RBFs Meshless Method of Lines for the Numerical Solution of Time-Dependent Nonlinear Coupled Partial Differential Equations

ABSTRACT

In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.

In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.

Cite this paper

nullS. Haq, A. Hussain and M. Uddin, "RBFs Meshless Method of Lines for the Numerical Solution of Time-Dependent Nonlinear Coupled Partial Differential Equations,"*Applied Mathematics*, Vol. 2 No. 4, 2011, pp. 414-423. doi: 10.4236/am.2011.24051.

nullS. Haq, A. Hussain and M. Uddin, "RBFs Meshless Method of Lines for the Numerical Solution of Time-Dependent Nonlinear Coupled Partial Differential Equations,"

References

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[11] M. Uddin, S. Haq and S. Islam, “Numerical Solution of Complex Modified Kortewegde Vries Equation by Mesh-Free Collocation Method,” Computers & Mathematics with Applications, Vol. 58, No. 3, 2009, pp. 566-578. doi:10.1016/j.camwa.2009.03.104

[12] S. Islam, S. Haq and M. Uddin, “A Mesh Free Interpolation Method for the Numerical Solution of the Coupled Nonlinear Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 33, No. 3, 2009, pp. 399-409.

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[16] F. J. Hickernell and Y. C. Hon, “Radial Basis Function Approximation of the Surface Wind Field from Scattered Data,” International Journal of Applied Science and Computers, Vol. 4, No. 3, 1998, pp. 221-247.

[17] M. A. Golberg, C. S. Chen and S. Karur, “Improved Multiquadric Apporoxmation for Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 18, No. 1, 1996, pp. 9-17. doi:10.1016/S0955-7997(96)00033-1

[18] I. Babuska, U. Banerjee and J. E. Osborn, “Survey of Meshless and Generalized Finite Element Methods: A Unified Approach,” Acta Numerica, Vol. 12, 2003, pp. 1-125. doi:10.1017/S0962492902000090

[19] D. Brown, L. Ling, E. Kansa and J. Levesley, “On Approximate Cardinal Preconditioning Methods for Solving PDEs with Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 29, No. 4, 2005, pp. 343-353. doi:10.1016/j.enganabound.2004.05.006

[20] G. E. Fasshauer, A. Q. M. Khaliq and D. A. Voss, “Using Meshfree Approximation for Multi-Asset American Option Problems,” Journal of the Chinese Institute of Eng, Vol. 27, No. 4, 2004, pp. 563-571. doi:10.1080/02533839.2004.9670904

[21] B. Fornberg, E. Larsson and G. Wright, “A New Class of Oscillatory Radial Basis Functions,” Computers & Mathematics with Applications, Vol. 51, No. 8, 2006, pp. 1209-1222. doi:10.1016/j.camwa.2006.04.004

[22] E. J. Kansa and Y. C. Hon, “Circumventing the Illconditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Differential Equations,” Computers & Mathematics with Applications, Vol. 39, No. 7-8, 2000, pp. 123-137. doi:10.1016/S0898-1221(00)00071-7

[23] I. Dag and Y. Dereli, “Numerical Solutions of KdV Equation Using Radial Basis Functions,” Applied Mathematical Modelling, Vol. 32, No. 4, 2008, pp. 535-546. doi:10.1016/j.apm.2007.02.001

[24] M. Dehghan and M. Tatari, “Determination of a Control Parameter in a One-Dimensional Parabolic Equation Using the Method of Radial Basis Functions,” Mathematical and Computer Modelling, Vol. 44, No. 11-12, 2006, pp. 1160-1168. doi:10.1016/j.mcm.2006.04.003

[25] M. Uddin, S. Haq and S. Islam, “A Mesh-Free Numerical Method for Solution of the Family of Kuramoto-Siva- shinsky Equations,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 458-469. doi:10.1016/j.amc.2009.02.037

[26] W. E. Schiesser, “The Numerical Method of Lines: Integration of Partial Differential Equations,” Academic Press, San Diego, 1991.

[27] Q. Shen, “A Meshless Method of Lines for the Numerical Solution of KdV Equation Using Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 33, No. 10, 2009, pp. 1171-1180. doi:10.1016/j.enganabound.2009.04.008

[28] S. Gottlieb and C. W. Shu, “Total Variation Diminishing Runge-Kutta Schemes,” Mathematics of Computation, Vol. 67, No. 221, 1998, pp. 73-85. doi:10.1090/S0025-5718-98-00913-2

[29] R. Hirota and J. Satsuma, “A Coupled KdV Equation is One Case of the Four-Reduction of the KP Hierarchy,” Journal of the Physical Society of Japan, Vol. 51, No. 10, 1982, pp. 3390-3397. doi:10.1143/JPSJ.51.3390

[1] A. Osborne, “The Inverse Scattering Transform: Tools for the Nonlinear Fourier Analysis and Filtering of Ocean Surface Water Waves,” Chaos. Solitons & Fractals, Vol. 5, No. 12, 1995, pp. 2623-37. doi:10.1016/0960-0779(94)E0118-9

[2] L. Ostrovsky and Y. A. Stepanyants, “Do Internal Solutions Exist in the Ocean,” Reviews of Geophysics, Vol. 27, No. 3, 1989, pp. 293-310. doi:10.1029/RG027i003p00293

[3] G. C. Das and J. Sarma “Response to ‘Comment on’ a New Mathematical Approach for Finding the Solitary Waves in Dusty Plasma,” Physics of Plasmas, Vol. 6, No. 11, 1999, pp. 4394-4397. doi:10.1063/1.873705

[4] R. Hirota and J. Satsuma, “Soliton Solutions of a Coupled Kortewege-de Vries Equation,” Physics Letters A, Vol. 85, No. 8-9, 1981, pp. 407-408. doi:10.1016/0375-9601(81)90423-0

[5] Y. C. Hon and X. Z. Mao, “An Efficient Numerical Scheme for Burgers Equation,” Applied Mathematics and Computation, Vol. 95, No. 1, 1998, pp. 37-50. doi:10.1016/S0096-3003(97)10060-1

[6] R. Chen and Z. Wu, “Solving Partial Differential Equation by Using Multiquadric Quasiinterpolation,” Applied Mathematics and Computation, Vol. 186, No. 2, 2007, pp. 1502-1510. doi:10.1016/j.amc.2006.07.160

[7] D. Kaya and E. I. Inan, “Exact and Numerical Traveling Wave Solutions for Nonlinear Coupled Equations Using Symbolic Computation,” Applied Mathematics and Computation, Vol. 151, No. 3, 2004, pp. 775-787. doi:10.1016/S0096-3003(03)00535-6

[8] Y. Xu, C.-W. Shu, “Local Discontinuous Galerkin Methods for the Kuramoto-Sivashinsky Equations and the Ito-Type Coupled Equations,” Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 25-28, 2006, pp. 3430-3447. doi:10.1016/j.cma.2005.06.021

[9] L. M. B. Assas, “Variational Iteration Method for Solving Coupled-KdV Equations,” Chaos, Solitons & Fractals, Vol. 38, No. 4, 2008, pp. 1225-1228. doi:10.1016/j.chaos.2007.02.012

[10] A. H. Khater, R. S. Temsah and M. M. Hassan, “A Chebyshev Spectral Collocation Method for Solving Burgers’ Type Equations,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 333-350. doi:10.1016/j.cam.2007.11.007

[11] M. Uddin, S. Haq and S. Islam, “Numerical Solution of Complex Modified Kortewegde Vries Equation by Mesh-Free Collocation Method,” Computers & Mathematics with Applications, Vol. 58, No. 3, 2009, pp. 566-578. doi:10.1016/j.camwa.2009.03.104

[12] S. Islam, S. Haq and M. Uddin, “A Mesh Free Interpolation Method for the Numerical Solution of the Coupled Nonlinear Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 33, No. 3, 2009, pp. 399-409.

[13] R. L. Hardy, “Multiquadric Equations of Topography and Other Irregular Surfaces,” Journal of Geophysical Research, Vol. 76, No. 8, 1971, pp. 1905-1915. doi:10.1029/JB076i008p01905

[14] R. Franke, “Scattered Data Interpolation: Tests of Some Methods,” Mathematics of Computation, Vol. 38, No. 157, 1982, pp. 181-200.

[15] R. Franke, “A Critical Comparison of Some Methods for Interpolation of Scattered Data,” Technical Report NPS-53-79-003, Naval Postgraduate School, 1975.

[16] F. J. Hickernell and Y. C. Hon, “Radial Basis Function Approximation of the Surface Wind Field from Scattered Data,” International Journal of Applied Science and Computers, Vol. 4, No. 3, 1998, pp. 221-247.

[17] M. A. Golberg, C. S. Chen and S. Karur, “Improved Multiquadric Apporoxmation for Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 18, No. 1, 1996, pp. 9-17. doi:10.1016/S0955-7997(96)00033-1

[18] I. Babuska, U. Banerjee and J. E. Osborn, “Survey of Meshless and Generalized Finite Element Methods: A Unified Approach,” Acta Numerica, Vol. 12, 2003, pp. 1-125. doi:10.1017/S0962492902000090

[19] D. Brown, L. Ling, E. Kansa and J. Levesley, “On Approximate Cardinal Preconditioning Methods for Solving PDEs with Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 29, No. 4, 2005, pp. 343-353. doi:10.1016/j.enganabound.2004.05.006

[20] G. E. Fasshauer, A. Q. M. Khaliq and D. A. Voss, “Using Meshfree Approximation for Multi-Asset American Option Problems,” Journal of the Chinese Institute of Eng, Vol. 27, No. 4, 2004, pp. 563-571. doi:10.1080/02533839.2004.9670904

[21] B. Fornberg, E. Larsson and G. Wright, “A New Class of Oscillatory Radial Basis Functions,” Computers & Mathematics with Applications, Vol. 51, No. 8, 2006, pp. 1209-1222. doi:10.1016/j.camwa.2006.04.004

[22] E. J. Kansa and Y. C. Hon, “Circumventing the Illconditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Differential Equations,” Computers & Mathematics with Applications, Vol. 39, No. 7-8, 2000, pp. 123-137. doi:10.1016/S0898-1221(00)00071-7

[23] I. Dag and Y. Dereli, “Numerical Solutions of KdV Equation Using Radial Basis Functions,” Applied Mathematical Modelling, Vol. 32, No. 4, 2008, pp. 535-546. doi:10.1016/j.apm.2007.02.001

[24] M. Dehghan and M. Tatari, “Determination of a Control Parameter in a One-Dimensional Parabolic Equation Using the Method of Radial Basis Functions,” Mathematical and Computer Modelling, Vol. 44, No. 11-12, 2006, pp. 1160-1168. doi:10.1016/j.mcm.2006.04.003

[25] M. Uddin, S. Haq and S. Islam, “A Mesh-Free Numerical Method for Solution of the Family of Kuramoto-Siva- shinsky Equations,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 458-469. doi:10.1016/j.amc.2009.02.037

[26] W. E. Schiesser, “The Numerical Method of Lines: Integration of Partial Differential Equations,” Academic Press, San Diego, 1991.

[27] Q. Shen, “A Meshless Method of Lines for the Numerical Solution of KdV Equation Using Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 33, No. 10, 2009, pp. 1171-1180. doi:10.1016/j.enganabound.2009.04.008

[28] S. Gottlieb and C. W. Shu, “Total Variation Diminishing Runge-Kutta Schemes,” Mathematics of Computation, Vol. 67, No. 221, 1998, pp. 73-85. doi:10.1090/S0025-5718-98-00913-2

[29] R. Hirota and J. Satsuma, “A Coupled KdV Equation is One Case of the Four-Reduction of the KP Hierarchy,” Journal of the Physical Society of Japan, Vol. 51, No. 10, 1982, pp. 3390-3397. doi:10.1143/JPSJ.51.3390