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A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem

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A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem
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  Research Article  A Meshless Method Based on the FundamentalSolution and Radial Basis Function for Solving an Inverse Heat Conduction Problem Muhammad Arghand and Majid Amirfakhrian Department of Mathematics, Central ehran Branch, Islamic Azad University, ehran, Iran Correspondence should be addressed to Majid Amirakhrian; amirakhrian@iauctb.ac.irReceived 󰀲󰀵 November 󰀲󰀰󰀱󰀴; Accepted 󰀱󰀱 March 󰀲󰀰󰀱󰀵Academic Editor: Alkesh PunjabiCopyright © 󰀲󰀰󰀱󰀵 M. Arghand and M. Amirakhrian. Tis is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work isproperly cited.We propose a new meshless method to solve a backward inverse heat conduction problem. Te numerical scheme, based on theundamental solution o the heat equation and radial basis unctions (RBFs), is used to obtain a numerical solution. Since thecoefficients matrix is ill-conditioned, the ikhonov regularization (R) method is employed to solve the resulted system o linearequations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problemdemonstrates the stability, accuracy, and efficiency o the proposed method. 1. Introduction ransient heat conduction phenomena are generally described by the parabolic heat conduction equation, andi the initial temperature distribution and the boundary conditions are speci󿬁ed, then this, in general, leads to awell-posed problem which may easily be solved numerically by using various numerical methods. However, in many practicalsituationswhendealingwithaheatconductingbody itisnotalwayspossibletospeciytheboundaryconditionsorthe initial temperature. Hence, we are aced with an inverseheat conduction problem. Inverse heat conduction problems(IHCPs) occur in many branches o engineering and science.Mechanical and chemical engineers, mathematicians, andspecialistsinothersciencesbranchesareinterestedininverseproblems. From another point o view, since the existence,uniqueness, and stability o the solutions o these problemsare not usually con󿬁rmed, they are generally identi󿬁edas ill-posed [󰀱–󰀴]. According to the act that unknown solutions o inverse problems are determined throughindirect observable data which contain measurement errors,such problems are naturally unstable. In other words, themain difficulty in the treatment o inverse problems is theunstability o their solution in the presence o noise inthe measured data. Hence, several numerical methods havebeen proposed or solving the various kinds o inverseproblems. In addition to, ill-posedness o these kinds o prob-lems,ill-conditioningotheresultingdiscretizedmatrixromthe traditional methods like the 󿬁nite differences method(FDM) [󰀵], the 󿬁nite element method (FEM), and so orth[󰀶, 󰀷], is the main problem making all numerical algorithms or determining the solution o these kinds o problems.Accordingly, within recent years, meshless methods, as themethod o undamental solution (MFS), radial basis unc-tions (RBFs) method, and some other methods, have beenapplied by many scientists in the 󿬁eld o applied sciencesand engineering [󰀸–󰀱󰀵]. Kupradze and Aleksidze [󰀱󰀶] 󿬁rst introducedMFSwhichde󿬁nesthesolutionotheproblemasalinearcombinationoundamentalsolutions.Honetal.[󰀱󰀷–󰀲󰀰] applied the MFS to solve some inverse heat conductionproblems. In 󰀱󰀹󰀹󰀰s, Kansa applied RBFs method to solve thedifferent types o partial differential equations [󰀲󰀱, 󰀲󰀲]. Afer that, Kansa and many scientists regarded RBFs method tosolve different types o mathematical problems rom partialor ordinary differential equations to integral equations[󰀲󰀳–󰀲󰀶]. Following their works, during recent years, many  researchers 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 pageshttp://dx.doi.org/10.1155/2015/256726  󰀲 Advances in Mathematical Physicssome o these kinds problems [󰀲󰀷, 󰀲󰀸]. Consequently, in this work, we will present a meshless numerical scheme, basedon combining the radial basis unction and the undamentalsolution o the heat equation, in order to approximate thesolution o a backward inverse heat conduction problem(BIHCP),theprobleminwhichanunknowninitialconditionor/and temperature distribution in previous time will bedetermined. Tis kind o problem may emerge in many practical application areas such as archeology and mantleplumes[󰀲󰀹].Ontheotherhand,sincethesystemothelinearequations obtained rom discretizing the problem in thepresentedmethodisill-conditioned,ikhonovregularization(R) method is applied in order to solve it. Te generalizedcross-validation (GCV) criterion has been assigned to adoptan optimum amount o the regularization parameter. Testructure o the rest o this work is organized as ollows: InSection 󰀲, we represent the mathematical ormulation o theproblem. Te method o undamental solutions, radial basisunctions method, and method o undamental solution-radial basis unctions (MFSRBF) are described in Section 󰀳.Section 󰀴 embraces ikhonov regularization method with arule or choosing an appropriate regularization parameter.In Section 󰀵, we present the obtained numerical results o solving a test problem. Section 󰀶 ends in a brie conclusionand some suggestions. 2. Mathematical Formulation of the Problem In this section, we consider the ollowing one-dimensionalinverse heat conduction problem:    − 󽠵 2  󽠵󽠵  = 0, (􍠵,󝠵) ∈ Ω = (0,) × 󰀨0,󝠵 max 󰀩,  (󰀱)with the ollowing initial and boundary conditions: (􍠵,0) = 󰁻􀁻(􍠵), 􍠵 ∈ 󐁛0,􍠵 0 󰀩,(􍠵), 􍠵 ∈ 󐁛􍠵 0 ,󐁝,(0,󝠵) = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,(,󝠵) = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝, (󰀲)where  (􍠵)  and  (󝠵)  are considered as known unctions and 󝠵 max  is a given positive constant, while  (􍠵)  and  (󝠵)  areregardedasunknownunctions.So,inordertoestimate (􍠵) and  (󝠵) , we consider additional temperature measurementsand heat 󿬂ux given at a point  􍠵 0 ,  􍠵 0  ∈ (0,1) , as overspeci󿬁edconditions:  󽠵  󰀨􍠵 0 ,󝠵󰀩 = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,󰀨􍠵 0 ,󝠵󰀩 = ℎ(󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝.  (󰀳)o solve the above problem, at 󿬁rst, we divide the problem(󰀱)–(󰀳) into two separate problems. Te problem    is asollows:    − 󽠵 2  󽠵󽠵  = 0, (􍠵,󝠵) ∈ Ω 􍠵  = 󰀨0,􍠵 0 󰀩 × 󰀨0,󝠵 max 󰀩,(􍠵,0) = (􍠵), 􍠵 ∈ 󐁛0,􍠵 0 󐁝,(0,󝠵) = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝, 󽠵  󰀨􍠵 0 ,󝠵󰀩 = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,󰀨􍠵 0 ,󝠵󰀩 = ℎ(󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,  (󰀴)and the problem    is considered as ollows:    − 󽠵 2  󽠵󽠵  = 0, (􍠵,󝠵) ∈ Ω 󝠵  = 󰀨􍠵 0 ,󰀩 × 󰀨0,󝠵 max 󰀩,(􍠵,0) = (􍠵), 􍠵 ∈ 󐁛􍠵 0 ,󐁝, 󽠵  󰀨􍠵 0 ,󝠵󰀩 = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,󰀨􍠵 0 ,󝠵󰀩 = ℎ(󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝,(,󝠵) = (󝠵), 󝠵 ∈ 󐁛0,󝠵 max 󐁝. (󰀵)Obviously, the problems A and B are considered as IHCP,where  (󝠵) ,  (􍠵,󝠵)  and  (􍠵) ,  (􍠵,󝠵)  are unknown unctionsin the problems A and B, respectively. 3. Method of Fundamental Solutions andMethod of Radial Basis Functions In this section, we introduce the numerical scheme orsolving the problem (󰀱)–(󰀳) using the undamental solutions and radial basis unctions. 󰀳.󰀱.MethodofFundamentalSolutions.  Teundamentalsolu-tion o  (󰀱) is presented as below: (􍠵,󝠵) = 12󽠵√ 󝠵 −󽠵 2 /(4 2 ) (󝠵),  (󰀶)where  (󝠵)  is Heaviside unit unction. Assuming that   >󝠵 max  is a constant, it can be demonstrated that the time shifunction (􍠵,󝠵) = (􍠵,󝠵 + )  (󰀷)is also a nonsingular solution o  (󰀱) in the domain  Ω .In order to solve an IHCP by MFS, as the problem   ,since the basis unction    satis󿬁es the heat equation (󰀱)automatically, we assume that  (􍠵  ,󝠵  )} =1  is a given seto scattered points on the boundary   Ω 􍠵 . An approximatesolution is de󿬁ned as a linear combination o     as ollows: Φ(􍠵,󝠵) =   󲈑 =1   󰀨􍠵 − 􍠵  ,󝠵 − 󝠵  󰀩,  (󰀸)where (􍠵,󝠵) isgivenby (󰀷)and   ’sareunknowncoefficientswhich can be determined by solving the ollowing matrixequation: A Λ =  b ,  (󰀹)where  Λ = [ 1 ,...,  ]  and  b  is a   × 1  known vector.Also, as the undamental unctions are the solution o theheat equation, only the initial and boundary conditions arepracticed to make the system o linear equations; that is,  A  is  Advances in Mathematical Physics 󰀳 󰁡󰁢󰁬󰁥 󰀱: Some well-known radial basis unctions.In󿬁nitely smooth RBFs  () Gaussian (GA)   (− 2 ) Multiquadrics (MQ)  √  2 +  2 Inverse multiquadrics (IMQ)  􀀨√  2 +  2 􀀩 −1 Inverse quadric (IQ)  ( 2 +  2 ) −1 a   ×   square matrix which is de󿬁ned using the initial andthe boundary conditions as ollows: A  = 󰁛󽠵  󰁝, 󽠵   = Φ(􍠵,󝠵), (􍠵,󝠵) ∈ Ω 􍠵 , , = 1,...,. (󰀱󰀰)For more details, see [󰀱󰀳, 󰀱󰀸]. 󰀳.󰀲. Method of Radial Basis Functions.  In this section, weconsider RBF method or interpolation o scattered data.Supposethat x  ∗ and x   area󿬁xedpointandanarbitrarypointin R  , respectively. A radial unction   ∗ is de󿬁ned via   ∗ = ∗ () , where   = ‖ x   −  x  ∗ ‖ 2 . Tat is, the radial unction   ∗ dependsonlyonthedistancebetween x  and x  ∗ .Tisproperty implies that the RBFs   ∗ are radially symmetric about  x  ∗ .Somewell-knownin󿬁nitelysmoothRBFsaregiveninable󰀱.As it is observed, these unctionsdepend on a ree parameter  ,knownastheshapeparameter,whichhasanimportantrolein approximation theory using RBFs.Let   x   } =1  be a given set o distinct points in domain  Ω in R  . Te main idea o using the RBFs is interpolation witha linear combination o RBFs o the same types as ollows: Ψ( x  ) =   󲈑 =1      ( x  ),  (󰀱󰀱)where    ( x  ) = (‖ x  − x   ‖)  and    ’s are unknown coefficientsor   = 1,2,..., . Assume that we want to interpolate thegiven values     = ( x   ) ,   = 1,2,..., . Te unknowncoefficients   ’sareobtained,sothat Ψ( x   ) =   ,  = 1,2,..., , which results in the ollowing system o linear equations: A Λ =  b ,  (󰀱󰀲)where  Λ = [ 1 ,...,  ]  ,  b  = [ 1 ,...,  ]  , and  A  = [󽠵  ] or  , = 1,2,...,  with entries  󽠵   =   ( x   ) . Generally,the matrix  A  has been shown to be positive de󿬁nite (andthereore nonsingular) or distinct interpolation points orin󿬁nitely smooth RBFs by Schoenberg’s theorem [󰀳󰀰]. Also,using Micchelli’s theorem [󰀳󰀱], it is shown that  A  is aninvertible matrix or a distinct set o scattered nodes in thecase o MQ-RBF. 4. Method of Fundamental Solutions-RadialBasis Functions Tis section is speci󿬁ed to introduce the numerical schemeor solving the problem (󰀱)–(󰀳) using the undamental solu- tions and radial basis unctions. At 󿬁rst, we are going to󿬁gure out how radial basis unctions and the undamentalsolutions are applied to approximate the solution o theproblem   , 󰀲. Similarly, this method will be used or solvingthe problem    on domain  Ω 󝠵 . Because the one-dimensionalheat conduction equation depends on both parameters o   􍠵 and  󝠵 ,    scattered nodes are considered in the domain  Ω 􍠵  =[0,􍠵 0 ] × [0,󝠵 max  + ] . We assume that Ξ   = 󐁻󰀨􍠵  ,󝠵  󰀩󐁽   =1  , Ξ   = 󐁻󰀨􍠵  ,󝠵  󰀩󐁽  󽠵 =1  (󰀱󰀳)aretwosetsoscatteredpointsinthedomain Ω 􍠵 ,where  =   +  󝠵  and  Ξ   and  Ξ   are the interior and the boundary points in domains  Ω 􍠵  = [0,􍠵 0 ] × [0,󝠵 max  + ]  and  Ω   =[0,􍠵 0 ] × [0,󝠵 max ] , respectively. Also, we assume that Ξ   = Γ 1  ∪ Γ 2  ∪ Γ 3 ,  (󰀱󰀴)where Γ 1  = 󐁻󰀨􍠵  ,󝠵  󰀩 : 󝠵   = 0, 0 ⩽ 􍠵   ⩽ 􍠵 0 ,  = 1,...,󐁽,Γ 2  = {􀀨􍠵  ,󝠵  􀀩 : 􍠵   = 􍠵 0 , 0 < 󝠵   ⩽ 󝠵 max ,  = 1,...,󰁽,Γ 3  = 󐁻󰀨􍠵  ,󝠵  󰀩 : 􍠵   = 􍠵 0 , 0 < 󝠵   ⩽ 󝠵 max ,  = 1,...,󐁽, (󰀱󰀵)so that   󝠵  =  +  +  .We suppose that the solution o the problem 󰀲 in  Ω 􍠵  canbe expressed as ollows: ̃(􍠵,󝠵) =   󽠵 󲈑 =1      (􍠵,󝠵) +   󲈑 = 󽠵 +1      (􍠵,󝠵),  (󰀱󰀶)where    (􍠵,󝠵) = (􍠵 − 􍠵  ,󝠵 − 󝠵   + )  and  (􍠵,󝠵)  is theundamentalsolutionotheheatequationwhichisdescribedin Section 󰀳.󰀲. Also,     = (‖(􍠵,󝠵) − (􍠵  ,󝠵  )‖ 2 )  and    is aradial unction which is de󿬁ned in Section 󰀳.󰀱. o determinethecoefficientsin(󰀱󰀶),weimposetheapproximatesolution  ̃ tosatisythegivenpartialdifferentialequationwiththeotherconditions at any point  (􍠵,󝠵) ∈ Ξ   ∪ Ξ 󝠵 , so we achieve theollowing system o linear equations: A Λ =  b ,  (󰀱󰀷)where Λ = [ 1 ,...,  ]  , b  = [0,(􍠵  ),(󝠵  ),ℎ(󝠵  )]  ,and󰀰is   ×1 zeromatrix.Also,the × matrix A canbesubdividedinto two submatrices as ollows: A  = 􀁛 A  A 󝠵 􀁝,  (󰀱󰀸)where  A   = [󽠵 (1)  ]  is the     ×   obtained submatrix o applying the interior points whose entries are de󿬁ned asollows: 󽠵 (1)  = 0,  = 1,...,  ,  = 1,..., 󝠵 ,  (󰀱󰀹)because as already mentioned above, the undamental unc-tions     satisy the heat equation and also 󽠵 (1)  = 󐀨 󝠵 − 󽠵 2   2 􍠵 2 󐀩   (􍠵,󝠵), = 1,...,  , , =  󝠵  + 1,...,, (􍠵,󝠵) ∈ Ξ  .  (󰀲󰀰)  󰀴 Advances in Mathematical PhysicsUsing the boundary points o domain  Ω  , namely,  Ξ  , thesubmatrix  A 󝠵  = [󽠵 (2)  ]  results, where 󽠵 (2)  =    (􍠵,󝠵), = 1,...,, , = 1,..., 󝠵 , (􍠵,󝠵) ∈ Γ 1 ,󽠵 (2)  =    (􍠵,󝠵), = 1,...,, , =  󝠵  + 1,...,, (􍠵,󝠵) ∈ Γ 1 ,󽠵 (2)  = 􍠵   (􍠵,󝠵), =  + 1,..., + , , = 1,..., 󝠵 , (􍠵,󝠵) ∈ Γ 2 ,󽠵 (2)  = 􍠵   (􍠵,󝠵), =  + 1,..., + , , =  󝠵  + 1,...,, (􍠵,󝠵) ∈ Γ 2 ,󽠵 (2)  =    (􍠵,󝠵),= +  + 1,..., +  + , ,=1,..., 󝠵 , (􍠵,󝠵)∈Γ 3 ,󽠵 (2)  =    (􍠵,󝠵), =  +  + 1,..., +  + ,, =  󝠵  + 1,...,,(􍠵,󝠵) ∈ Γ 3 . (󰀲󰀱)It is essential to keep in mind that inherent errors inmeasurement data are inevitable. On the other hand, as itwas mentioned, a radial basis unction depends on the shapeparameter   , which has an effect on the condition number o  A . Tereore, the obtained system o linear equations (󰀱󰀷) isill-conditioned. So, it cannot be solved, directly. Since, smallperturbation in initial data may produce a large amount o perturbation in the solution, we use the rand unction inmatlab in the numerical example presented in Section 󰀶 andwe produce noisy data as the ollows: ̃   =    (1 +  ⋅  rand ()),  = 1,...,,  (󰀲󰀲)where     is the exact data, rand ()  is a random numberuniormlydistributedin [−1,1] ,andthemagnitude  displaysthe noise level o the measurement data. 5. Regularization Method Solving the system o linear equations (󰀱󰀷) usually does notlead to accurate results by most numerical methods, becausecondition number o matrix  A  is large. It means that theill-conditioning o matrix  A  makes the numerical solutionunstable. Now, in methods as the proposed method in thisstudy, which is based on RBFs, the condition number o   A depends on some actors such as the shape parameter   , aswell. On the other hand, or 󿬁xed values o the shape param-eter   , the condition number increases with the number o scattered nodes   . In practice, the shape parameter mustbe adjusted with the number o interpolating points. Also,the accuracy o radial basis unctions relies on the shapeparameter. So, in case a suitable amount o it is chosen,the accuracy o the approximate solution will be increased.Despite various research works which are done, 󿬁ndingthe optimal choice o the shape parameter is still an openproblem [󰀳󰀲–󰀳󰀵]. Accordingly, some regularization methods arepresentedtosolvesuchill-conditionedsystems.ikhonov regularization (R) method is mostly used by researchers[󰀳󰀶]. In this method, the regularized solution  Λ  or thesystem o linear equations (󰀱󰀷) is explained as the solution o the ollowing minimization problem:min Λ  A Λ − ̃ b  2 +  2 ‖Λ‖ 2 􀁽,  > 0,  (󰀲󰀳)where  ‖ ⋅ ‖  denotes the Euclidean norm and    is called theregularization parameter. Some methods such as   -curve[󰀳󰀷], cross-validation (CV), and generalized cross-validation(GCV) [󰀳󰀸] are carried out to determine the regularizationparameter    or the R method. In this work, we apply (GCV) to obtain regularization parameter. In this method,regularization parameter    minimizes the ollowing (GCV)unction: () =  A Λ  − ̃ b  2 󰀨 trace 󰀨 I   −  AA  󰀩󰀩 2 ,  > 0,  (󰀲󰀴)where  A  = ( A tr A  +  2 I  ) −1 A tr .Te regularized solution (󰀱󰀷) is shown by   Λ  ∗ = [  ∗ 1  ,...,  ∗  ]  ,inwhich  ∗ isaminimizero   .Tentheapproximatesolution or the problem 󰀲 is written as ollows: ̃ ∗  (􍠵,󝠵) =   󽠵 󲈑 =1   ∗      (􍠵,󝠵) +   󲈑 = 󽠵 +1   ∗      (􍠵,󝠵),(󝠵) =   󽠵 󲈑 =1   ∗   󰀨0 − 􍠵  ,󝠵 − 󝠵  󰀩 +   󲈑 = 󽠵 +1   ∗   󰀨0 − 􍠵  ,󝠵 − 󝠵  󰀩. (󰀲󰀵) 6. Numerical Experiment Inthissection,weinvestigatetheperormanceandtheability o the present method by giving a test problem. Tereore,in order to illustrate the efficiency and the accuracy o theproposed method along with the R method, initially, wede󿬁ne the root mean square (RMS) error, the relative rootmean square (RES) error, and the maximum absolute error  ∞ ()  as ollows:RMS () = 󲈚 1  􍠵 󲈑 =1 󰀨󰀨􍠵  ,󝠵  󰀩 − ̃ ∗ 󰀨􍠵  ,󝠵  󰀩󰀩 2 , RES () = 􂈚∑  􍠵 =1  󰀨󰀨􍠵  ,󝠵  󰀩 − ̃ ∗ 󰀨􍠵  ,󝠵  󰀩󰀩 2 􂈚∑  􍠵 =1  󰀨󰀨􍠵  ,󝠵  󰀩󰀩 2 , ∞  () =  max 1⩽⩽ 􍠵 󽮔󽮔󽮔󽮔󰀨􍠵  ,󝠵  󰀩 − ̃ ∗ 󰀨􍠵  ,󝠵  󰀩󽮔󽮔󽮔󽮔, (󰀲󰀶)  Advances in Mathematical Physics 󰀵 󰁡󰁢󰁬󰁥 󰀲: Te obtained values o RMS (), RES (), RMS () , and RES ()  using MFSRBF or various values o    ,  􍠵 0  = 0.5 , and   = 0.01  by solving problem  A .   RMS ()  RES ()  ∞ ()  RMS ()  RES ()  ∞ ()  cond () 󰀱.󰀱 󰀰.󰀰󰀱󰀷󰀳󰀹󰀱 󰀰.󰀰󰀱󰀲󰀶󰀰󰀳 󰀰.󰀰󰀴󰀴󰀵󰀱󰀱 󰀰.󰀰󰀱󰀴󰀹󰀶󰀸 󰀰.󰀰󰀱󰀲󰀷󰀰󰀱 󰀰.󰀰󰀲󰀴󰀵󰀸󰀸  1.6111 + 092.5  󰀰.󰀰󰀰󰀵󰀰󰀸󰀱 󰀰.󰀰󰀰󰀳󰀶󰀸󰀲 󰀰.󰀰󰀱󰀹󰀱󰀴󰀱 󰀰.󰀰󰀰󰀷󰀵󰀲󰀸 󰀰.󰀰󰀰󰀶󰀳󰀸󰀸 󰀰.󰀰󰀱󰀹󰀱󰀴󰀱  3.5475 + 105  󰀰.󰀰󰀰󰀵󰀰󰀶󰀱 󰀰.󰀰󰀰󰀳󰀶󰀶󰀸 󰀰.󰀰󰀱󰀴󰀴󰀸󰀴 󰀰.󰀰󰀰󰀵󰀶󰀸󰀲 󰀰.󰀰󰀰󰀴󰀸󰀲󰀱 󰀰.󰀰󰀱󰀴󰀴󰀸󰀵  5.3298 + 1110  󰀰.󰀰󰀰󰀲󰀹󰀰󰀷 󰀰.󰀰󰀰󰀲󰀱󰀰󰀷 󰀰.󰀰󰀰󰀸󰀵󰀱󰀰 󰀰.󰀰󰀰󰀳󰀵󰀸󰀷 󰀰.󰀰󰀰󰀳󰀰󰀴󰀳 󰀰.󰀰󰀰󰀸󰀵󰀱󰀰  2.3557 + 1320  󰀰.󰀰󰀰󰀳󰀴󰀶󰀳 󰀰.󰀰󰀰󰀲󰀵󰀱󰀰 󰀰.󰀰󰀱󰀳󰀱󰀴󰀸 󰀰.󰀰󰀰󰀴󰀸󰀶󰀴 󰀰.󰀰󰀰󰀴󰀱󰀲󰀸 󰀰.󰀰󰀱󰀳󰀱󰀴󰀷  7.9497 + 1330  󰀰.󰀰󰀰󰀴󰀴󰀷󰀲 󰀰.󰀰󰀰󰀳󰀲󰀴󰀱 󰀰.󰀰󰀲󰀱󰀲󰀶󰀴 󰀰.󰀰󰀰󰀷󰀹󰀵󰀴 󰀰.󰀰󰀰󰀶󰀷󰀴󰀹 󰀰.󰀰󰀲󰀱󰀲󰀶󰀴  1.1148 + 1150  󰀰.󰀰󰀰󰀳󰀶󰀱󰀳 󰀰.󰀰󰀰󰀲󰀶󰀱󰀸 󰀰.󰀰󰀱󰀲󰀳󰀴󰀳 󰀰.󰀰󰀰󰀵󰀷󰀸󰀱 󰀰.󰀰󰀰󰀴󰀹󰀰󰀵 󰀰.󰀰󰀱󰀲󰀳󰀴󰀳  1.4673 + 11 where     is the total number o testing points in the domain Ω  ,  (􍠵  ,󝠵  )  and  ̃ ∗ (􍠵  ,󝠵  )  are the exact and the approxi-matedvaluesatthesepoints,respectively.RMS,RES,and  ∞ errors o the unctions  (󝠵)  and  (􍠵)  are similarly de󿬁ned,as well. In our computation, the Matlab code developed by Hansen [󰀳󰀹] is used or solving the discrete ill-conditionedsystem o linear equations (󰀱󰀷). Example . For simplicity, we assume that  􍠵 0  = 0.5  and  󽠵 = = 󝠵 max  = 1 . By these assumptions, the exact solution o theproblem (󰀱)–(󰀳) is given as ollows [󰀴󰀰]: (􍠵,󝠵) =  − sin (􍠵) + 􍠵 2 + 2󝠵,(󝠵) = 2󝠵,(􍠵) =  sin (􍠵) + 􍠵 2 . (󰀲󰀷)Since, or a large 󿬁xed number o scattered points   ,the matrix  A  will be more ill-conditioned and also, smaller values o the shape parameter    generate more accurateapproximations, we suppose that   = 0.1  and the numbero interpolating points is   = 20 . Te obtained values o RMS () , RES () ,   ∞ () , RMS () , RES () ,   ∞ () , RMS () ,RES () , and   ∞ () , as well as condition number o   A  or  = 0.01  and     = 208  and various values o    , are giveninables󰀲and󰀴.Numericalresultsindicatethatthismethod is not depended on parameter   . By the assumptions   = 20 and   = 4.1  and with various levels o noise added into thedata, ables 󰀳 and 󰀵 illustrate the relative root mean square error o   (󝠵)  and  (􍠵)  at three points  􍠵 0  = 0.1 , 󰀰.󰀵, and 􍠵 0  = 0.9  o the interval  [0,1] , respectively. Figures 󰀱 and 󰀳 eatureoutacomparisonbetweentheexactsolutionsandtheapproximate solutions or  􍠵 0  = 0.5  and   = 3  and variouslevels o noise added into the data. It is observed that as thenoise level increases, the approximated unctions will haveacceptable accuracy. Figures 󰀲 and 󰀴 display the relationship betweentheaccuracyandtheparameter  withnoiseolevel  = 0.01  added into the data. Tey elucidate not only that thenumerical results are stable with respect to parameter   , butalso that they retain at the same level o accuracy or a widerange o values   . Tereore, the accuracy o the numericalsolutions is not relatively dependent on the parameter   .In addition, the study o the presented numerical results, via MFS in [󰀴󰀰], indicates that with noise o level   = 0 ,namely, with the noiseless data and or  􍠵 0  = 0.5  o theinterval  [0,1] , RMS () = 1.7 × 10 −3 with   = 1.9  and 󰁡󰁢󰁬󰁥 󰀳: Te resulted values o RES ()  using MFSRBF or variouslevels o noise   , different values o   􍠵 0 , and   = 4.1 . 􍠵 0   = 0.1  = 0.01  = 0.001  = 0.00010.1 3.491 − 02 2.949 − 03 5.095 − 04 4.129 − 040.5 2.703 − 02 3.306 − 03 7.707 − 04 3.069 − 040.9 6.242 − 02 9.011 − 03 3.292 − 03 1.126 − 03 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.522.5Exact p(t)  and its approximation for,  x 0  = 0.5,T = 3      p      (     t      )   a  n    d      p     ∗       (     t      )  = 0.01 = 0.03 = 0.05t F󰁩󰁧󰁵󰁲󰁥 󰀱: Te exact    and its approximation   ∗ obtained with   =20 ,   = 3 ,   = 0.01 , and  􍠵 0  = 0.5  and or various levels o noiseadded into the data using MQ-RBF with   = 0.1 . RMS () = 1.89 × 10 −2 or   = 3 . Also, when the data areconsidered with noise, RMS () = 2.84×10 −2 and RMS () =3.53 × 10 −2 , while by the proposed method in this work andwith the same assumptions, we achieve RMS () = 3.0479 ×10 −5 and RMS () = 3.5691×10 −6 or   = 0 . Also, with noiseo level   = 10 −4 , we obtain RMS () = 1.1021 × 10 −4 andRMS () = 1.9204 × 10 −4 . Accordingly, the MFSRBF, whichis based on the undamental solutions o the heat equationand radial basis unctions, is more accurate in comparison toMFS.able 󰀳 indicates the values o the obtained RES ()  or various levels o noise, different values o   􍠵 0  and   = 4.1  by MFSRBF.
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