work andwith the same assumptions we achieve RMS(119901) = 30479 times10minus5 and RMS(119891) = 35691 times 10minus6 for 120575 = 0 Also with noise
of level 120575 = 10minus4 we obtain RMS(119901) = 11021 times 10minus4 andRMS(119891) = 19204 times 10minus4 Accordingly the MFSRBF whichis based on the fundamental solutions of the heat equationand radial basis functions is more accurate in comparison toMFS
Table 3 indicates the values of the obtained RES(119901) forvarious levels of noise different values of 119909
0and 119879 = 41 by
MFSRBF
6 Advances in Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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 Mathematical Physics 5
Table 2 The obtained values of RMS(119906)RES(119906)RMS(119901) and RES(119901) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem A
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119901) RES(119901) 119864
infin(119901) cond(119860)
11 0017391 0012603 0044511 0014968 0012701 0024588 16111119890 + 09
25 0005081 0003682 0019141 0007528 0006388 0019141 35475119890 + 10
5 0005061 0003668 0014484 0005682 0004821 0014485 53298119890 + 11
10 0002907 0002107 0008510 0003587 0003043 0008510 23557119890 + 13
20 0003463 0002510 0013148 0004864 0004128 0013147 79497119890 + 13
30 0004472 0003241 0021264 0007954 0006749 0021264 11148119890 + 11
50 0003613 0002618 0012343 0005781 0004905 0012343 14673119890 + 11
where119873119905is the total number of testing points in the domain
Ω119887 119906(119909119896 119905119896) and lowast(119909
119896 119905119896) are the exact and the approxi-
mated values at these points respectively RMS RES and 119864infin
errors of the functions 119901(119905) and 119891(119909) are similarly definedas well In our computation the Matlab code developed byHansen [39] is used for solving the discrete ill-conditionedsystem of linear equations (17)
Example For simplicity we assume that 1199090= 05 and 119886 =
119897 = 119905max = 1 By these assumptions the exact solution of theproblem (1)ndash(3) is given as follows [40]
119906 (119909 119905) = 119890minus119905 sin (119909) + 1199092 + 2119905
119901 (119905) = 2119905
119891 (119909) = sin (119909) + 1199092
(27)
Since for a large fixed number of scattered points 119873the matrix A will be more ill-conditioned and also smallervalues of the shape parameter 119888 generate more accurateapproximations we suppose that 119888 = 01 and the numberof interpolating points is 119873 = 20 The obtained values ofRMS(119906) RES(119906) 119864
infin(119906) RMS(119901) RES(119901) 119864
infin(119901) RMS(119891)
RES(119891) and 119864infin(119891) as well as condition number of A for
120575 = 001 and 119873119905= 208 and various values of 119879 are given
in Tables 2 and 4 Numerical results indicate that this methodis not depended on parameter 119879 By the assumptions119873 = 20and 119879 = 41 and with various levels of noise added into thedata Tables 3 and 5 illustrate the relative root mean squareerror of 119901(119905) and 119891(119909) at three points 119909
0= 01 05 and
1199090= 09 of the interval [0 1] respectively Figures 1 and 3
feature out a comparison between the exact solutions and theapproximate solutions for 119909
0= 05 and 119879 = 3 and various
levels of noise added into the data It is observed that as thenoise level increases the approximated functions will haveacceptable accuracy Figures 2 and 4 display the relationshipbetween the accuracy and the parameter 119879with noise of level120575 = 001 added into the dataThey elucidate not only that thenumerical results are stable with respect to parameter 119879 butalso that they retain at the same level of accuracy for a widerange of values 119879 Therefore the accuracy of the numericalsolutions is not relatively dependent on the parameter 119879In addition the study of the presented numerical resultsvia MFS in [40] indicates that with noise of level 120575 = 0namely with the noiseless data and for 119909
0= 05 of the
interval [0 1] RMS(119901) = 17 times 10minus3 with 119879 = 19 and
Table 3 The resulted values of RES(119901) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 3491119890 minus 02 2949119890 minus 03 5095119890 minus 04 4129119890 minus 04
05 2703119890 minus 02 3306119890 minus 03 7707119890 minus 04 3069119890 minus 04
09 6242119890 minus 02 9011119890 minus 03 3292119890 minus 03 1126119890 minus 03
0 01 02 03 04 05 06 07 08 09 10
05
1
15
2
25
Exact
p(t) and its approximation for x0 = 05 T = 3p(t)
andplowast(t)
120575 = 001
120575 = 003
120575 = 005
t
Figure 1 The exact 119901 and its approximation 119901lowast obtained with119873 =
20 119879 = 3 120575 = 001 and 1199090= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
RMS(119891) = 189 times 10minus2 for 119879 = 3 Also when the data areconsidered with noise RMS(119901) = 284 times 10minus2 and RMS(119891) =353 times 10
minus2 while by the proposed method in this work andwith the same assumptions we achieve RMS(119901) = 30479 times10minus5 and RMS(119891) = 35691 times 10minus6 for 120575 = 0 Also with noise
of level 120575 = 10minus4 we obtain RMS(119901) = 11021 times 10minus4 andRMS(119891) = 19204 times 10minus4 Accordingly the MFSRBF whichis based on the fundamental solutions of the heat equationand radial basis functions is more accurate in comparison toMFS
Table 3 indicates the values of the obtained RES(119901) forvarious levels of noise different values of 119909
0and 119879 = 41 by
MFSRBF
6 Advances in Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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 Mathematical Physics
Table 4 The obtained values of RMS(119906)RES(119906)RMS(119891) and RES(119891) using MFSRBF for various values of 119879 1199090= 05 and 120575 = 001 by
solving problem B
119879 RMS(119906) RES(119906) 119864infin(119906) RMS(119891) RES(119891) 119864
infin(119891) cond(119860)
11 0023701 0011307 0109059 0053926 0041309 0109060 29272119890 + 08
25 0003687 0001759 0012035 0005482 0004199 0008311 11421119890 + 08
5 0003769 0001798 0011163 0006226 0004769 0011164 31868119890 + 10
10 0005477 0002613 0019829 0001900 0001455 0003587 10342119890 + 12
20 0005138 0002451 0018525 0003612 0002767 0006490 14588119890 + 14
30 0005022 0002036 0013977 0005012 0002917 0008241 52325119890 + 13
50 0006734 0003212 0012163 0004516 0003459 0006278 38672119890 + 13
2 3 4 5 6 7 8 9 100
0005
001
0015
002
0025
003
0035
0
0005
001
0015
002
0025
003
0035
004
0045
Parameter T
RES(p
) and
RM
S(p
)
RES(p)RMS(p)
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
RMS(u
) and
RES
(u)
Figure 2 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 with respect to the parameter 119879
05 06 07 08 09 10
05
1
15
2
25
Exact120575 = 001
120575 = 003
120575 = 005
f(x) and its approximation for x0 = 05 T = 3
f(x)
andflowast(x)
x
Figure 3 The exact 119891 and its approximation 119891lowast obtained with119873 =20 119879 = 3 120575 = 001 and 119909
0= 05 and for various levels of noise
added into the data using MQ-RBF with 119888 = 01
Table 5 The obtained values of RES(119891) using MFSRBF for variouslevels of noise 120575 different values of 119909
0 and 119879 = 41
1199090
120575 = 01 120575 = 001 120575 = 0001 120575 = 00001
01 5611119890 minus 02 1006119890 minus 02 6731119890 minus 03 1066119890 minus 03
05 2032119890 minus 02 2677119890 minus 03 6219119890 minus 04 2954119890 minus 04
09 2012119890 minus 02 1327119890 minus 03 5852119890 minus 04 1965119890 minus 04
7 Conclusion
Thepresent study successfully applies a newmeshless schemebased on the fundamental solution of the heat equation andthe radial basis function with the Tikhonov regularizationmethod in order to solve a backward inverse heat conductionproblem At first we have divided the inverse problem intotwo separate problems and then by applying the MFSRBFwith the TRmethod on the obtained problems two unknownfunctions in this IHCP are approximated simultaneouslyNumerical results clarify that the presented method is anexact and reliable numerical technique to solve a BIHCPand also the accuracy of the numerical solution is not
Advances in Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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 Mathematical Physics 7
0
001
002
003
004
005
006
007
008
009
0
001
002
003
004
005
006
2 3 4 5 6 7 8 9 10Parameter T
RES(u)RMS(u)
2 3 4 5 6 7 8 9 10Parameter T
RES(f)
RMS(u
) and
RES
(u)
RMS(f)
RES(f
) and
RM
S(f
)
Figure 4 The accuracy of the obtained numerical results by MFSRBF for 120575 = 001 and 1199090= 05 versus the parameter 119879
relatively depended on the parameter 119879 Accordingly notonly this method can be applied to solve a BIHCP in higherdimensions but also it is possible to practice it in otherinverse problems Hence this will be contemplated more infuture researches
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] J V Beck B Black-Well and C R St-Clair Inverse Heat Con-duction Ill-Posed Problems John Wiley amp Sons 1985
[2] J R Cannon The One-Dimensional Heat Equation Addison-Wesley 1984
[3] O M Alifanov Inverse Heat Conduction Problems Springer1994
[4] M N Ozisik Boundary Value Problems of Heat ConductionDover Publications New York NY USA 1989
[5] J R Cannon and J A Vander Hoke ldquoImplicit finite differencescheme for the diffusion ofmass in porousmediardquo inNumericalSolution of Partial Differential Equations vol 527 pp 527ndash539North-Holland 1982
[6] B T Johansson and D Lesnic ldquoDetermination of a spacewisedependent heat sourcerdquo Journal of Computational and AppliedMathematics vol 209 no 1 pp 66ndash80 2007
[7] A Farcas and D Lesnic ldquoThe boundary-element method forthe determination of a heat source dependent on one variablerdquoJournal of Engineering Mathematics vol 54 no 4 pp 375ndash3882006
[8] M D Buhmann Radial Basis Functions Cambridge UniversityPress Cambridge UK 2003
[9] M D J Powell ldquoRadial basis functions for multivariable inter-polation a reviewrdquo in Numerical Analysis D F Griffths and GA Watson Eds pp 223ndash241 Longman Scientific amp TechnicalHarlow UK 1987
[10] Z M Wu ldquoHermite-Birkhoff interpolation of scattered data byradial basis functionsrdquo Approximation Theory and Its Applica-tions vol 8 no 2 pp 1ndash10 1992
[11] M A Golberg and C S Chen ldquoThe method of fundamen-tal solutions for potential hemholtz and diffusion problemsrdquoin Boundary Integral Methods Numerical and MathematicalAspects pp 103ndash176 Computational Mechanics PublicationSouthampton UK 1999
[12] B Jin and L Marin ldquoThe method of fundamental solutions forinverse source problems associated with the steady-state heatconductionrdquo International Journal for Numerical Methods inEngineering vol 69 no 8 pp 1570ndash1589 2007
[13] L Yan C-L Fu and F-L Yang ldquoThe method of fundamentalsolutions for the inverse heat source problemrdquo EngineeringAnalysis with Boundary Elements vol 32 no 3 pp 216ndash2222008
[14] S Chantasiriwan ldquoMethods of fundamental solutions for time-dependent heat conduction problemsrdquo International Journal forNumerical Methods in Engineering vol 66 no 1 pp 147ndash1652006
[15] W Chen and M Tanaka ldquoA meshless integration-free andboundary-only RBF techniquerdquoComputersampMathematics withApplications vol 43 no 3ndash5 pp 379ndash391 2002
[16] V D Kupradze andM A Aleksidze ldquoThemethod of functionalequations for the approximate solution of certain boundary-value problemsrdquo USSR Computational Mathematics and Math-ematical Physics vol 4 pp 82ndash126 1964
[17] Y C Hon and T Wei ldquoA fundamental solution method forinverse heat conduction problemrdquo Engineering Analysis withBoundary Elements vol 28 no 5 pp 489ndash495 2004
[18] N SMera ldquoThemethod of fundamental solutions for the back-ward heat conduction problemrdquo Inverse Problems in Science andEngineering vol 13 no 1 pp 65ndash78 2005
[19] L Marin and D Lesnic ldquoThe method of fundamental solutionsfor the Cauchy problem in two-dimensional linear elasticityrdquoInternational Journal of Solids and Structures vol 41 no 13 pp3425ndash3438 2004
8 Advances in Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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 Mathematical Physics
[20] L Marin and D Lesnic ldquoThe method of fundamental solu-tions for the Cauchy problem associated with two-dimensionalHelmholtz-type equationsrdquoComputers amp Structures vol 83 no4-5 pp 267ndash278 2005
[21] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamics ISurface approximations and partial derivative estimatesrdquo Com-puters ampMathematics with Applications vol 19 no 8-9 pp 127ndash145 1990
[22] E J Kansa ldquoMultiquadricsmdasha scattered data approximationscheme with applications to computational fluid-dynamicsmdashII Solutions to parabolic hyperbolic and elliptic partial differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 19 no 8-9 pp 147ndash161 1990
[23] E J Kansa ldquoExact explicit time integration of hyperbolic partialdifferential equations with mesh free radial basis functionsrdquoEngineering Analysis with Boundary Elements vol 31 no 7 pp577ndash585 2007
[24] G E Fasshauer ldquoSolving partial differential equations bycollocation with radial basis functionsrdquo in Surface Fitting andMultires-Olution Methods A Le Mhaur C Rabut and L LSchumaker Eds Vanderbilt University Press Nashville TennUSA 1997
[25] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[26] M Tatari and M Dehghan ldquoA method for solving partialdifferential equations via radial basis functions applicationto the heat equationrdquo Engineering Analysis with BoundaryElements vol 34 no 3 pp 206ndash212 2010
[27] M Dehghan and V Mohammadi ldquoThe numerical solutionof Fokker-Planck equation with radial basis functions (RBFs)based on the meshless technique of Kansarsquos approach and Gal-erkin methodrdquo Engineering Analysis with Boundary Elementsvol 47 pp 38ndash63 2014
[28] J A Kolodziei and A UsciLowska ldquoApplication of MFS fordetermination of effective thermal conductivity of unidirec-tional composites with linearly temperature dependent con-ductivity of constituentsrdquo Engineering Analysis with BoundaryElements vol 36 no 3 pp 293ndash302 2012
[29] L Hui and L Jijun ldquoSolution of Backward Heat problem byMorozov principle and conditional stabilityrdquoNumerical Mathe-matics vol 14 no 2 pp 180ndash192 2005
[30] I J Schoenberg ldquoMetric spaces and completly monotonicfunctionsrdquo Annals of Mathematics vol 39 pp 811ndash841 1938
[31] H Wendland ldquoPiecewise polynomial positive definite andcompactly supported radial functions of minimal degreerdquoAdvances in Computational Mathematics vol 4 no 4 pp 389ndash396 1995
[32] G E Fasshauer and J G Zhang ldquoOn choosing lsquooptimalrsquo shapeparameters for RBF approximationrdquoNumerical Algorithms vol45 no 1ndash4 pp 345ndash368 2007
[33] R ECarlson andTA Foley ldquoTheparameterR2 inmultiquadricinterpolationrdquoComputersampMathematics withApplications vol21 no 9 pp 29ndash42 1991
[34] A H Cheng ldquoMultiquadric and its shape parametermdasha numer-ical investigation of error estimate condition number andround-off error by arbitrary precision computationrdquo Engineer-ing Analysis with Boundary Elements vol 36 no 2 pp 220ndash2392012
[35] L-T Luh ldquoThe shape parameter in the Gaussian function IIrdquoEngineering Analysis with Boundary Elements vol 37 no 6 pp988ndash993 2013
[36] H W Engl M Hanke and A Neubauer Regularization ofInverse Problems vol 375 of Mathematics and its ApplicationsKluwer Academic New York NY USA 1996
[37] P C Hansen Rank-Defficient and Discrete Ill-Posed ProblemsSIAM Philadelphia Pa USA 1998
[38] G H Golub M Heath and G Wahba ldquoGeneralized cross-validation as a method for choosing a good ridge parameterrdquoTechnometrics vol 21 no 2 pp 215ndash223 1979
[39] P C Hansen ldquoRegularization tools a Matlab package foranalysis and solution of discrete ill-posed problemsrdquoNumericalAlgorithms vol 6 no 1-2 pp 1ndash35 1994
[40] A Shidfar Z Darooghehgimofrad and M Garshasbi ldquoNoteon using radial basis functions and Tikhonov regularizationmethod to solve an inverse heat conduction problemrdquoEngineer-ing Analysis with Boundary Elements vol 33 no 10 pp 1236ndash1238 2009
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
![WebGraphEd, an open source graph drawing editor for the web...simplest layout is the 3D version of the radial layout ( gure 11). Another is the cone tree [12] ( gure 12), but in this](https://static.fdocuments.net/doc/165x107/5f1f7f1ee0deea55c57139a4/webgraphed-an-open-source-graph-drawing-editor-for-the-web-simplest-layout.jpg)
