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A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. (English) Zbl 1244.65193
Summary: This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the coupled Korteweg-de Vries (KdV) equations, coupled Burgers’ equations, and quasi-nonlinear hyperbolic equations. Contrary to the mesh oriented methods such as the finite-difference and finite element methods, the new technique does not require mesh to discretize the problem domain, and a set of scattered nodes provided by initial data is required for realization of solution of the problem. Accuracy of the method is assessed in terms of the error norms $$L_{2},L_{\infty }$$, number of nodes in the domain of influence, time step length, parameter free and parameter dependent RBFs. Numerical experiments are performed to demonstrate the accuracy and robustness of the method for the three classes of partial differential equations (PDEs).

##### MSC:
 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 76M25 Other numerical methods (fluid mechanics) (MSC2010)
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##### References:
 [1] Osborne, A., The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of Ocean surface water waves, Chaos solitons fractals, 5, 2623-2637, (1995) · Zbl 1080.86502 [2] Ostrovsky, L.; Stepanyants Yu, A., Do internal solutions exist in the Ocean, Rev geophys, 27, 293-310, (1989) [3] Das, G.C.; Sarma, J., Response to a new mathematical approach for finding the solitary waves in dusty plasma, Phys plasmas, 6, 4394-4397, (1999) [4] Hirota, R.; Satsuma, J., Soliton solutions of a coupled kortewege-de Vries equation, Phys lett A, 85, 407-408, (1981) [5] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for Burgers equation, Appl math comput, 95, 37-50, (1998) · Zbl 0943.65101 [6] Chen, R.; Wu, Z., Solving partial differential equation by using multiquadric quasi-interpolation, Appl math comput, 186, 1502-1510, (2007) · Zbl 1117.65134 [7] Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the ito-type coupled equations, Comput methods appl mech, 195, 3430-3447, (2006) · Zbl 1124.76035 [8] Kaya, D.; Inan, E.I., Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation, Appl math comput, 151, 775-787, (2004) · Zbl 1048.65096 [9] Laila Assas, M.B., Variational iteration method for solving coupled- KdV equations, Chaos solitons fractals, 38, 1225-1228, (2008) · Zbl 1152.35466 [10] Zayed, E.M.E.; Zedan, H.A.; Gepreel, K.A., On the solitary wave solutions for nonlinear Hirota Satsuma coupled-KdV of equations, Chaos solitons fractals, 22, 285-303, (2004) · Zbl 1069.35080 [11] Caom, D.B.; Yan, J.R.; Zang, Y., Exact solutions for a new coupled mkdv equations and a coupled KdV equations, J phys lett A, 297, 68-74, (2002) · Zbl 0994.35104 [12] Zhang, J.L.; Wang, M.L.; Feng, Z.D., The improved F-expansion method and its applications, Phys lett A, 350, 103-109, (2006) · Zbl 1195.65211 [13] Khater AH, Temsah RS, Hassan MM. A Chebyshev spectral collocation method for solving Burgers’ type equations. J. Appl. Math. Comput, doi:10.1016/j.cam.2007.11.007. [14] Ganji, D.D.; Rafei, M., Solitary wave equations for a generalized hirota – satsuma coupled-KdV equation by homotopy perturbation method, Phys lett A, 356, 131-137, (2006) · Zbl 1160.35517 [15] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, Geo phys res, 176, 1905-1915, (1971) [16] Kansa, E.J., Multiquadrics scattered data approximation scheme with applications to computational fluid-dynamics I, surface approximations and partial derivative estimates, Computers & math. with appl., 19, 127-145, (1990) · Zbl 0692.76003 [17] Micchelli, C.A., Interpolation of scatterted data: distance matrix and conditionally positive definite functions, Constr approx, 2, 11-22, (1986) · Zbl 0625.41005 [18] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions II, Math comput, 54, 211-230, (1990) · Zbl 0859.41004 [19] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv comput math, 8, 381-399, (1998) · Zbl 0909.65088 [20] Tarwater AE. A parameter study of Hardy’s multiquadric method for scattered data interpolation. Technical Report UCRL-54670, Lawrence Livermore National Laboratory, 1985. [21] Golberg, M.A.; Chen, C.S; Karur, S.R., Improved multiquadric approximation for partial differential equations, Eng anal bound elem, 18, 9-17, (1996) [22] Khattak, A.J.; Siraj-ul-Islam, A comparative study of numerical solutions of a class of KdV equation, Appl. math. comput., 199, 425-434, (2008) · Zbl 1143.65078 [23] Hirota, R.; Satsuma, J., Phys soc jpn, 51, 3390-3397, (1982)
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