Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 978310, 11 pages http://dx.doi.org/10.1155/2014/978310 Research Article A Meshfree Method for Numerical Solution of Nonhomogeneous Time-Dependent Problems Ziwu Jiang,1 Lingde Su,1,2 and Tongsong Jiang1 1 Department of Mathematics, Linyi University, Linyi 276005, China 2 College of Mathematics, Shandong Normal University, Jinan 250014, China Correspondence should be addressed to Tongsong Jiang; [email protected] Received 17 April 2014; Revised 2 July 2014; Accepted 2 July 2014; Published 21 July 2014 Academic Editor: Ali H. Bhrawy Copyright © 2014 Ziwu Jiang et al. This 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 numerical meshfree scheme to solve time-dependent problems with variable coefficient governed by telegraph and wave equations which are more suitable than ordinary diffusion equations in modelling reaction diffusion for such branches of sciences. Finite difference method is adopted to deal with time variable and its derivative, and radial basis functions methodis developed for spatial discretization. The results of numerical experiments are presented and are compared with analytical solutions to confirm the accuracy of our scheme. 1. Introduction thefinitedifference,finiteelements,andmultigridmethods. As one kind of mesh method, finite difference methods are This paper is devoted to the numerical computation of the adoptedtosolvethispartialdifferentialequations[1, 2]. nonhomogeneous time-dependent problem with the follow- Although these methods are effective for solving various ing form: kinds of partial differential equations, conditional stability of explicit finite difference procedures and the need to + =1 () ++2 (,) , (1) use large amount of CPU time in implicit finite difference where , , are constants, 1() and 2(, ) are given schemes limit the applicability of these methods. Further- analytic functions, and ∈{0,1}.Thecases=0and = more, numerical solution can be provided only on mesh 1 correspond to the telegraph problem and wave problem, points from these methods [3],andtheaccuracyofthesewell- respectively. known techniques is reduced in nonsmooth and nonregular Telegraph equations describe various phenomena in domains. Some authors used Legendre-Gauss-Lobatto collo- many applied fields, such as a planar random motion of cation method and Chebyshev-tau method to solve the space- aparticleinfluidflow,transmissionofelectricalimpulses fractional advection diffusion equation and got high accuracy intheaxonsofnerveandmusclecells,propagationof results [4, 5]. electromagnetic waves in superconducting media, and prop- Recently, meshfree techniques have attracted attention agation of pressure waves occurring in pulsatile blood flow of researchers in order to avoid the mesh generation. Some in arteries. The wave equation is also an important second- meshfree schemes are the element free Galerkin method, order linear partial differential equation for the description thereproducingkernelparticle,thepointinterpolation,and of waves, such as sound waves, light waves, and water waves. so forth. For more description see [6] and the references It arises in fields like acoustics, electromagnetics, and fluid therein. Dehghan and Shokri [7]havesolvedthesecondorder dynamics. telegraph equations with constant coefficients using meshfree Over the past several decades, many numerical methods method. In this paper, we extend this problem considered in have been developed to solve boundary-value problems [7] to one kind of partial differential equations with variable involving ordinary and partial differential equations, such as coefficients. 2 Abstract and Applied Analysis Radical basis functions (RBFs) method is known as a influenced by the choice of parameter ,sinceunsuitable powerful approximating tool for scattered data interpolation parameter will produce the singular interpolation matrix. problem. As a meshfree method, the usage of RBFs to solve Moreover, the number of the chosen nodes can also affect numerical solution of partial differential equations is based the accuracy. Further learning about RBFs method can be got on the collocation scheme. The major advantage of numerical from [22, 23]. proceduresbyusingRBFsismeshfreecomparedwiththe If P denotes the space of -variate polynomial of order traditional techniques. RBFs are used actively for solving not exceeding more than andlettingthepolynomials PDEs, and the examples can be found in [10–14]. 1,2,..., be the basis of in , then the polynomial Inthelastdecade,thedevelopmentoftheRBFsmethodas (x) in (2)isusuallywritteninthefollowingform: a truly meshfree approach for approximating the solutions of partial derivative equations has drawn the attention of many (x) = ∑ (x ), researchers in science and engineering. Meshfree method has (4) become an important numerical computation method, and =1 there are many academic monographs published [15–21]. where = ( − 1 + )!/(!(. −1)!) In this paper, we present an effective numerical scheme Toget the coefficients (1,2,...,) and (1,2,...,), to solve time-dependent problems governed by telegraph the collocation method is used. However, in addition to the andwaveequationsusingthemeshfreemethodwithRBFs. equations resulting from collocating (2)atthe points, extra The results of numerical experiments are presented and are equations are required. This is ensured by the conditions compared with analytical solutions to confirm the good accuracy of the presented scheme. The layout of the paper is as follows. In Section 2,the ∑ (x)=0, =1,2,...,. (5) overview about RBFs and the numerical scheme of our =1 method on the time-dependent problems are introduced. The Supposed that L, is a linear partial differential operator, then results of numerical experiments are presented in Section 3. L can be approximated by Section 4 is dedicated to a brief conclusion. L (x) ≃ ∑L(x, x)+L (x) . (6) 2. The Meshfree Method =1 2.1. Radial Basis Function Approximation. The approxima- tion of a distribution (x),usingRBFs,maybewrittenasa 2.2. Nonhomogeneous Time-Dependent Problems. Let ue linear combination of radial basis functions, and usually it consider the following time-dependent problem: takes the following form: + =1 () ++2 (,) , (7) (x) ≃ ∑(x, x)+(x) , for x ∈Ω⊂ , (2) ∈Ω∪Ω=[1,1]⊂, 0<≤, =1 with initial conditions where is the number of data points, x =(1,2,...,), (,) 0 = () ,∈Ω, is the dimension of the problem, the ’s are coefficients to 1 be determined, and is the radial basis function. Equation (8) (,) 0 =2 () ,∈Ω, (2) can be written without the additional polynomial .In that case, must be a positive definite function to guarantee and Dirichlet boundary conditions the solvability of the resulting system. However, is usually required when is conditionally positive definite, that is, (,) =ℎ(,) ,∈Ω,0<≤, (9) when hasapolynomialgrowthtowardsinfinity.Wewilluse () (, ) RBFs, which defined as where , ,and are constant coefficients, 1 , 2 , 1(), 2(),andℎ(, ) are given functions, and (, ) is the 1 unknown function. Inverse multiquadric (IMQ): (x, x)= , 2 2 √ + Equation (7) is discretized according to the following - weighted scheme: >0, (3) (, )+ −2(,) +(, )− 2 ()2 Thin plate splines (TPS): (x, x)= log (), (, )+ −(, )− = 1, 2, 3, . ., + 2 where =‖x − x‖ is the Euclidean norm. Since given by ∞ =[1 () Δ (, )+ +(, )+ ] (3)is continuous, we can use it directly. √ 2 2 The IMQ radial basis function takes the form 1/ + , + (1−) [1 () Δ (,) +(,) ]+2 (, )+ , >0. The accuracy of the numerical solution is severely (10) Abstract and Applied Analysis 3 where 0≤≤1, isthetimestepsize,andΔ is the Laplace Using the notation LA to designate the matrix of the −1 operator. By using the notation =(,) with = + same dimension as A and containing the elements ̂ where ,wecanget ̂ = L , 1≤,≤,then(11) together with the boundary conditions (9)canbewritteninmatrixformas (1 + −2)+1 −2 () Δ+1 1 2 []+1 =[] +( −1)[ ]−1 2 2 2 (19) =(2+(1−) ) +(1−) 1 () Δ (11) +2[ ]+1 + []+1, 2 + ( −1)−1 +2+1. 2 2 where 2 2 Supposethatthereareatotalof−2interpolation points, =(2+(1−) ) A +(1−) ([1]∗ΔA), and () canbeapproximatedby 2 −2 =(1+ − ) A 2 () = ∑ ()+−1+. (12) =1 2 − ([1]∗ΔA)+A + A, ( , ,..., , ) In order to determine the coefficients 1 2 −1 , −1 −1 −1 the collocation method is used by applying (12)ateverypoint [] =[1 ,..., ,0,...,0] , , =1,2,...,−2.Thusweobtain +1 +1 +1 −2 [] =[0,...,0,ℎ+1,...,ℎ−2,0,0] , ( )= ∑ ( )+ + , −1 (13) +1 +1 +1 =1 [2] =[2 (1),...,2 (),0,...,0] , = √( −)2 where .Theadditionalconditionsdueto(5) [1]=[1 (1),...,1 (),0,...,0] . canbewrittenas (20) −2 −2 The operator “∗” means that the th component of vector ∑ = ∑ =0. (14) =1 =1 [1] is multiplied to all components of th row of matrix Δ.Equation(19)isobtainedbycombining(11)appliedto Writing (13)togetherwith(14)inamatrixform the domain points, and (9)appliedtotheboundarypoints [] = A [] , (15) meanwhile. At =0,(19)hasthefollowingform: where 1 0 −1 2 1 1 [] =[] +( −1)[] + [2] + [] . [] =[1,2,...,−2,0,0] , 2 (16) (21) [] =[1,2,...,] , −1 A =[ ,1≤,≤] To approximate , the second initial condition can be used. and is given as follows: For this purpose, the second initial condition is discretized as ⋅⋅⋅ 1 11 1(−2) 1 1 () −−1 () . =2 () ,∈Ω. (22) . d . 2