Solving a Class of PDEs by a Local Reproducing Kernel Method with An Adaptive Residual Subsampling Technique

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1 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 Solving a Class of PDEs by a Local Reprodcing Kernel Method with An Adaptive Residal Sbsampling Techniqe H. Rafieayan Zadeh, M. Mohammadi,2 and E. Babolian Abstract: A local reprodcing kernel method based on spatial trial space spanned by the Newton basis fnctions in the native Hilbert space of the reprodcing kernel is proposed. It is a trly meshless approach which ses the local sb clsters of domain nodes for approimation of the arbitrary field. It leads to a system of ordinary differential eqations (ODEs) for the time-dependent partial differential eqations (PDEs). An adaptive algorithm, so-called adaptive residal sbsampling, is sed to adjst nodes in order to remove oscillations which are cased by a sharp gradient. The method is applied for solving the Allen-Cahn and Brgers eqations. The nmerical reslts show that the proposed method is efficient, accrate and be able to remove oscillations cased by sharp gradient. Keywords: Local reprodcing kernel method, method of lines, Newton basis fnctions, adaptive residal sbsampling algorithm. Introdction Dring last decades, meshless methods are considered by many researchers. Unlike traditional nmerical methods in solving PDEs, meshless methods [Belytschko, Krongaz, Organ, Fleming, and Krysl (996)] need no mesh generation. Taking translates of kernels as trial fnctions in collocation methods, which are trly meshless, leads to a highly sccessfl method [C. Franke (998); Hang, Ren, and Rssell (99a); Hang, Ren, and Rssell (99b)]. Implementation of these methods is easy and allow good accracy at low comptational cost. In addition, it was proven recently [Schaback (25)] that symmetric collocation sing kernels is optimal along all linear PDE solvers sing the same inpt data. There are plenty of papers which se kernel-based methods for solving varios problems [Abbas- Faclty of Mathematical Sciences and Compter, Kharazmi University, 5 Taleghani Ave., Tehran , Iran 2 Corresponding athor

2 376 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 bandy, Azarnavid, and Alhthali (24); Atlri and Shen (23); Dong, Alotaibi, Mohiddine, and Atlri (24); Elgohary, Dong, Jnkins, and Atlri (24); Han and Atlri (24); Hon and Schaback (28); Mohammadi and Mokhtari (24); Mohammadi and Mokhtari (2); Mohammadi and Mokhtari (23); Mohammadi, Mokhtari, and Panahipor (23); Mohammadi, Mokhtari, and Panahipor (24); Mohammadi, Mokhtari, and Schaback (24); Mohammadi, Mokhtari, and Isfahani (24); Mokhtari, Isfahani, and Mohammadi (22)]. It is well known that representations of kernel-based approimants in terms of the standard basis of translated kernels are notoriosly nstable. A more sefl basis, so-called Newton basis, is offered in [Mller and Schaback (29)]. The Newton basis trns ot to be orthonomal in the reprodcing native Hilbert space, and it is complete, if infinitely many data locations are reasonably chosen. A timesaving calclation of Newton basis arising from a pivoted Cholesky factorization has been introdced in [Pazoki and Schaback (2)]. For the time-dependent PDEs, a spatial interpolation is applied by sing epansion in terms of kernel fnctions. Bt since the coefficients are time-dependent, a system of ODEs is obtained. This is the well-known method of lines (MOL), and it trns ot to be approimately sefl in varios cases. In this paper, we se epansion in terms of the Newton basis fnctions (NBFs) for constrcting the spatial interpolation. In global collocation methods, collocation matri is made by considering the whole domain, since the obtained matri will be fll. This limits the applicability of those methods to solve large scale problems. Bt local methods [Chen, Ganesh, Golberg, and Cheng (22); Lee, Li, and Fan (23); Mai-Dy and Tran-Cong (22)] se overlapping sb-domains of the whole domain, ths a sparse matri is obtained. A general idea behind the local reprodcing kernel method is the se of the local sb clsters of domain nodes, named local inflence domain, for approimation of the arbitrary field. With the selected inflence domain, an approimation fnction is introdced as a sm of weighted NBFs. Then the collocation approach is sed to determine weights. After the sccessfl approimation fnction creation all the needed differential operators can be constrcted by applying an arbitrary operator on the approimation fnction. The main advantage to sing the local method is that less compter storage and flops are needed. The kernel fnctions are defined by sing a set of points called centers or trial points. The centers can set anywhere of given domain independently, therefore, center points can be moved or added or removed in the domain and this is the base of adaptive algorithms. Positions of the centers set inflence on approimation qality and stability of the interpolation [Schaback (995)], in addition, increasing nmber of centers cases large condition nmeber of interpolation matri. Adap-

3 Solving a Class of PDEs by a Local Reprodcing Kernel Method 377 tive algorithms are often applied for problems containing rapid variations in the given domain. Different kinds of adaptive methods have been developed in the literatre, e.g. Greedy algorithm [Ling, Opfe, and Schaback (26); Ling and Schaback (29)], pwind techniqe [Lin and Atlri (2); Lin and Atlri (2)], r-adaptive mesh method (moving mesh strategy) [Hang, Ren, and Rssell (994)]. For kinds of center choosing algorithms with some algorithmic analysis are introdced in [Gong, Wei, Wang, Feng, and Wang (2)]. In this paper, we se adaptive residal sbsampling method [Driscoll and Herydono (27)], where nodes can be added or removed based on interpolation residals evalated at a finer point set. In the adaptive residal sbsampling algorithm, an interpolant has been compted for the center set, the residal of the reslting approimation is sampled on a finer node set. Nodes from the finer set are added to or removed from the set of centers based on the size of the residal of the interpolation at those points. The interpolant is then recompted sing the new center set for a new approimation. In this stdy, we se a local reprodcing kernel method based on spatial trial space spanned by the NBFs accompanied with an adaptive residal sbsampling techniqe for the nmerical soltion of a class of PDEs inclding the Allen-Cahn and Brgers eqations with different kinds of initial and bondary conditions. The Allen-Cahn eqation describes the motion of anti-phase bondaries in crystalline solids. It has been widely sed in material science applications. The Brgers eqation is the simplest nonlinear model eqation for diffsive waves in flid dynamics. The Brgers eqation arises in many physical problems inclding one- dimensional trblence, sond waves in a viscos medim, shock waves in a viscos medim, waves in flid filled viscos elastic tbes, and magneto-hydrodynamic waves in a medim with finite electrical condctivity. The Brgers eqation is similar to the one dimensional Navier-Stokes eqation withot the stress term. The rest of the paper is organized as follows. In section 2, the kernel-based trial fnctions and the NBFs are reviewed. The proposed local reprodcing kernel method is given in section 3. The method is illstrated in section 4 by solving the nonlinear Allen-Cahn eqation with mied bondary conditions. In section 5, the adaptive residal sbsampling algorithm is illstrated. Nmerical eperiments are given in section 6. The last section is devoted to a brief conclsion. 2 Kernel-based trial fnctions Let Ω be a nonempty set. A fnction K : Ω Ω R is called a kernel on Ω. Let K : Ω Ω R be a symmetric positive definite kernel on Ω. This means that for all finite sets { i Ω,i =,...,n} the kernel matri A = [K( i, j )] i, j=,...,n is symmetric and positive definite. These kernels are reprodcing in native Hilbert

4 378 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 space N K = span{k(, ) Ω} of fnctions on Ω in the sense f,k(, ) NK = f () for all Ω, f N K. The most important eamples are the Whittle-Matern kernels r m d/2 K m d/2 (r), r = y,,y R d, reprodcing in the Sobolev space W2 m(rd ) for m > d/2, where K ν is the modiffed Bessel fnction of the second kind [Scahabck (2)]. The following will be independent of the kernel chosen, bt ones shold be aware that the kernel shold be smooth enogh to allow sffciently many derivatives for the PDE and additional smoothness for fast convergence [Wendland (25)]. For scattered nodes { i Ω,i =,...,n} the translates K j () = K(, j ) are the trial fnctions. The Newton basis fnctions (NBFs) {N k ()} n k= can be epressed by N k () = n K(, j )c jk, k =,...,n. () j= Considering N() = [N () N n ()], T () = [K(, ) K(, n )] and C = [c jk ] j,k=,...,n, Eq. () can be written as N() = T () C. Sbseqently, we have N = A C, where, N = [N j ( i )] i, j=,...,n and A = [K( i, j )] i, j=,...,n. It has been proved [Pazoki and Schaback (2)] that the Cholesky decomposition A = L L T with a nonsinglar lower trianglar matri L leads to the Newton basis N() = T () (L T ), (2) with N = L, C = (L T ). (3) Hence the condition nmber of the collocation matri corresponding to the NBFs is smaller than the one corresponding to translated kernels. Conseqently, sing the NBFs for collocation will lead to more stable methods than sing the basis of translates. Note that, L N() = L T () (L T ), (4) where, L can be any linear operator like derivatives. Moreover, for compting the vales of the Newton basis at the other points, for eample {y i Ω,i =,...,n y },

5 Solving a Class of PDEs by a Local Reprodcing Kernel Method 379 (2) is sed. The Newton basis commonly is constrcted by positive definite kernels like radial basis fnctions (RBFs). The RBF is defined as φ j () = K(, j ) = φ( j 2 ), where { j Ω, j =,...,n} is a set of distinct points called centers. Some kinds of RBFs are given in Tab.. The RBFs may have a free parameter, called the shape parameter, denoted by ε. As the shape parameter changes, the shape of the RBFs changes, and sbseqently the accracy of interpolant and the condition nmber of the interpolation matri will change. For interpolation of scattered data by RBFs, an ncertainty relation between the error and the condition of the interpolation matri is proven. It states that the error and the condition nmber cannot both be kept small [Schaback (995)] and there is a trade-off between the accracy and the condition nmber. Note that the GA, IMQ and MS RBFs are eamples of positive definite kernels and can be sed for constrcting the NBFs. Table : Some kinds of RBFs Gassian (GA) φ(r) = e (εr)2 Powers (P) φ(r) = r ε Mltiqadric (MQ) φ(r) = + (εr) 2 Inverse Mltiqadric (IMQ) φ(r) = / + (εr) 2 Thin plate splines (TPS) φ(r) = r 2ε log(r), 2ε > Matern Sobolov (MS) φ(r) = r ε K ε (r), K ε is the modified Bessel fnction of second kind 3 Local reprodcing kernel method Consider the following time-dependent PDE L (,t) = f (,t), Ω, t [,T ], (5) with bondary condition B(,t) = g(,t), Ω, (6) and initial condition I (,) = (), Ω, (7) where L : H F is a differential operator, H and F are Hilbert spaces of fnctions on Ω, B is Dirichlet or Nemann or mied bondary condition operator and I is

6 38 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 a linear operator. It is assmed that the problem (5) (7) is well-posed. We choose discrete points X = { i Ω,i =,...,n} and a symmetric positive definite kernel K : Ω Ω R. Let X I = { i,i =,...,ni} be the interior points and X B = { i,i = ni+,...,n} be the bondary points, where ni is the nmber of interior points. For each i X, we consider a stencil Ω i = {k i }m k= which contains the center i and its m nearest neighboring points. In addition, N i,...,ni m are considered as the NBFs corresponding to the stencil {k i }m k=. To approimate the soltion (,t), we consider kernel-based approimant in terms of the NBFs on the local domain Ω i instead of the whole domain Ω. Then the approimate soltion in the local domain Ω i will be in the following form (,t) = m k= αk i (t) Ni k (). (8) Therefore, for i Ω i we have ( i,t) = m k= α i k (t) Ni k ( i). (9) Conseqently, (9) can be written as the following vector form ( i,t) = N i α i, i Ω i, () where, N i = [N i ( i) N i m( i )] and α i = [α i (t) αi m(t)] T. By sing (8) the linear system U i = N i α i is obtained, where U i = [( i,t) ( i m,t)] T, and N i = [N i k (i p)] k,p=,...,m, ths α i = (N i ) U i. () By sbstitting () in () we have ( i,t) = N i (N i ) U i.

7 Solving a Class of PDEs by a Local Reprodcing Kernel Method 38 We now write the PDE (5) at a point i X I as follows L ( N i (N i ) U i) = f ( i,t), i =,...,ni. (2) The bondary condition (6) implies that ( B N i (N i ) U i) = g( i,t), i = ni +,...,n. (3) Moreover, (7) reslts I U() = U, (4) where I U() = [I (,) I ( n,)] T and U = [ ( ) ( n )] T. The e- qations (2) (3) with the initial condition (4) lead to a system of ODEs at which the nknown vector U = [(,t) ( n,t)] T is to be determined. In order to obtain a matri form for the ODE system, matrices of spatial derivatives mst be calclated. For the spatial partial derivatives, we have s s ( i,t) = N i (s) (Ni ) U i = C i (s) Ui, where N i (s) = [ s s Ni ( i ) s and C i (s) = Ni (s) (Ni ). s Ni m( i )], Now for constrcting global derivative matrices from local contribtions, we define global ni-by-n sparse matrices D (s), according to derivatives appeared in (5), of the form D (s) (i,i i ) = C i (s), i =,...,ni, where I i is a vector that contains the indices of center i and it s m nearest neighboring points. To clarify proposed method, it is applied for solving the Allen- Cahn eqation in the net section.

8 382 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 4 Method validation In this section, we apply the proposed method for solving the Allen-Cahn eqation of the form t ( 2 ) = ν, [a,b], t [,T ], (5) with mied bondary conditions, β (a,t) + γ (a,t) = g (t), t [,T ], (6) β 2 (b,t) + γ 2 (b,t) = g 2(t), t [,T ], (7) and initial condition, (,) = (), [a,b], (8) where β, β 2, γ, and γ 2 are known parameters, g (t) and g 2 (t) are known fnctions, and ν is the kinematics viscosity. Let X = {, 2,..., n, n } be discretization points in the interval [a,b] where = a, and n = b. Based on the methodology and notation described in the previos section, we write the PDE (5) at a point i,i = 2,...,n, as follows t ( i,t) ( i,t) ( (( i,t)) 2) = νn i (2) (Ni ) U i, i = 2,...,n. (9) Then the eqation (9) leads to the following matri form U I = U I. ( U I. 2) + νd (2) U, (2) where. and. denote the pointwise prodct and power between two matrices or vectors, D (2) (i,i i+ ) = C i+ (2), i =,...,n 2, U I = [( 2,t) ( n,t)] T, U I = [ t ( 2,t) t ( n,t)] T, and is a (n 2)-by- vector with in its entries. Now we implement the bondary conditions (6) (7) at the points and n as follows β (,t) + γ C () U = g (t), (2) β 2 ( n,t) + γ 2 C n () Un = g 2 (t). (22)

9 Solving a Class of PDEs by a Local Reprodcing Kernel Method 383 Let the 2-by-n sparse matri W be as follows: W(,I ) = β ( ) + γ (C () ), W(2,I n ) = β 2 ( n ) + γ 2 (C n () ), where i is a n vector with in the ith entry and zero elsewhere. Then the Eqs. (2) (22) lead to W U = (g i (t),i =,2) T. So the nknown vector (( i,t),i =,n) T can be written in terms of the nknown vector U I by solving the following eqations: W(:,[,n]) (( i,t),i =,n) T = (g i (t),i =,2) T W(:,2 : n ) U I. (23) By sbsitting (23) in (2), we get the system of ODEs with the initial conditions U I () = ( ( i ),2 i n ). Note that the nonlinearity of the PDE is preserved, and a good ODE solver will atomatically se a reasonable time-stepping and detect stiffness of the ODE system. 5 Adaptive residal sbsampling algorithm Since kernel-based methods are completely meshfree, some adaptive algorithms for finding optimal point sets may be devised. For eample, in problems that eist rapid variations in given domain, sch as steep gradients, corners, and topological changes reslting from nonlinearity, adaptive methods may be preferred over fied grid methods. In order to achieve accracy and stability, adaptive methods select optimal centers by moving, adding or removing points. In adaptive residal sbsampling algorithm, some points may be added or removed by sing compted residals [Driscoll and Herydono (27)]. In this section, we describe adaptive residal sbsampling algorithm for a local reprodcing kernel method based on spatial trial space spanned by the NBFs. Implementation of adaptive residal sbsampling techniqe for time-dependent PDEs is to alternate time stepping with adaptation. First, initial centers { i,i =,...,n} are generated sing n eqally spaced points in the given domain. Then, sing NBF interpolation at centers and initial condition of the PDE, nknown coefficients vector α = [α α n ] T is calclated from the linear system Nα = U,

10 384 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 where, N = [N j ( i )] i, j=,...,n is the NBF matri, U = [ ( ) ( n )] T and () is the initial condition fnction of the PDE. Now, the set {y i = 2 ( i+ i ),i =,...,n } is considered halfway between the centers. The residals vector r is calclated by r = N y α U y, where, N y is the NBF matri for the points {y i } n i=, which is calclated by (2) (3), U y = [ (y ) (y n )] T, and r = [r r n ] T. Points at which the residal eceeds a threshold θ r are to become centers, and centers that lie between two points whose error is below a smaller threshold θ c are removed. This means, if r i > θ r, then y i will be added to centers set, and if r i < θ c and r i+ < θ c, then i+ will be removed. This process is called coarse refine (coarse for removing centers and refine for adding new points). Therefore, a new centers set is given and the coarse refine process is repeated while any new point can not be added. After ending coarse refine processes, a new centers set is obtained. These new centers are sed to advance the discrete soltion p to a predetermined time t = τ by sing the local method, which is described in the previos section. τ mst be large enogh to avoid ecessive adaptation steps, while keeping it small enogh that the adaptation can keep p with emerging or changing featres in the soltion. So soltion is obtained at the time t = τ. Now residal sbsampling algorithm is applied by sing the soltion at this time level as a new initial state for frther time. This process contines to achieve t = T. 6 Nmerical reslts In this section, we present the reslts of or scheme for the nmerical soltion of some eqations. Fnctions with steep variation in the domain are often employed. The obtained reslts state the ability of the method for adapting centers to the regions with steep variations. Some reslts withot sing adaptive method are presented to show nstability in regions with rapid variations. In this work, we take the MS and IMQ RBFs for constrcting the Newton basis, in addition, we take τ =.. In eamples, ν = /Re, where Re is the Reynold nmber and ν is the kinematics viscosity. Eample : Consider the Allen-Cahn eqation [Driscoll and Herydono (27)] t ( 2 ) = ν, [,], t [,T ], (24) with Re = 6, Dirichlet bondary condition, (±,t) = ±, (25)

11 Solving a Class of PDEs by a Local Reprodcing Kernel Method 385 and initial condition, π ) (,) = sin( 2 (2 3 ). (26) As shown in Fig., the soltion of the eqations (24) (26), withot sing the adaptive residal sbsampling method, have small oscillations in regions with steep gradients that are corrected by sing the adaptive method. Adapting the nodes in steep gradients is shown in Fig. 2. In this eample, we se the MS RBF with the shape parameter ε =, θ r = 3 and θ c = 9. In addition, the nmber of nodes starts with 3 and finally grows to 76 at T = Figre : Soltion of the Allen-Cahn eqation, (left) withot the adaptive method, (right) with the adaptive method. Eample 2: Consider the moving front problem given by the Brgers eqation [Driscoll and Herydono (27)] t = + ν, [,], t [,T ], (27) with Re =, Dirichlet bondary condition, (,t) = (,t) =, (28) and initial condition, (,) = sin(2π) + sin(π). (29) 2

12 386 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 Figre 2: Soltion of the Allen-Cahn eqation with the adaptive method. Nmber of nodes increases in region with steep gradients. As the previos eample, the soltion of the eqations (27) (29) generate steep front and conseqently have nstabilities in steep front. Correction of these nstabilities and adaption of nodes by sing the adaptive method are shown in Figres 3 and 4 respectively. Here, we employed the MS RBF with the shape parameter ε =, θ r = 4 and θ c = 8. Frthermore, nmber of nodes at T = grows to Figre 3: Soltion of the eqations (27) (29), (left) whitot the adaptive method, (right) with the adaptive method.

13 Solving a Class of PDEs by a Local Reprodcing Kernel Method 387 Figre 4: Soltion of the eqations (27) (29); Adaption of nodes. Table 2: Nmerical reslts of the eqations (3) (32) for Re =.,,. Re =. (T =.2) Re = (T = ) Re = (T = ) Eact Appro Eact Appro Eact Appro θ r θ c Eample 3: Consider the Brgers eqation [Hon and Mao (998)] t = + ν, [,], t [,T ], (3) with Dirichlet bondary condition, (,t) = (,t) =, (3)

14 388 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , (a) (b) (c) (d) Figre 5: Soltion of the eqations (3) (32), (a) Re =. for T =.2, (b) Re = for T =.25, (c) Re = for T =, (d) Re = for T =. and initial condition, (,) = sin(π). (32) The eact soltion of eqations (3) (32) is given by [Caldwell and Smith (982)] (,t) = 4πν n= ni n(/2πν) sin(nπ) e ( n2 νπ 2 t) I (/2πν) + 2 n= I n(/2πν) cos(nπ) e ( n2 νπ 2 t), where I n denotes the modiffed Bessel fnction of order n. In this eample, we se the IMQ RBF with the shape parameters ε = 5,5,5,5.8 for Re =.,,, respectively. The nmerical soltions are shwon in Fig. 5. Frthermore, for Re =.,,, comparison with the eact soltions and the thersholds are represented in Tab. 2.

15 Solving a Class of PDEs by a Local Reprodcing Kernel Method 389 For Re =,, the soltion of eqations (3) (32) generates steep gradient toward =, sbseqently the approimate soltion has oscillations in regions with steep gradient. For Re =, by implementing the proposed method with the MS RBF with the shape parameter ε =.5, withot sing the adaptive algorithm, the oscillations are gradally disappeared when the nmber of nodes increases (Fig. 6). The adaptive residal sbsampling method starts with 3 nodes and gradally grows to maimm of 53 nodes, and conseqently obtains soltion at T = with 5 nodes (Fig. 7) Figre 6: Soltion of the eqations (3) (32), for Re = whit MS RBF, (left) 2 nodes, (right) nodes Figre 7: Soltion of the eqations (3) (32) by sing the adaptive method with 5 nodes at T =. For Re =, the oscillations do not vanish even if the nmber of nodes be very large. Fig. 8 shows the oscillations with nodes and 2 nodes in this case. As

16 39 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 shown in Fig. 9, by sing the adaptive residal sbsampling algorithm and MS RBF with ε =, the oscillations are disappeared at T = with 8 nodes (Fig. 9). For Re =, the nmerical soltion has been compared with the Christie accrate soltion given in [Hon and Mao (998)]. Absolte error graph of implementation of the proposed method for Re = is shown in Fig. 9. In addition, adaption of nodes are shown in Fig Figre 8: Soltion of the eqations (3) (32), for Re = with MS RBF, (left) nodes, (right) 2 nodes Absolte error Figre 9: (left) Soltion of the eqations (3) (32) for Re = with the adaptive method, (right) Absolte error graph at =,.5,.,.6,.22,.27,.33,.38,.44,.5,.55,.6,.66,.72,.77,.83,.88,.94,. Eample 4: Consider the Brgers eqation [Pgh (995)] t = + ν + f (,t), [,], t [,T ], (33)

17 Solving a Class of PDEs by a Local Reprodcing Kernel Method 39 Figre : Soltion of the eqations (3) (32); Adaption of nodes. with Nemann bondary condition, (,t) = (,t) =, (34) initial condition, (,) = 4 cos(π), (35) and f (,t) = 4 e νt cos(π) and the eact soltion (,t) = 4 e νt cos(π). ( ν + π 4 e νt sin(π) νπ 2), (36) The eqations (33) (35) are solved for Re = 6, 2, 24,, by the proposed method with the MS RBF with the shape parameter ε =.5. The absolte error graphs at T =.5 are also shown in Fig.. 7 Conclsion In this paper, a local reprodcing kernel method based on spatial trial spaces s- panned by the Newton basis fnctions was sed for solving some time-dependent

18 392 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 Absolte error (a) Re = 6 Absolte error (b) Re = 2 Absolte error Absolte error (c) Re = (d) Re = Absolte error (e) Re = Figre : Absolte error graphs for soltion of the eqations (33) (35) at T =.5.

19 Solving a Class of PDEs by a Local Reprodcing Kernel Method 393 PDEs. In addition, the adaptive residal sbsampling algorithm was employed to correct oscillations. Nmerical reslts show that the method works properly for problems with oscillatory behavior de to steep gradients. It seems that this method can be applied for solving higher dimensional time dependent PDEs. We leave this to or frther work. References Abbasbandy, S.; Azarnavid, B.; Alhthali, M. (24): A shooting reprodcing kernel hilbert space method for mltiple soltions of nonlinear bondary vale problems. J. Compt. Appl. Math. Atlri, S.; Shen, S. (23): Method. Tech. Science Press. The Meshless Local Petrov-Galerkin (MLPG) Belytschko, T.; Krongaz, Y.; Organ, D.; Fleming, M.; Krysl, P. (996): Meshless methods: an overview and recent developments. Compt. Methods Appl. Mech., Engrg., vol. 39, pp C. Franke, R. S. (998): Solving partial differential eqations by collocation sing radial basis fnctions. Appl. Math. Compt., vol. 93, pp Caldwell, J.; Smith, P. (982): Soltion of brgers eqation with a large reynolds nmber. Appl. Math. Modelling., vol. 6, pp Chen, C.; Ganesh, M.; Golberg, M.; Cheng, A. (22): Mltilevel compact radial basis fnctions based comptational scheme for some elliptic problems. Compt. Math. Appl, vol. 43, pp Dong, L.; Alotaibi, A.; Mohiddine, S.; Atlri, S. N. (24): Comptational methods in engineering: A variety of primal & mied methods, with global & local interpolations, for well-posed or ill-posed bcs. CMES Compt. Model. Eng. Sci., vol. 99, pp. 85. Driscoll, T.; Herydono, A. (27): Adaptive residal sbsampling methods for radial basis fnction interpolation and collocation problems. Compt. Math. Appl, vol. 53, pp Elgohary, T.; Dong, L.; Jnkins, J.; Atlri, S. N. (24): Time domain inverse problems in nonlinear systems sing collocation & radial basis fnctions. CMES Compt. Model. Eng. Sci., vol., pp Gong, D.; Wei, C.; Wang, L.; Feng, L.; Wang, L. (2): adaptive methods for center choosing of radial basis fnction interpolation: a review. Lectre Notes in Compt. Sci., vol. 6377, pp

20 394 Copyright 25 Tech Science Press CMES, vol.8, no.6, pp , 25 Han, Z.; Atlri, S. (24): On the (meshless local petrov-galerkin) mlpg-eshelby method in comptational finite deformation solid mechanics - part ii. CMES Compt. Model. Eng. Sci., vol. 97, pp Hon, Y.; Mao, X. (998): An efficient nmerical scheme for brgers eqation. Appl. Math. Compt., vol. 95, pp Hon, Y.; Schaback, R. (28): Solvability of partial differential eqations by meshless kernel methods. Adv. Compt. Math., vol. 28, pp Hang, W.; Ren, Y.; Rssell, R. D. (99): Mltiqadrics a scattered data approimation scheme with applications to comptational flid dynamics i: srface approimations and partial derivative estimates. Compters Math. Appl., vol. 9, pp Hang, W.; Ren, Y.; Rssell, R. D. (99): Mltiqadrics a scattered data approimation scheme with applications to comptational flid-dynamicsii: soltions to parabolic, hyperbolic and elliptic partial differential eqations. Compters Math. Appl., vol. 9, pp Hang, W.; Ren, Y.; Rssell, R. D. (994): Moving mesh partial differential eqations (mmpdes) based on the eqidistribtion principle. SIAM J. Nmer. Anal., vol. 3, pp Lee, C.; Li, X.; Fan, S. (23): Local mltiqadric approimation for solving bondary vale problems. Compt. Mech., vol. 3, pp Lin, H.; Atlri, S. (2): Meshless local petrov-galerkin (mlpg) method for convection-diffsion problems. CMES Compt. Model. Eng. Sci., vol., pp Lin, H.; Atlri, S. (2): The meshless local petrov-galerkin (mlpg) method for solving incompressible navier-stokes eqations. CMES Compt. Model. Eng. Sci., vol. 2, pp Ling, L.; Opfe, R.; Schaback, R. (26): Reslts on meshless collocation techniqes. Eng. Anal. Bond. Elem., vol. 3, pp Ling, L.; Schaback, R. (29): An improved sbspace selection algorithm for meshless collocation methods. Int. J. Nmer. Meth. Engng., vol. 8, pp Mai-Dy, N.; Tran-Cong, T. (22): Mesh-free radial basis fnction network methods with domain decomposition for approimation of fnctions and nmerical soltion of poisson eqation. Eng. Anal. Bond. Elem., vol. 26, pp

21 Solving a Class of PDEs by a Local Reprodcing Kernel Method 395 Mohammadi, M.; Mokhtari, R. (2): Solving the generalized reglarized long wave eqation on the basis of a reprodcing kernel space. J. Compt. Appl. Math., vol. 235, pp Mohammadi, M.; Mokhtari, R. (23): A new algorithm for solving nonlinear schrödinger eqation in the reprodcing kernel space. Iranian J. Sci. Tech., vol. 37, pp Mohammadi, M.; Mokhtari, R. (24): A reprodcing kernel method for solving a class of nonlinear systems of pdes. Math. Model. Anal., vol. 9, pp Mohammadi, M.; Mokhtari, R.; Isfahani, F. T. (24): Solving an inverse problem for a parabolic eqation with a nonlocal bondary condition in the reprodcing kernel space. Iranian J. Nmer. Anal. Optimization., vol. 4, pp Mohammadi, M.; Mokhtari, R.; Panahipor, H. (23): A galerkinreprodcing kernel method: application to the 2d nonlinear copled brgers e- qations. Eng. Anal. Bond. Elem., vol. 37, pp Mohammadi, M.; Mokhtari, R.; Panahipor, H. (24): Solving two parabolic inverse problems with a nonlocal bondary condition in the reprodcing kernel space. Appl. Compt. Math., vol. 3, pp Mohammadi, M.; Mokhtari, R.; Schaback, R. (24): A meshless method for solving the 2d brsselator reaction-diffsion system. CMES: Compt. Model. Eng. Sci., vol., pp Mokhtari, R.; Isfahani, F. T.; Mohammadi, M. (22): Reprodcing kernel method for solving nonlinear differential-difference eqations. Abstr. Appl. Anal., vol. 22, pp.. Mller, S.; Schaback, R. (29): Theory., vol. 6, pp Pazoki, M.; Schaback, R. (2): Appl. Math., vol. 236, pp A newton basis for kernel spaces. J. Appro. Bases for kernel-based spaces. J. Compt. Pgh, S. (995): Finite element approimations of brgers eqation. Scahabck, R. (2): Kernel-based meshless methods. Lectre Note, Gottingen. Schaback, R. (995): Error estimates and condition nmbers for radial basis fnction interpolation. Adv. Compt. Math., vol. 3, pp Schaback, R. (25): A comptational tool for comparing all linear pde solvers. Adv. Compt. Math., vol. 4, pp Wendland, H. (25): Press. Scattered Data Approimation. Cambridge University

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