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Application of meshfree collocation method to a class of nonlinear partial differential equations. (English) Zbl 1244.65149
Eng. Anal. Bound. Elem. 33, No. 5, 661-667 (2009); corrigendum ibid. 33, No. 5, 661-667 (2009).
Summary: An algorithm for the numerical solution of the generalized Hirota-Satsuma equations and Jaulent-Miodek equations based on meshless radial basis functions (RBFs) method using collocation points, called Kansa’s method, is presented. Four model problems with six different initial conditions are considered for the computation. A fairly explicit scheme is used to approximate the solution. The comparison is made with the exact solutions of each problem of the generalized Hirota-Satsuma coupled Korteweg-de Vries equations. A system consisting highly nonlinear partial differential equations known as Jaulent-Miodek equations and generalized Hirota-Satsuma coupled modified-Korteweg-de Vries equations are considered for comparison with the work already published. The multiquadric RBF results are compared with homotopy perturbation method (HPM) and variational iteration method (VIM) to highlight the excellent performance of the method.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Korteweg, D.J.; De-Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave, Philos mug, 39, 422-443, (1895) · JFM 26.0881.02
[2] Zakharov, V.E.; Manakov, S.V.; Novikov, S.P.; Itayevsky, S.P., Theory of solitons, (1980), Nauk Moscow
[3] Ablowitz, M.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0472.35002
[4] Zabusky, N.J.; Kruskal, M.D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys rev lett, 15, 240-243, (1965) · Zbl 1201.35174
[5] Hirota, R.; Satsuma, J., Soliton solutions of a coupled korteweg – de Vries equation, Phys lett A, 85, 407-408, (1981)
[6] Wu, Y.T.; Geng, X.G.; Hu, X.B.; Zhu, S.M., A generalized hirota – satsuma coupled korteweg – de Vries equation and miura transformations, Phys lett A, 255, 259-264, (1999) · Zbl 0935.37029
[7] Fan, E., Soliton solutions for a generalized hirota – satsuma coupled KdV equation and a coupled mkdv equation, Phys lett A, 282, 18-22, (2001) · Zbl 0984.37092
[8] Zhang, H., New exact solutions for two generalized hirota – satsuma coupled KdV systems, Comm nonlinear sc numer simul, 12, 1120-1127, (2007) · Zbl 1350.35177
[9] Kaya, D., Solitary wave solutions for a generalized hirota – satsuma coupled KdV equation, Appl math comput, 147, 69-78, (2004) · Zbl 1037.35069
[10] Xie, Fuding, New traveling wave solutions of the generalized coupled hirota – satsuma KdV system, Chaos, solitons fractals, 20, 1005-1012, (2004) · Zbl 1049.35164
[11] Ganji, D.D.; Rafei, M., Solitary wave solutions for a generalized hirota – satsuma coupled KdV equation by homotopy perturbation method, Phys lett A, 356, 131-137, (2006) · Zbl 1160.35517
[12] Baldwin, D., Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear pdes, J sym comput, 37, 669-705, (2004) · Zbl 1137.35324
[13] Halim, A.A.; Leble, S.B., Analytical and numerical solution of a coupled KdV-mkdv system, Chaos solitons fractals, 19, 99-108, (2004) · Zbl 1068.35128
[14] Kansa, E.J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics-I, Comput math appl, 19, 127-145, (1990) · Zbl 0692.76003
[15] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions applications to elliptic partial differential equations, Comput math appl, 39, 123-137, (2000) · Zbl 0955.65086
[16] 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
[17] Siraj-ul-Islam, A meshfree method for numerical solution of KdV equation, Eng anal bound elem, 32, 849-855, (2008) · Zbl 1244.76087
[18] Dağ, Idris; Dereli, Yilmaz, Numerical solutions of KdV equation using radial basis functions, Appl math model, 32, 535-546, (2008) · Zbl 1132.65096
[19] Siraj-ul-Islam, et al. A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. Eng Anal Bound Elem 2008, in press, doi:10.1016/j.enganabound.2008.06.005. · Zbl 1357.65120
[20] Wu, Zongmin, Dynamical knot and shape parameter setting for simulating shock wave by using multi-quadric quasi-interpolation, Eng anal bound elem, 29, 354-358, (2005) · Zbl 1182.76933
[21] Khater AH, et al. Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method. Math Comput Modelling 2008;9, in press, doi:10.1016/j.mcm.2008.02.001.
[22] Carlson, R.E.; Foley, T.A., The parameter R2 in multiquadric interpolation, Comput math appl, 21, 29-42, (1991) · Zbl 0725.65009
[23] Tarwater AE. A parameter study of Hardy’s multiquadric method for scattered data interpolation, Lawrence Livermore National Laboratory. Technical Report UCRL-54670, 1985.
[24] Madych, W.R.; Nelson, S.A., Multivariate interpolation and conditionally positive definite functions II, Math comp, 54, 211-230, (1990) · Zbl 0859.41004
[25] Madych, W.R.; Nelson, S.A., Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J approx theory, 70, 94-114, (1992) · Zbl 0764.41003
[26] Cheng, A.H.D., Exponential convergence and h-c multiquadric collocation method for partial differential equations, Numer meth partial diff eq, 19, 571-594, (2003) · Zbl 1031.65121
[27] Hardy, R.L., Multiquadric equations of topography and other irregular surfaces, J geo-phys res, 176, 1905-1915, (1971)
[28] Franke, R., Scattered data interpolation: test of some methods, Math comput, 38, 181-200, (1982) · Zbl 0476.65005
[29] Ganji, D.D., Application of He’s variational iteration method to nonlinear jaulent – miodek equations and comparing it with ADM, Comput appl math, 207, 35-45, (2007) · Zbl 1120.65107
[30] Fan, E., Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos solitons fractals, 16, 819-839, (2003) · Zbl 1030.35136
[31] Jalili M, et al. Application of He’s homotopy-perturbation method to strongly nonlinear coupled systems. J Phys: Conf Ser 2008;96:012078, doi:10.1088/1742-6596/96/1/012078.
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