Meshless Method of Lines for Numerical Solution of Kawahara Type Equations

ABSTRACT

In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method.

In this work, an algorithm based on method of lines coupled with radial basis functions namely meshless method of lines (MMOL) is presented for the numerical solution of Kawahara, modified Kawahara and KdV Kawahara equations. The motion of a single solitary wave, interaction of two and three solitons and the phenomena of wave generation is discussed. The results are compared with the exact solution and with the results in the relevant literature to show the efficiency of the method.

Cite this paper

nullN. Bibi, S. Tirmizi and S. Haq, "Meshless Method of Lines for Numerical Solution of Kawahara Type Equations,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 608-618. doi: 10.4236/am.2011.25081.

nullN. Bibi, S. Tirmizi and S. Haq, "Meshless Method of Lines for Numerical Solution of Kawahara Type Equations,"

References

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[2] T. Bridges and G. Derks, “Linear Instability of Solitary Wave Solutions of the Kawahara Equation and its Generalizations,” Society for Industrial and Applied Mathematics: Journal on Mathematical Analysis, Vol. 33, No. 6, 2002, pp. 1356-1378. doi:10.1137/S0036141099361494

[3] J. K. Hunter and J. Scheurle, “Existence of Perturbed Solitary Wave Solutions to a Model Equation for Water Waves,” Physica D, Vol. 32, No. 2, 1988, pp. 253-268. doi:10.1016/0167-2789(88)90054-1

[4] D. J. Benny, “Long Waves on Liquid Film,” Journal of Mathematical Physics, Vol. 45, 1966, pp. 150-155.

[5] S .P. Lin, “Finite Amplitude Side-Band Stability of a Viscous Film,” Journal of Fluid Mechanics, Vol. 63, No. 3, 1974, pp. 417-429. doi:10.1017/S0022112074001704

[6] E. Yusufoglu and A. Bekir, “Symbolic Computation and New Families of Exact Travelling Solutions for the Kawahara and Modified Kawahara Equations,” Computers and Mathematics with Applications, Vol. 55, No. 6, 2008, pp. 1113-1121. doi:10.1016/j.camwa.2007.06.018

[7] D. Kaya, “An Explicit and Numerical Solutions of Some Fifth-Order KdV Equation by Decomposition Method,” Applied Mathematics and Computation, Vol. 144, No. 2-3, 2003, pp. 353-363. doi:10.1016/S0096-3003(02)00412-5

[8] E. Yusufoglu, A. Bekir and M. Alp, “Periodic and Solitary Wave Solutions of Kawahara and Modified Kawahara Equations by Sine-Cosine Method,” Chaos, Solitons and Fractals, Vol. 37, No. 4, 2008, pp. 1193-1197.

[9] L. Jin, “Application of Variational Iteration Method and Homotopy Perturbation Method to the Modified Kawahara Equation,” Mathematical and Computer Modeling, Vol. 49, No. 3-4, 2009, pp. 573-578. doi:10.1016/j.mcm.2008.06.017

[10] A. Korkmaz and I. Dag, “Crank-Nicolson-Differential Quadrature Algorithms for the Kawahara Equation,” Chaos Solitons and Fractals, Vol. 42, No. 1, 2009, pp. 65-73. doi:10.1016/j.chaos.2008.10.033

[11] K. Djidjeli, W. G. Price, E. H. Twizell and Y. Wang, “Numerical Methods for the Solution of the Third and Fifth-Order Dispersive Korteweg de Vries Equations,” Journal of Computational and Applied Mathematics, Vol, 58, No. 3, 1995, pp. 307-336. doi:10.1016/0377-0427(94)00005-L

[12] J. M. Yuan, J. Shen, W. Jiahong, “A Dual Petrov-Galerkin Method for the Kawahara-Type Equations,” Journal of Scientific Computing, Vol. 34, No. 1, 2008, pp. 48-63. doi:10.1007/s10915-007-9158-4

[13] S. Haq and M. Uddin, “RBFs Approximation Method for Kawahara Equation,” Engineering Analysis with Boundary Elements, Vol. 35, No. 3, 2011, pp.575-580. doi:10.1016/j.enganabound.2010.07.009

[14] 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

[15] S. Haq, N. Bibi, S. I. A. Tirmizi and M. Usman, “Meshless Method of Lines for the Numerical Solution of Generalized Kuramoto-Sivashinsky Equation,” Applied Mathematics and Computation, Vol. 217, No. 6, 2010, pp. 2404-2413.

[16] W. E. Schiesser, “The Numerical Method of Lines: Integration of Partial Differential Equations,” 1st Edition, Academic Press; San Diego California, 1991.

[17] E. J. Kansa, “Multiquadrics a Scattered Data Approximation Scheme with Application to Computational Fluid Dynamics-I, ” Computers and Mathematics with Applications, Vol. 19, No. 8-9, 1990, pp. 127-145. doi:10.1016/0898-1221(90)90270-T

[18] E. J. Kansa, “Multiquadrics a Scattered Data Approximation Scheme with Application to Computational Fluid Dynamics-II,” Computers and MaFFthematics with Applications, Vol. 19, No. 8-9, 1990, pp.147-161. doi:10.1016/0898-1221(90)90271-K

[19] B. Fornberg, T. A. Driscoll, G. Wright and R. Charles, “Observations on the Behavior of Radial Basis Function Approximations near Boundaries,” Computers and Mathematics with Applications, Vol. 43, No. 3-5, 2002, pp. 473-490.

[20] Y. C. Hon and Z. Wu, “A Quasi-Interpolation Method for Solving Stiff Ordinary Differential Equations,” International Journal of Numerical Methods for Engineering, Vol. 48, No. 8, 2000, pp. 1187-97. doi:10.1002/(SICI)1097-0207(20000720)48:8<1187::AID-NME942>3.0.CO;2-K

[21] C. Franke and R. Schaback, “Solving Partial Differential Equations by Collocation Using Radial Basis Functions,” Applied Mathematics and Computation, Vol. 93, No. 1, 1998, pp. 73-82.doi:10.1016/S0096-3003(97)10104-7

[22] A. S. M. Wong, Y. C. Hon, T. S. Li, S. L. Chug and E. J. Kansa, “Multizone Decomposition of Time Dependent Problems Using the Multiquadric Scheme,” Computers and Mathematics with Applications, Vol. 37, No. 8, 1999, pp. 23-43. doi:10.1016/S0898-1221(99)00098-X

[23] W. Chen and M. Tanaka, “A Meshless Exponential Convergence, Integration-Free and Boundary-Only RBF Technique,” Computers and Mathematics with Applications, Vol. 43, No. 3-5, 2002, pp. 379-391. doi:10.1016/S0898-1221(01)00293-0

[24] G. E. Fasshauer, “Solving Partial Differential Equations by Collocation with Radial Basis Functions,” Nashville Vanderbilt University Press, Proceeding of Chamonix, Nashville, 1996.

[25] S. Ul. Islam, A. J. Khattak and I. A. Tirmizi, “A Meshfree Method for Numerical Solution of KdV Equation,” Engineering Analysis with Boundary Elements, Vol. 32, No. 10, 2008, pp. 849-855.

[26] A. J. Khattak, S. I. A. Tirmizi and S. Ul. Islam, “Application of Meshfree Collocation Method to a Class of Nonlinear Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 33, No. 5, 2009, pp. 661-667. doi:10.1016/j.enganabound.2008.10.001

[27] 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

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

[29] Z. M. Wu, “Solving PDE with Radial Basis Function and the Error Estimation,” In: Z. Chen, Y. Li, C. A. Micchelli, Y. Xu and M. Dekker, Ed., Advances in Computational Mathematics, Lecture Notes on Pure and Applied Mathematics, Volume 202, GuangZhou, 1998.

[30] A. E. Tarwater, “A Parameter Study of Hardy’s Multiquadric Method for Scattered Data Interpolation,” Technical Report, Lawrence Livermore National Laboratory, Livermore, 1985.

[31] M. A. Golberg, C. S. Chen and S. R. Karur, “Improved Multiquadric Approximation 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

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

[33] A. S. M. Wong, Y. C. Hon, T. S. Li, S. L. Chug and E. J. Kansa, “Multizone Decomposition of Time Dependent Problems Using the Multiquadric Scheme,” Computers and Mathematics with Applications, Vol. 37, No. 8, 1999, pp. 23-43. doi:10.1016/S0898-1221(99)00098-X

[34] L. Ling and E. J. Kansa, “A Least-Square Preconditioner for Radial Basis Functions Collocation Methods,” Advances in Computational Mathematics, Vol 23, No. 1-2, 2005, pp. 31-54. doi:10.1007/s10444-004-1809-5

[35] H. Wendland, “Piecewise Polynomial, Positive Definite and Compactly Supported Radial Functions of Minimal Degree,” Advances in Computational Mathematics, Vol. 4, No. 1, 1995, pp. 389-96. doi:10.1007/BF02123482

[36] Y. L. Wu and G. R. Liu, “A Meshfree Formulation of Local Radial Point Interpolation Method (LRPIM) for Incompressible Flow Simulation,” Computational Mechanics, Vol. 30, No. 5-6, 2003, pp. 355-365.

[37] C. K. Lee, X. Liu and S. C. Fan, “Local Multiquadric Approximation for Solving Boundary Value Problems,” Computational Mechanics, Vol. 30, No. 5-6, 2003, pp. 396-409. doi:10.1007/s00466-003-0416-5

[38] R. Vertnik and B. Sarler, “Meshless Local Radial Basis Function Collocation Method for Convective-Diffusive Solid-Liquid Phase Change Problems,” International Journal of Numerical Methods for Heat & Fluid flow, Vol. 16, No. 5, 2006, pp. 617-640. doi:10.1108/09615530610669148

[39] M. Buhmann and N. Dyn, “Spectral Convergence of Multiquadric Interpolation,” Proceedings of the Edinburgh Mathematical Society, Vol. 36, No. 2, 1993, pp. 319-333. doi:10.1017/S0013091500018411

[40] M. D. Buhmann, “Radial Basis Functions,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511543241

[41] W. R. Madych and S. A. Nelson, “Error Bounds for Multiquadric Interpolation,” In: C. Chui, L. Schumaker and J. Ward Eds., Approximation Theory VI, Academic Press, New York, 1989, pp. 413-416.

[42] W. R. Madych and S. A. Nelson, “Multivariate Interpolation and Conditionally Positive Definite Functions, II,” Mathematics of Computation, Vol. 54, No. 189, 1990, pp. 211-230.doi:10.1090/S0025-5718-1990-0993931-7

[43] S. A. Sarra, “Adaptive Radial Basis Function Method for Time Dependent Partial Differential Equations,” Applied Numerical Mathematics, Vol. 54, No. 1, 2005, pp. 79-94. doi:10.1016/j.apnum.2004.07.004

[44] G. E. Fasshauer, “On the Numerical Solution of Differential Equations with Radial Basis Functions,” Boundary Element Technology XIII, Waterford Institute of Technology Press, Waterford, 1999, pp. 291-300.

[45] L. Collatz, “The Numerical Treatment of Differential Equations,” 3rd Edition, Springer, Verlag, 1966.

[46] J. D. Lambert, “Computational Methods in Ordinary Differential Equations”, John Wiley and Sons, J. W. Arrow- smith Ltd, Bristol, 1983.

[47] M. K. Jain, “Numerical Solution of Differential Equations,” New Age International (P) Ltd., Publishers, New Delhi, 1984.

[48] L. N Trefeten, “Spectral Methods in MATLAB,” Society for Industrial and Applied Mathematics, 2000. doi:10.1137/1.9780898719598

[49] R. P. Malik, “On Fifth Order KdV-Type Equation,” Bogoliubov laboratory of theoretical physics, Joint Institute for Nuclear Research, Dubna, 1997.

[50] A. M. Wazwaz, “New Solitary Wave Solutions to the Modified Kawahara Equation,” Physics Letters A, Vol. 360, No. 4-5, 2007, pp. 588-592. doi:10.1016/j.physleta.2006.08.068

[51] A. M. Wazwaz, “Abundant Solitons Solutions for Several Forms of the Fifth-Order KdV Equation by Using the Tanh Method,” Applied Mathematics and Computation, Vol. 182, No. 1, 2006, pp. 283-300. doi:10.1016/j.amc.2006.02.047

[52] J. C. Ceballos, M. Sepulveda and O. P. V. Villagran, “The Korteweg-de Vries-Kawahara Equation in a Bounded Domain and Some Numerical Results,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 912-936. doi:10.1016/j.amc.2007.01.107

[1] T. Kawahara, “Oscillatory Solitary Waves in Dispersive Media,” Journal of the Physical Society of Japan, Vol. 33, No. 1, 1972, pp. 260-264. doi:10.1143/JPSJ.33.260

[2] T. Bridges and G. Derks, “Linear Instability of Solitary Wave Solutions of the Kawahara Equation and its Generalizations,” Society for Industrial and Applied Mathematics: Journal on Mathematical Analysis, Vol. 33, No. 6, 2002, pp. 1356-1378. doi:10.1137/S0036141099361494

[3] J. K. Hunter and J. Scheurle, “Existence of Perturbed Solitary Wave Solutions to a Model Equation for Water Waves,” Physica D, Vol. 32, No. 2, 1988, pp. 253-268. doi:10.1016/0167-2789(88)90054-1

[4] D. J. Benny, “Long Waves on Liquid Film,” Journal of Mathematical Physics, Vol. 45, 1966, pp. 150-155.

[5] S .P. Lin, “Finite Amplitude Side-Band Stability of a Viscous Film,” Journal of Fluid Mechanics, Vol. 63, No. 3, 1974, pp. 417-429. doi:10.1017/S0022112074001704

[6] E. Yusufoglu and A. Bekir, “Symbolic Computation and New Families of Exact Travelling Solutions for the Kawahara and Modified Kawahara Equations,” Computers and Mathematics with Applications, Vol. 55, No. 6, 2008, pp. 1113-1121. doi:10.1016/j.camwa.2007.06.018

[7] D. Kaya, “An Explicit and Numerical Solutions of Some Fifth-Order KdV Equation by Decomposition Method,” Applied Mathematics and Computation, Vol. 144, No. 2-3, 2003, pp. 353-363. doi:10.1016/S0096-3003(02)00412-5

[8] E. Yusufoglu, A. Bekir and M. Alp, “Periodic and Solitary Wave Solutions of Kawahara and Modified Kawahara Equations by Sine-Cosine Method,” Chaos, Solitons and Fractals, Vol. 37, No. 4, 2008, pp. 1193-1197.

[9] L. Jin, “Application of Variational Iteration Method and Homotopy Perturbation Method to the Modified Kawahara Equation,” Mathematical and Computer Modeling, Vol. 49, No. 3-4, 2009, pp. 573-578. doi:10.1016/j.mcm.2008.06.017

[10] A. Korkmaz and I. Dag, “Crank-Nicolson-Differential Quadrature Algorithms for the Kawahara Equation,” Chaos Solitons and Fractals, Vol. 42, No. 1, 2009, pp. 65-73. doi:10.1016/j.chaos.2008.10.033

[11] K. Djidjeli, W. G. Price, E. H. Twizell and Y. Wang, “Numerical Methods for the Solution of the Third and Fifth-Order Dispersive Korteweg de Vries Equations,” Journal of Computational and Applied Mathematics, Vol, 58, No. 3, 1995, pp. 307-336. doi:10.1016/0377-0427(94)00005-L

[12] J. M. Yuan, J. Shen, W. Jiahong, “A Dual Petrov-Galerkin Method for the Kawahara-Type Equations,” Journal of Scientific Computing, Vol. 34, No. 1, 2008, pp. 48-63. doi:10.1007/s10915-007-9158-4

[13] S. Haq and M. Uddin, “RBFs Approximation Method for Kawahara Equation,” Engineering Analysis with Boundary Elements, Vol. 35, No. 3, 2011, pp.575-580. doi:10.1016/j.enganabound.2010.07.009

[14] 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

[15] S. Haq, N. Bibi, S. I. A. Tirmizi and M. Usman, “Meshless Method of Lines for the Numerical Solution of Generalized Kuramoto-Sivashinsky Equation,” Applied Mathematics and Computation, Vol. 217, No. 6, 2010, pp. 2404-2413.

[16] W. E. Schiesser, “The Numerical Method of Lines: Integration of Partial Differential Equations,” 1st Edition, Academic Press; San Diego California, 1991.

[17] E. J. Kansa, “Multiquadrics a Scattered Data Approximation Scheme with Application to Computational Fluid Dynamics-I, ” Computers and Mathematics with Applications, Vol. 19, No. 8-9, 1990, pp. 127-145. doi:10.1016/0898-1221(90)90270-T

[18] E. J. Kansa, “Multiquadrics a Scattered Data Approximation Scheme with Application to Computational Fluid Dynamics-II,” Computers and MaFFthematics with Applications, Vol. 19, No. 8-9, 1990, pp.147-161. doi:10.1016/0898-1221(90)90271-K

[19] B. Fornberg, T. A. Driscoll, G. Wright and R. Charles, “Observations on the Behavior of Radial Basis Function Approximations near Boundaries,” Computers and Mathematics with Applications, Vol. 43, No. 3-5, 2002, pp. 473-490.

[20] Y. C. Hon and Z. Wu, “A Quasi-Interpolation Method for Solving Stiff Ordinary Differential Equations,” International Journal of Numerical Methods for Engineering, Vol. 48, No. 8, 2000, pp. 1187-97. doi:10.1002/(SICI)1097-0207(20000720)48:8<1187::AID-NME942>3.0.CO;2-K

[21] C. Franke and R. Schaback, “Solving Partial Differential Equations by Collocation Using Radial Basis Functions,” Applied Mathematics and Computation, Vol. 93, No. 1, 1998, pp. 73-82.doi:10.1016/S0096-3003(97)10104-7

[22] A. S. M. Wong, Y. C. Hon, T. S. Li, S. L. Chug and E. J. Kansa, “Multizone Decomposition of Time Dependent Problems Using the Multiquadric Scheme,” Computers and Mathematics with Applications, Vol. 37, No. 8, 1999, pp. 23-43. doi:10.1016/S0898-1221(99)00098-X

[23] W. Chen and M. Tanaka, “A Meshless Exponential Convergence, Integration-Free and Boundary-Only RBF Technique,” Computers and Mathematics with Applications, Vol. 43, No. 3-5, 2002, pp. 379-391. doi:10.1016/S0898-1221(01)00293-0

[24] G. E. Fasshauer, “Solving Partial Differential Equations by Collocation with Radial Basis Functions,” Nashville Vanderbilt University Press, Proceeding of Chamonix, Nashville, 1996.

[25] S. Ul. Islam, A. J. Khattak and I. A. Tirmizi, “A Meshfree Method for Numerical Solution of KdV Equation,” Engineering Analysis with Boundary Elements, Vol. 32, No. 10, 2008, pp. 849-855.

[26] A. J. Khattak, S. I. A. Tirmizi and S. Ul. Islam, “Application of Meshfree Collocation Method to a Class of Nonlinear Partial Differential Equations,” Engineering Analysis with Boundary Elements, Vol. 33, No. 5, 2009, pp. 661-667. doi:10.1016/j.enganabound.2008.10.001

[27] 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

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

[29] Z. M. Wu, “Solving PDE with Radial Basis Function and the Error Estimation,” In: Z. Chen, Y. Li, C. A. Micchelli, Y. Xu and M. Dekker, Ed., Advances in Computational Mathematics, Lecture Notes on Pure and Applied Mathematics, Volume 202, GuangZhou, 1998.

[30] A. E. Tarwater, “A Parameter Study of Hardy’s Multiquadric Method for Scattered Data Interpolation,” Technical Report, Lawrence Livermore National Laboratory, Livermore, 1985.

[31] M. A. Golberg, C. S. Chen and S. R. Karur, “Improved Multiquadric Approximation 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

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

[33] A. S. M. Wong, Y. C. Hon, T. S. Li, S. L. Chug and E. J. Kansa, “Multizone Decomposition of Time Dependent Problems Using the Multiquadric Scheme,” Computers and Mathematics with Applications, Vol. 37, No. 8, 1999, pp. 23-43. doi:10.1016/S0898-1221(99)00098-X

[34] L. Ling and E. J. Kansa, “A Least-Square Preconditioner for Radial Basis Functions Collocation Methods,” Advances in Computational Mathematics, Vol 23, No. 1-2, 2005, pp. 31-54. doi:10.1007/s10444-004-1809-5

[35] H. Wendland, “Piecewise Polynomial, Positive Definite and Compactly Supported Radial Functions of Minimal Degree,” Advances in Computational Mathematics, Vol. 4, No. 1, 1995, pp. 389-96. doi:10.1007/BF02123482

[36] Y. L. Wu and G. R. Liu, “A Meshfree Formulation of Local Radial Point Interpolation Method (LRPIM) for Incompressible Flow Simulation,” Computational Mechanics, Vol. 30, No. 5-6, 2003, pp. 355-365.

[37] C. K. Lee, X. Liu and S. C. Fan, “Local Multiquadric Approximation for Solving Boundary Value Problems,” Computational Mechanics, Vol. 30, No. 5-6, 2003, pp. 396-409. doi:10.1007/s00466-003-0416-5

[38] R. Vertnik and B. Sarler, “Meshless Local Radial Basis Function Collocation Method for Convective-Diffusive Solid-Liquid Phase Change Problems,” International Journal of Numerical Methods for Heat & Fluid flow, Vol. 16, No. 5, 2006, pp. 617-640. doi:10.1108/09615530610669148

[39] M. Buhmann and N. Dyn, “Spectral Convergence of Multiquadric Interpolation,” Proceedings of the Edinburgh Mathematical Society, Vol. 36, No. 2, 1993, pp. 319-333. doi:10.1017/S0013091500018411

[40] M. D. Buhmann, “Radial Basis Functions,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511543241

[41] W. R. Madych and S. A. Nelson, “Error Bounds for Multiquadric Interpolation,” In: C. Chui, L. Schumaker and J. Ward Eds., Approximation Theory VI, Academic Press, New York, 1989, pp. 413-416.

[42] W. R. Madych and S. A. Nelson, “Multivariate Interpolation and Conditionally Positive Definite Functions, II,” Mathematics of Computation, Vol. 54, No. 189, 1990, pp. 211-230.doi:10.1090/S0025-5718-1990-0993931-7

[43] S. A. Sarra, “Adaptive Radial Basis Function Method for Time Dependent Partial Differential Equations,” Applied Numerical Mathematics, Vol. 54, No. 1, 2005, pp. 79-94. doi:10.1016/j.apnum.2004.07.004

[44] G. E. Fasshauer, “On the Numerical Solution of Differential Equations with Radial Basis Functions,” Boundary Element Technology XIII, Waterford Institute of Technology Press, Waterford, 1999, pp. 291-300.

[45] L. Collatz, “The Numerical Treatment of Differential Equations,” 3rd Edition, Springer, Verlag, 1966.

[46] J. D. Lambert, “Computational Methods in Ordinary Differential Equations”, John Wiley and Sons, J. W. Arrow- smith Ltd, Bristol, 1983.

[47] M. K. Jain, “Numerical Solution of Differential Equations,” New Age International (P) Ltd., Publishers, New Delhi, 1984.

[48] L. N Trefeten, “Spectral Methods in MATLAB,” Society for Industrial and Applied Mathematics, 2000. doi:10.1137/1.9780898719598

[49] R. P. Malik, “On Fifth Order KdV-Type Equation,” Bogoliubov laboratory of theoretical physics, Joint Institute for Nuclear Research, Dubna, 1997.

[50] A. M. Wazwaz, “New Solitary Wave Solutions to the Modified Kawahara Equation,” Physics Letters A, Vol. 360, No. 4-5, 2007, pp. 588-592. doi:10.1016/j.physleta.2006.08.068

[51] A. M. Wazwaz, “Abundant Solitons Solutions for Several Forms of the Fifth-Order KdV Equation by Using the Tanh Method,” Applied Mathematics and Computation, Vol. 182, No. 1, 2006, pp. 283-300. doi:10.1016/j.amc.2006.02.047

[52] J. C. Ceballos, M. Sepulveda and O. P. V. Villagran, “The Korteweg-de Vries-Kawahara Equation in a Bounded Domain and Some Numerical Results,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 912-936. doi:10.1016/j.amc.2007.01.107