Research Article a Meshfree Method for Numerical Solution of Nonhomogeneous Time-Dependent Problems

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 𝑢𝑡𝑡 +𝑎𝛼𝑢𝑡 =𝑏𝑓1 (𝑥) 𝑢𝑥𝑥 +𝑐𝛼𝑢+𝑓2 (𝑥,) 𝑡 , (1) use large amount of CPU time in implicit finite difference where 𝑎, 𝑏, 𝑐 are constants, 𝑓1(𝑥) and 𝑓2(𝑥, 𝑡) are given schemes limit the applicability of these methods. Further- analytic functions, and 𝛼∈{0,1}.Thecases𝛼=0and 𝛼= 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 𝑐,sinceunsuitable powerful approximating tool for scattered data interpolation parameter 𝑐 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]. 𝑑 proceduresbyusingRBFsismeshfreecomparedwiththe If P𝑞 denotes the space of 𝑑-variate polynomial of order traditional techniques. RBFs are used actively for solving not exceeding more than 𝑞 andlettingthepolynomials 𝑑 𝑑 PDEs, and the examples can be found in [10–14]. 𝑃1,𝑃2,...,𝑃𝑚 be the basis of 𝑃𝑞 in 𝑅 , then the polynomial Inthelastdecade,thedevelopmentoftheRBFsmethodas 𝜓(x) in (2)isusuallywritteninthefollowingform: a truly meshfree approach for approximating the solutions of partial derivative equations has drawn the attention of many 𝑚 𝜓 (x) = ∑𝜁 𝑃 (x ), researchers in science and engineering. Meshfree method has 𝑖 𝑖 𝑗 (4) become an important numerical computation method, and 𝑖=1 there are many academic monographs published [15–21]. where 𝑚 = (𝑞 − 1 + 𝑑)!/(𝑑!(𝑞. −1)!) In this paper, we present an effective numerical scheme Toget the coefficients (𝜆1,𝜆2,...,𝜆𝑁) and (𝜁1,𝜁2,...,𝜁𝑚), to solve time-dependent problems governed by telegraph the collocation method is used. However, in addition to the 𝑁 andwaveequationsusingthemeshfreemethodwithRBFs. equations resulting from collocating (2)atthe𝑁 points, extra The results of numerical experiments are presented and are 𝑚 equations are required. This is ensured by the 𝑚 conditions compared with analytical solutions to confirm the good accuracy of the presented scheme. 𝑁 The layout of the paper is as follows. In Section 2,the ∑𝜆𝑗𝑃𝑖 (x𝑗)=0, 𝑖=1,2,...,𝑚. (5) overview about RBFs and the numerical scheme of our 𝑗=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. L𝑢 can be approximated by Section 4 is dedicated to a brief conclusion. 𝑁 L𝑢 (x) ≃ ∑𝜆𝑗L𝜑(x, x𝑗)+L𝜓 (x) . (6) 2. The Meshfree Method 𝑗=1 2.1. Radial Basis Function Approximation. The approximation of a distribution 𝑢(x),usingRBFs,maybewrittenasa 2.2. Nonhomogeneous Time-Dependent Problems. Let ue linear combination of 𝑁 radial basis functions, and usually it consider the following time-dependent problem: takes the following form: 𝑢𝑡𝑡 +𝑎𝛼𝑢𝑡 =𝑏𝑓1 (𝑥) 𝑢𝑥𝑥 +𝑐𝛼𝑢+𝑓2 (𝑥,) 𝑡 , 𝑁 𝑑 (7) 𝑢 (x) ≃ ∑𝜆𝑗𝜑(x, x𝑗)+𝜓(x) , for x ∈Ω⊂𝑅 , (2) 𝑥∈Ω∪𝜕Ω=[𝑎1,𝑏1]⊂𝑅, 0<𝑡≤𝑇, 𝑗=1 with initial conditions where 𝑁 is the number of data points, x =(𝑥1,𝑥2,...,𝑥𝑑), 𝑢 (𝑥,) 0 =𝑔 (𝑥) ,𝑥∈Ω, 𝑑 is the dimension of the problem, the 𝜆’s are coefficients to 1 be determined, and 𝜑 is the radial basis function. Equation (8) 𝑢𝑡 (𝑥,) 0 =𝑔2 (𝑥) ,𝑥∈Ω, (2) can be written without the additional polynomial 𝜓.In that case, 𝜑 must be a positive definite function to guarantee and Dirichlet boundary conditions the solvability of the resulting system. However, 𝜓 is usually required when 𝜑 is conditionally positive definite, that is, 𝑢 (𝑥,) 𝑡 =ℎ(𝑥,) 𝑡 ,𝑥∈𝜕Ω,0<𝑡≤𝑇, (9) when 𝜑 hasapolynomialgrowthtowardsinfinity.Wewilluse 𝑎 𝑏 𝑐 𝑓 (𝑥) 𝑓 (𝑥, 𝑡) RBFs, which defined as where , ,and are constant coefficients, 1 , 2 , 𝑔1(𝑥), 𝑔2(𝑥),andℎ(𝑥, 𝑡) are given functions, and 𝑢(𝑥, 𝑡) is the 1 unknown function. Inverse multiquadric (IMQ): 𝜑(x, x𝑗)= , 2 2 𝜃 √𝑟𝑗 +𝑐 Equation (7) is discretized according to the following - weighted scheme: 𝑐>0, (3) 𝑢 (𝑥, 𝑡 )+ 𝑑𝑡 −2𝑢(𝑥,) 𝑡 +𝑢(𝑥, 𝑡 )− 𝑑𝑡 2𝑚 (𝑑𝑡)2 Thin plate splines (TPS): 𝜑(x, x𝑗)=𝑟𝑗 log (𝑟𝑗), 𝑢 (𝑥, 𝑡 )+ 𝑑𝑡 −𝑢(𝑥, 𝑡 )− 𝑑𝑡 𝑚 = 1, 2, 3, . . ., +𝑎𝛼 2𝑑𝑡 where 𝑟𝑗 =‖x − x𝑗‖ is the Euclidean norm. Since 𝜑 given by ∞ =𝜃[𝑏𝑓1 (𝑥) Δ𝑢 (𝑥, 𝑡 )+ 𝑑𝑡 +𝑐𝛼𝑢(𝑥, 𝑡 )+ 𝑑𝑡 ] (3)is𝐶 continuous, we can use it directly. √ 2 2 The IMQ radial basis function takes the form 1/ 𝑟 +𝑐 , + (1−𝜃) [𝑏𝑓1 (𝑥) Δ𝑢 (𝑥,) 𝑡 +𝑐𝛼𝑢(𝑥,) 𝑡 ]+𝑓2 (𝑥, 𝑡 )+ 𝑑𝑡 , 𝑐>0. The accuracy of the numerical solution is severely (10) Abstract and Applied Analysis 3 where 0≤𝜃≤1, 𝑑𝑡 isthetimestepsize,andΔ is the Laplace Using the notation LA to designate the matrix of the 𝑛 𝑛 𝑛 𝑛−1 operator. By using the notation 𝑢 =𝑢(𝑥,𝑡) with 𝑡 =𝑡 + same dimension as A and containing the elements 𝑎̂𝑖𝑗 where 𝑑𝑡,wecanget 𝑎̂𝑖𝑗 = L𝑎𝑖𝑗 , 1≤𝑖,𝑗≤𝑁,then(11) together with the boundary 𝑑𝑡 conditions (9)canbewritteninmatrixformas (1 + 𝑎𝛼 −𝑐𝛼𝜃𝑑𝑡2)𝑢𝑛+1 −𝑏𝜃𝑑𝑡2𝑓 (𝑥) Δ𝑢𝑛+1 1 𝑎𝛼𝑑𝑡 2 𝐵[𝜆]𝑛+1 =𝐶[𝜆]𝑛 +( −1)[𝑢 ]𝑛−1 2 𝑑 2 𝑛 2 𝑛 (19) =(2+𝑐𝛼(1−𝜃) 𝑑𝑡 )𝑢 +𝑏(1−𝜃) 𝑑𝑡 𝑓1 (𝑥) Δ𝑢 (11) +𝑑𝑡2[𝑓 ]𝑛+1 + [𝐻]𝑛+1, 𝑑𝑡 2 + (𝑎𝛼 −1)𝑢𝑛−1 +𝑑𝑡2𝑓𝑛+1. 2 2 where 2 2 Supposethatthereareatotalof𝑁−2interpolation points, 𝐶=(2+𝑐𝛼(1−𝜃) 𝑑𝑡 ) A𝑑 +𝑏(1−𝜃) 𝑑𝑡 ([𝑓1]∗ΔA𝑑), 𝑛 and 𝑢 (𝑥) canbeapproximatedby 𝑑𝑡 2 𝑁−2 𝐵=(1+𝑎𝛼 −𝑐𝛼𝜃𝑑𝑡 ) A𝑑 𝑛 𝑛 𝑛 𝑛 2 𝑢 (𝑥) = ∑ 𝜆𝑗𝜑(𝑟𝑗)+𝜆𝑁−1𝑥+𝜆𝑁. (12) 𝑗=1 2 −𝑏𝜃𝑑𝑡 ([𝑓1]∗ΔA𝑑)+A𝑏 + A𝑒, (𝜆 ,𝜆 ,...,𝜆 ,𝜆 ) In order to determine the coefficients 1 2 𝑁−1 𝑁 , 𝑛−1 𝑛−1 𝑛−1 𝑇 the collocation method is used by applying (12)ateverypoint [𝑢𝑑] =[𝑢1 ,...,𝑢𝑝 ,0,...,0] , 𝑥𝑖, 𝑖=1,2,...,𝑁−2.Thusweobtain 𝑛+1 𝑛+1 𝑛+1 𝑇 𝑁−2 [𝐻] =[0,...,0,ℎ𝑝+1,...,ℎ𝑁−2,0,0] , 𝑢𝑛 (𝑥 )= ∑ 𝜆𝑛𝜑(𝑟 )+𝜆𝑛 𝑥 +𝜆𝑛 , 𝑖 𝑗 𝑖𝑗 𝑁−1 𝑖 𝑁 (13) 𝑛+1 𝑛+1 𝑛+1 𝑇 𝑗=1 [𝑓2] =[𝑓2 (𝑥1),...,𝑓2 (𝑥𝑝),0,...,0] , 𝑟 = √(𝑥 −𝑥)2 𝑇 where 𝑖𝑗 𝑖 𝑗 .Theadditionalconditionsdueto(5) [𝑓1]=[𝑓1 (𝑥1),...,𝑓1 (𝑥𝑝),0,...,0] . canbewrittenas (20) 𝑁−2 𝑁−2 𝑛 𝑛 The operator “∗” means that the 𝑖th component of vector ∑ 𝜆𝑗 = ∑ 𝜆𝑗𝑥𝑗 =0. (14) 𝑗=1 𝑗=1 [𝑓1] is multiplied to all components of 𝑖th row of matrix Δ𝐴𝑑.Equation(19)isobtainedbycombining(11)appliedto Writing (13)togetherwith(14)inamatrixform the domain points, and (9)appliedtotheboundarypoints 𝑛 𝑛 [𝑢] = A [𝜆] , (15) meanwhile. At 𝑛=0,(19)hasthefollowingform: where 1 0 𝑎𝛼𝑑𝑡 −1 2 1 1 𝑛 𝑛 𝑛 𝑛 𝑇 𝐵[𝜆] =𝐶[𝜆] +( −1)[𝑢𝑑] +𝑑𝑡 [𝑓2] + [𝐻] . [𝑢] =[𝑢1,𝑢2,...,𝑢𝑁−2,0,0] , 2 (16) 𝑛 𝑛 𝑛 𝑛 𝑇 (21) [𝜆] =[𝜆1,𝜆2,...,𝜆𝑁] , 𝑢−1 A =[𝑎 ,1≤𝑖,𝑗≤𝑁] To approximate , the second initial condition can be used. and 𝑖𝑗 is given as follows: For this purpose, the second initial condition is discretized as 𝜑 ⋅⋅⋅ 𝜑 𝑥 1 11 1(𝑁−2) 1 𝑢1 (𝑥) −𝑢−1 (𝑥) . . . . =𝑔2 (𝑥) ,𝑥∈Ω. (22) . d . . . 2𝑑𝑡 A =(𝜑 ⋅⋅⋅ 𝜑 𝑥 1). (17) (𝑁−2)1 (𝑁−2)(𝑁−2) 𝑁−2 Writing (21)togetherwith(22), we have 𝑥1 ⋅⋅⋅ 𝑥𝑁−2 00 1 ⋅⋅⋅ 1 0 0 𝑎𝛼𝑑𝑡 (𝐵 + (1 − )𝐴 ) [𝜆]1 2 𝑑 Assuming that there are 𝑝<𝑁−2internal points and (23) 𝑁−2−𝑝 𝑁×𝑁 A 1 boundary points, then the matrix can be =𝐶[𝜆]0 + (2−𝑎𝛼𝑑𝑡) 𝑑𝑡 [𝐺] + (𝑑𝑡)2[𝑓] + [𝐻]1, split into A = A𝑑 + A𝑏 + A𝑒,where 𝑇 where [𝐺] = [(𝑔2)1,...,(𝑔2)𝑝,0,...,0] . Together with the A𝑑 =[𝑎𝑖𝑗 for (1≤𝑖≤𝑝,1≤𝑗≤𝑁),0elsewhere], initial condition (8)and(19), we can get all the 𝜆;thuswecan get the numerical solutions. A𝑏 =[𝑎𝑖𝑗 for (𝑝+1≤𝑖≤𝑁−2,1≤𝑗≤𝑁), Since the coefficient matrix is unchanged in each tempo- 0 elsewhere], ral step, we use the LU factorization to the coefficient matrix only once and use this factorization in our algorithm.

A𝑒 =[𝑎𝑖𝑗 for (𝑁−1≤𝑖≤𝑁,1≤𝑗≤𝑁),0elsewhere]. Remark. Although (19)isvalidforanyvalueof𝜃∈[0,1],we (18) will use 𝜃=1/2(the famous Crank-Nicolson scheme). 4 Abstract and Applied Analysis

Space-time exact t=1 Space-time estimated t=1 Space-time exact t=2 Space-time estimated t=2 0 0 0 0 0.09 0.09 0.1 0.1 0.1 0.1 0.08 0.08 0.12 0.12 0.2 0.2 0.2 0.2 0.07 0.07 0.1 0.1 0.3 0.3 0.3 0.3 0.06 0.06 0.4 0.4 0.4 0.08 0.4 0.08 0.05 0.05 x 0.5 x 0.5 x 0.5 x 0.5 0.04 0.04 0.06 0.06 0.6 0.6 0.6 0.6 0.03 0.03 0.7 0.7 0.7 0.04 0.7 0.04

0.8 0.02 0.8 0.02 0.8 0.8 0.02 0.02 0.9 0.01 0.9 0.01 0.9 0.9

1 0 1 0 1 1 0 0 0.5 1 0 0.5 1 0 12 012 Times Times Times Times

Space-time exact t=3 Space-time estimated t=3 Space-time exact t=4 Space-time estimated t=4 0 0 0 0

0.1 0.12 0.1 0.12 0.1 0.12 0.1 0.12 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3

0.4 0.08 0.4 0.08 0.4 0.08 0.4 0.08 x 0.5 x 0.5 x 0.5 x 0.5 0.06 0.06 0.06 0.06 0.6 0.6 0.6 0.6

0.7 0.04 0.7 0.04 0.7 0.04 0.7 0.04

0.8 0.8 0.8 0.8 0.02 0.02 0.02 0.02 0.9 0.9 0.9 0.9

1 1 0 1 1 0 0 123 0 123 024 0 2 4 Times Times Times Times

Figure 1: Space-time graph of numerical and analytical solutions at time 𝑡 = 1, 2, 3, 4 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001 and using IMQ (with 𝑐 = 0.09) as the radial basis function for Example 1.

2 𝑥 2 −𝑡 3. Numerical Example where 𝑓2(𝑥, 𝑡) = [𝑥(1 − 𝑥)(2 −2𝑡+𝑡 )+2𝑒 𝑡 ]𝑒 ,withthe boundary conditions In this section, we present some numerical results to test the efficiency of the new scheme for solving time-dependent 𝑢 (0, 𝑡) =0, 𝑡>0, (25) problems governed by telegraph and wave equations. 𝑢 (1, 𝑡) =0, 𝑡>0,

and the initial conditions 𝑥 3.1. Example 1. Let 𝑎=𝑏=1, 𝛼=1, 𝑐=−1,and𝑓1(𝑥) = 𝑒 ; 𝑢 (𝑥,) 0 =0, 0≤𝑥≤1, (1) becomes the telegraph equation (26) 𝑢𝑡 (𝑥,) 0 =0, 0≤𝑥≤1.

The analytical solution of the equation is 𝑥 2 −𝑡 𝑢𝑡𝑡 +𝑢𝑡 =𝑒 𝑢𝑥𝑥 −𝑢+𝑓2 (𝑥,) 𝑡 , 0<𝑥<1,𝑡>0, (24) 𝑢 (𝑥,) 𝑡 =𝑥(1−𝑥) 𝑡 𝑒 . (27) Abstract and Applied Analysis 5

Exact and estimated t=1 Exact and estimated t = 1.5 0.1 0.2 0.09 0.18 0.08 0.16 0.07 0.14 0.06 0.12 0.05 0.1 0.04 0.08 Solutions Solutions 0.03 0.06 0.02 0.04 0.01 0.02 0 0 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 100 Nodes Nodes

Exact Exact Estimated Estimated

Exact and estimated t=2 Exact and estimated t=3 0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15 Solutions Solutions

0.1 0.1

0.05 0.05

0 0 0 10 20 30 40 50 60 70 80 90 100 0204060 80 100 Nodes Nodes

Exact Exact Estimated Estimated

Figure 2: The numerical and analytical solutions at time 𝑡 = 1, 1.5, 2,3 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001 and using TPS (with 𝑚=2)astheradial basis function for Example 2.

We solve this problem by using the IMQ and TPS radial 3.2. Example 2. In this example, 𝛼=0, 𝑏=1,and𝑓1(𝑥) = 2 basis functions. These results are obtained for 𝑑𝑥 = 0.01 and 𝑥 +1(1) become the wave equation 𝑑𝑡 = 0.001.The𝐿∞, 𝐿2, and root-mean-square (RMS) errors are obtained in Table 1 for 𝑡 = 1, 2, 3,and4. 2 Wealso give the analysis of the parameter 𝑐 in IMQ for the 𝑢𝑡𝑡 =(𝑥 +1)𝑢𝑥𝑥 +𝑓2 (𝑥,) 𝑡 , 0<𝑥<1,𝑡>0, (28) results. In Table 2,the𝐿∞, 𝐿2, and RMS errors with different 𝑐 at time 𝑡=3are presented. 2 3 2 2 3 −𝑡 The space-time graph of analytical and numerical results where 𝑓2(𝑥, 𝑡) = [2(𝑥 +1)𝑡 +(𝑥−𝑥 )(6𝑡 − 6𝑡 +𝑡 )]𝑒 ,with for 𝑡 = 1, 2, 3, 4 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001, 𝜃=1/2 the boundary conditions by using IMQ (with 𝑐 = 0.09) as the RBF is given in Figure 1. The results obtained show the very good accuracy and efficiency of the new approximate scheme. Note that 𝑢 (0, 𝑡) =0, 𝑡>0, we cannot distinguish the exact solution from the estimated (29) 𝑢 (1, 𝑡) =0, 𝑡>0. solution in Figure 1. 6 Abstract and Applied Analysis

Exact and estimated t=1 Exact and estimated t=2 0.7 0.4

0.6 0.35 0.3 0.5 0.25 0.4 0.2 0.3 0.15 Solutions Solutions 0.2 0.1 0.1 0.05

0 0

−0.1 −0.05 010.2 0.4 0.6 0.8 010.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x x

Exact Exact Estimated Estimated

Exact and estimated t=3 Exact and estimated t=4 0.3 0.14

0.25 0.12

0.1 0.2 0.08 0.15 0.06 0.1 Solutions Solutions 0.04 0.05 0.02

0 0

−0.05 −0.02 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x

Exact Exact Estimated Estimated

Figure 3: Space-time graph of numerical and analytical solutions at time 𝑡 = 1, 2, 3, 4 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001 and using IMQ (with 𝑐 = 0.08) as the radial basis function for Example 3.

The initial conditions are given by The analytical and numerical results for 𝑡 = 1, 1.5,,and 2 3 𝑑𝑥 = 0.01 𝑑𝑡 = 0.001 𝜃=1/2 𝑢 (𝑥,) 0 =𝑔 (𝑥) =0, 0≤𝑥≤1, with , , are given in Figure 2 by 1 using TPS (with 𝑚=2)astheRBF. (30) 𝑢𝑡 (𝑥,) 0 =𝑔2 (𝑥) =0, 0≤𝑥≤1, 3.3. Example 3. In this example, we consider the following and the analytical solution of the equation is given as wave equation: 2 3 −𝑡 𝑢 (𝑥,) 𝑡 =(𝑥−𝑥)𝑡 𝑒 . (31) 𝑢𝑡𝑡 =𝑢𝑥𝑥 +𝑓2 (𝑥,) 𝑡 , 0<𝑥<1,𝑡>0, (32) IMQ and TPS are used as the radial basis function in the 𝑑𝑥 = 2 −𝑡/2 discussed scheme, and these results are obtained for where 𝑓2(𝑥, 𝑡) = ((1/4) +𝜋 )𝑒 sin(𝜋𝑥),withtheboundary 0.01 and 𝑑𝑡 = 0.001. conditions Table 3 presents the 𝐿∞ and 𝐿2 and RMS errors for 𝑡= 1, 1.5, 2,and3. 𝑢 (0, 𝑡) =0, 𝑡>0, Similar to Example 2, in Table 4,the𝐿∞, 𝐿2,andRMS (33) 𝑢 (1, 𝑡) =0, 𝑡>0, errors with different 𝑐 at time 𝑡=2are presented. Abstract and Applied Analysis 7

Table 1: Numerical errors using IMQ (𝑐 = 0.09)andTPS(𝑚=2) represented at different times, where 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2. 𝐿 𝐿 𝑡 ∞-error 2-error RMS IMQ TPS IMQ TPS IMQ TPS 1 6.103 × 10−5 6.576 × 10−5 4.401 × 10−5 4.849 × 10−5 4.336 × 10−5 4.778 × 10−5 2 3.130 × 10−5 2.403 × 10−5 2.267 × 10−5 1.604 × 10−5 2.233 × 10−5 1.580 × 10−5 3 9.933 × 10−6 4.192 × 10−6 8.155 × 10−6 2.195 × 10−6 8.035 × 10−6 2.163 × 10−6 4 1.792 × 10−5 1.393 × 10−5 1.351 × 10−5 9.809 × 10−6 1.331 × 10−5 9.665 × 10−6

Table 2: Numerical errors using IMQ represented with different parameter 𝑐 at time 𝑡=3,where𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2.

𝑐𝐿∞ 𝐿2 RMS 0.01 9.242 × 10−2 6.522 × 10−2 6.427 × 10−2 0.03 5.959 × 10−4 5.559 × 10−4 5.478 × 10−4 0.05 1.175 × 10−4 1.139 × 10−4 1.122 × 10−4 0.07 2.850 × 10−5 2.630 × 10−5 2.591 × 10−5 0.09 9.933 × 10−6 8.155 × 10−6 8.035 × 10−6

Table 3: Numerical errors using IMQ (𝑐 = 0.08)andTPS(𝑚=2) represented at different times, where 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2. 𝐿 𝐿 𝑡 ∞-error 2-error RMS IMQ TPS IMQ TPS IMQ TPS 1.0 7.456 × 10−5 7.997 × 10−5 5.319 × 10−5 5.896 × 10−5 5.292 × 10−5 5.867 × 10−5 1.5 1.711 × 10−5 3.650 × 10−5 1.019 × 10−5 2.839 × 10−5 1.014 × 10−5 2.825 × 10−5 2.0 5.011 × 10−5 8.559 × 10−5 3.346 × 10−5 6.314 × 10−5 3.329 × 10−5 6.283 × 10−5 3.0 5.832 × 10−5 2.328 × 10−5 4.935 × 10−5 1.562 × 10−5 4.910 × 10−5 1.554 × 10−5

Table 4: Numerical errors using IMQ represented with different parameter 𝑐 at time 𝑡=2,where𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2.

𝑐𝐿∞ 𝐿2 RMS 0.01 2.260 × 10−1 1.598 × 10−1 1.590 × 10−1 0.02 7.283 × 10−3 5.736 × 10−3 5.707 × 10−3 0.04 6.325 × 10−4 5.998 × 10−4 5.969 × 10−4 0.06 1.261 × 10−4 8.307 × 10−5 8.265 × 10−5 0.08 5.011 × 10−5 3.346 × 10−5 3.329 × 10−5

Table 5: Numerical errors using IMQ (𝑐 = 0.08)andTPS(𝑚=2) represented at different times, where 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2. 𝐿 𝐿 𝑡 ∞-error 2-error RMS IMQ TPS IMQ TPS IMQ TPS 1 5.304 × 10−4 3.784 × 10−4 4.016 × 10−4 2.614 × 10−4 3.996 × 10−4 2.601 × 10−4 2 2.078 × 10−4 1.479 × 10−4 1.567 × 10−4 1.022 × 10−4 1.559 × 10−4 1.017 × 10−4 3 4.024 × 10−4 2.867 × 10−4 3.039 × 10−4 1.980 × 10−4 3.624 × 10−4 1.917 × 10−4 4 2.883 × 10−4 2.015 × 10−4 2.134 × 10−4 1.392 × 10−4 2.124 × 10−4 1.385 × 10−4 5 3.545 × 10−4 2.522 × 10−4 2.671 × 10−4 1.742 × 10−4 2.658 × 10−4 1.734 × 10−4

Table 6: Numerical errors using IMQ represented with different parameter 𝑐 at time 𝑡=1,where𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2.

𝑐𝐿∞ 𝐿2 RMS 0.01 1.584 × 10−1 1.056 × 10−1 1.051 × 10−1 0.02 3.556 × 10−2 2.789 × 10−2 2.775 × 10−2 0.04 3.344 × 10−3 3.173 × 10−3 3.158 × 10−3 0.06 9.963 × 10−4 8.535 × 10−4 8.493 × 10−4 0.08 5.304 × 10−4 4.016 × 10−4 3.996 × 10−4 8 Abstract and Applied Analysis

Space-time exact t = 1.5 Space-time estimated t = 1.5

1 1

0.5 0.5

0 0 Exact Estimated

−0.5 −0.5

−1 0 −1 0 0 0 0.5 0.5 0.5 0.5 Times 1 x Times 1 x 1.5 1 1.5 1

Space-time exact t=3.5 Space-time estimated t=3.5

1 1

0.5 0.5

0 0 Exact Estimated

−0.5 −0.5

−1 0 −1 0 0 0 1 1 0.5 0.5 2 x 2 Times 3 Times 3 x 4 1 4 1

Figure 4: Space-time graph of numerical and analytical solutions at time 𝑡 = 1.5, 3.5 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001 and using TPS (𝑚=2)as the radial basis function for Example 4. and the initial conditions Wealso give the analysis of the parameter 𝑐 in IMQ for the results. In Table 6,the𝐿∞, 𝐿2, and RMS errors with different 𝑢 (𝑥,) 0 = (𝜋𝑥) , 0≤𝑥≤1, sin 𝑐 at time 𝑡=1are presented. 1 (34) The graph of analytical and numerical results for 𝑡= 𝑢 (𝑥,) 0 =− (𝜋𝑥) , 0≤𝑥≤1. 1, 2, 3 4 𝑑𝑥 = 0.01 𝑑𝑡 = 0.001 𝜃=1/2 𝑡 2 sin ,and with , ,and and using IMQ (with 𝑐 = 0.08)astheRBFaregiveninFigure3. The analytical solution of the equation is

−𝑡/2 𝑢 (𝑥,) 𝑡 =𝑒 sin (𝜋𝑥) . (35) 3.4. Example 4. In this example, 𝑏=1, 𝛼=0,and𝑓1(𝑥) =; 1 then (1) becomes the telegraph equation WealsouseIMQandTPSastheradialbasisfunctions in the discussed scheme, and these results are obtained for 𝑑𝑥 = 0.01 and 𝑑𝑡 = 0.001.The𝐿∞, 𝐿2,andRMSerrorsare 𝑢 =𝑢 +𝑓 (𝑥,) 𝑡 , 0<𝑥<1,𝑡>0, obtained in Table 5 for 𝑡 = 1, 2, 3, 4 and 5. 𝑡𝑡 𝑥𝑥 2 (36) Abstract and Applied Analysis 9

Space-time estimated t=1 Space-time estimated t=3

1 1

0.8 0.5 0.6 0 0.4 Estimated Estimated −0.5 0.2

0 −1 1 3 3 3

2 2 2 Times0.5 Times 1 x 1 1 x 0 0 0 0

Figure 5: Space-time graph of numerical solutions at time 𝑡=1,3with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001 by using IMQ (with 𝑐 = 0.09)inExample5.

Table 7: Numerical errors using IMQ (𝑐 = 0.06)andTPS(𝑚=2) represented at different times, where 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2. 𝐿 𝐿 𝑡 ∞-error 2-error RMS IMQ TPS IMQ TPS IMQ TPS 0.5 5.588 × 10−4 4.458 × 10−4 4.219 × 10−4 3.201 × 10−4 4.198 × 10−4 3.185 × 10−4 1.5 3.248 × 10−4 4.274 × 10−4 2.189 × 10−4 3.044 × 10−4 2.178 × 10−4 3.029 × 10−4 2.5 5.514 × 10−4 4.378 × 10−4 4.159 × 10−4 3.138 × 10−4 4.139 × 10−4 3.122 × 10−4 3.5 3.167 × 10−4 4.195 × 10−4 2.129 × 10−4 2.982 × 10−4 2.118 × 10−4 2.967 × 10−4

where 𝑓2(𝑥, 𝑡), =0 with the initial conditions where 𝑓2(𝑥, 𝑡) = −12 sin 𝑥 sin 𝑡+4cos 𝑡 sin 𝑥,withthe boundary conditions 𝑢 (𝑥,) 0 =0, 0≤𝑥≤1, 𝑢 (0, 𝑡) =0, 𝑡>0, (37) 𝑢 (𝑥,) 0 =𝜋 (𝜋𝑥) , 0≤𝑥≤1. (40) 𝑡 cos 𝑢 (1, 𝑡) = sin 1 cos 𝑡, 𝑡 > 0,

The analytical solution of the equation is and the initial conditions 𝑢 (𝑥,) 0 = sin 𝑥, 0≤𝑥≤1, 𝑢 (𝑥,) 𝑡 = cos (𝜋𝑥) sin (𝜋𝑡) . (38) (41) 𝑢𝑡 (𝑥,) 0 =−sin 𝑥 sin 𝑡, 0 ≤ 𝑥 ≤ 1. We extracted the boundary conditions from the exact solu- We use the radial basis functions (IMQ, TPS) for the tion. discussed scheme. These results are obtained for 𝑑𝑥 = 0.01 The IMQ and TPS radial basis functions utilized inour and 𝑑𝑡 = 0.001. scheme and these results are obtained for 𝑑𝑥 = 0.01,and𝑑𝑡 = The space-time graph of numerical results for 𝑡=1,3 0.001.The𝐿∞, 𝐿2, and RMS errors are obtained in Table 7 for with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2and using IMQ 𝑡 = 0.5, 1.5, 2.5 and 3.5. (with 𝑐 = 0.09)astheRBFaregiveninFigure5. The space-time graph of analytical and numerical results We also compared our methods with quartic B-spline col- for 𝑡 = 1.5, 3.5 with 𝑑𝑥 = ,0.01 𝑑𝑡 = 0.001,and𝜃=1/2and location method [8] and cubic B-spline quasi-interpolation using TPS (with 𝑚=2)astheRBFaregiveninFigure4. method [9], and Tables 8 and 9 give the comparison, respectively. 3.5. Example 5. In this example, we consider the following telegraph equation: 4. Conclusions In this paper, we proposed a numerical scheme to solve 𝑢 + 12𝑢 =𝑢 −4𝑢+𝑓 (𝑥,) 𝑡 , 0<𝑥<3,𝑡>0, 𝑡𝑡 𝑡 𝑥𝑥 2 (39) the nonhomogeneous time-dependent problems using the 10 Abstract and Applied Analysis

Table 8: The 𝐿∞ errors compared with quartic B-spline collection method at different times with 𝑑𝑥 = 0.005, 𝑑𝑡 = 0.001 in [0, 1] and IMQ RBF (𝑐 = 0.05).

Methods 𝑡 = 0.2 𝑡 = 0.4 𝑡 = 0.6 𝑡=0.8 −5 −4 −4 −4 Present Method 8.853 × 10 1.896 × 10 2.594 × 10 3.028 × 10 −5 −5 −4 −4 Quartic B-spline [8] 2.428 × 10 7.932 × 10 1.209 × 10 1.488 × 10

Table 9: The 𝐿∞ errors compared with cubic B-spline quasi-interpolation method at different times with 𝑑𝑥 = 0.002, 𝑑𝑡 = 0.0005 in [0, 1] and IMQ RBF (𝑐 = 0.02).

Methods 𝑡 = 0.4 𝑡 = 0.6 𝑡 = 0.8𝑡=1 −5 −4 −4 −4 Present Method 9.505 × 10 1.299 × 10 1.515 × 10 1.616 × 10 −5 −4 −5 −5 Cubic B-spline [9] 5.576 × 10 6.933 × 10 7.686 × 10 7.891 × 10 meshfree method with inverse multiquadric and generalized [5]R.F.Ren,H.B.Li,W.Jiang,andM.Y.Song,“Anefficient thin plate splines radial basis functions. The main advantage Chebyshev-tau method for solving the space fractional diffu- of this method over finite difference techniques is that the sion equations,” Applied Mathematics and Computation,vol. latter methods provide the solution of the problem on mesh 224, pp. 259–267, 2013. points only. Numerical examples are given to demonstrate [6]G.R.LiuandY.T.Gu,“Boundarymeshfreemethodsbased the validity and applicability of our method. The results show on the boundary point interpolation methods,” Engineering that our method is meshless and accurate. Indeed by selecting Analysis with Boundary Elements,vol.28,no.5,pp.475–487, an optimal parameter 𝑐 in the inverse multiquadric function, 2004. excellent numerical results are obtained. [7] M. Dehghan and A. Shokri, “A numerical method for solving the hyperbolic telegraph equation,” Numerical Methods for Partial Differential Equations,vol.24,no.4,pp.1080–1093,2008. Conflict of Interests [8] M. Dosti and A. Nazemi, “Quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation,” The authors declare that there is no conflict of interests Journal of Information and Computing Science,vol.7,no.2,pp. regarding the publication of this paper. 83–90, 2012. [9] M. Dosti and A. Nazemi, “Solving one-dimensional hyperbolic Acknowledgments telegraph equation using cubic B-spline quasi-interpolation,” World Academy of Science, Engineering and Technology,vol.5, The respected anonymous referees have carefully reviewed pp. 930–935, 2011. this paper. As a result of their careful analysis, our paper has [10] E. J. Kansa, “Multiquadrics—a scattered data approximation been improved. The authors would like to express their thank- scheme with applications to computational fluid-dynamics. fulness to them for their helpful constructive comments. I. Surface approximations and partial derivative estimates,” This work was supported by the National Natural Science Computers & Mathematics with Applications,vol.19,no.8-9,pp. Foundation of China (Grant nos. 11301252, 11301529, and 127–145, 1990. 11201212) and Applied Mathematics Enhancement Program [11] M. Zerroukat, H. Power, and C. S. Chen, “A numerical method of Linyi University. for heat transfer problems using collocation and radial basis functions,” International Journal for Numerical Methods in Engineering, vol. 42, no. 7, pp. 1263–1278, 1998. References [12] M. Li, T. Jiang, and Y. C. Hon, “A meshless method based on RBFs method for nonhomogeneous backward heat conduction [1] M. Dehghan, “Parameter determination in a partial differential problem,” Engineering Analysis with Boundary Elements,vol.34, equation from the overspecified data,” Mathematical and Com- no.9,pp.785–792,2010. puter Modelling,vol.41,no.2-3,pp.196–213,2005. [13]T.Jiang,M.Li,andC.S.Chen,“Themethodofparticu- [2] M. Dehghan, “Implicit collocation technique for heat equation lar solutions for solving inverse problems of a nonhomoge- with non-classic initial condition,” International Journal of neous convection-diffusion equation with variable coefficients,” Nonlinear Sciences and Numerical Simulation,vol.7,no.4,pp. Numerical Heat Transfer A: Applications,vol.61,no.5,pp.338– 447–450, 2006. 352, 2012. [3] M. Dehghan, “Finite difference procedures for solving a prob- [14] T. S. Jiang, Z. L. Jiang, and K. Jospeh, “A numerical method for lem arising in modeling and design of certain optoelectronic one-dimensional time-dependent Schrodinger equation using devices,” Mathematics and Computers in Simulation,vol.71,no. radial basis functions,” International Journal of Computational 1, pp. 16–30, 2006. Methods, vol. 10, no. 2, Article ID 1341010, 12 pages, 2013. [4]A.H.BhrawyandD.Baleanu,“AspectralLegendre—Gauss— [15] G. R. Liu, Mesh Free Method: Moving Beyond the Finite Element Lobatto collocation method for a space-fractional advection Methods, CRC Press, Boca Raton, Fla, USA, 2002. diffusion equations with variable coefficients,” Reports on Math- [16] G. R. Liu and M. B. Liu, Smooth Pariticle Hydrodynamics—A ematical Physics,vol.72,no.2,pp.219–233,2013. Meshfree Particle Method, World Scientific, Singapore, 2003. Abstract and Applied Analysis 11

[17]S.N.AtluriandS.B.Shen,The Meshless Local Petrov- Galerkin(MLPG) Method, Tech Science Press, Encino, Calif, USA, 2002. [18] G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer, New York, NY, USA, 2005. [19] M. Griebel and M. A. Schweitzer, Meshfree Methods for Partial Differential Equations III, Springer, New York, NY, USA, 2006. [20] G. E. Fasshauer, Meshfree Approximation Methods with MAT- LAB, Illinois Institute of Technology, Chicago, Ill, USA, 2008. [21]A.J.Terrira,E.J.Kansa,G.E.Fasshauer,andV.M.Leitao,˚ Progress on Meshless Methods,NewYork,NY,USA,Springer, 2009. [22] R. Schaback, “Error estimates and condition numbers for radial basis function interpolation,” Advances in Computational Mathematics,vol.3,no.3,pp.251–264,1995. [23] Z. M. Wu and R. Schaback, “Local error estimates for radial basis function interpolation of scattered data,” IMA Journal of Numerical Analysis,vol.13,no.1,pp.13–27,1993. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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