Numerical Solutions of the KdV Equation Using B-Spline Functions

Mehrdad Lakestani1
1Department of Applied Mathematics, University of Tabriz, Tabriz, Iran

Tóm tắt

A numerical technique based on the finite difference and collocation methods is presented for the solution of Korteweg–de Vries (KdV) equation. The integration relations between any two families of B-spline functions are presented and are utilized to reduce the solution of KdV equation to the solution of linear algebraic equations. Numerical simulations for three test examples have been demonstrated to validate the technique proposed in the current paper. It is found that the simulating results are in good agreement with the exact solutions.

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Tài liệu tham khảo

Aksan EN, Özer A (2006) Numerical solution of Korteweg–de Vries equation by Galerkin B-spline finite element method. Appl Math Comput 175:1256–1265 Alexander ME, Morries J Li (1979) Galerkin methods for some model equations for nonlinear dispersive waves. J Comput Phys 30:428–451 Bhatta DD, Bhatti MI (2006) Numerical solution of KdV equation using modified Bernstein polynomials. Appl Math Comput 174(2):1255–1268 Bhrawy AH (2014) An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl Math Comput 247:30–46 Bhrawy AH, Al-shomrani M (2012) A Jacobi Dual-Petrov Galerkin-Jacobi collocation method for solving Korteweg–de Vries equations. Abstr Appl Anal, Article ID 418943 Bhrawy AH, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time–space fractional partial differential equations. J Comput Phys 281:876–895 Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral method in fluid dynamics. Springer-Verlag, New York Chui CK (1992) An Introduction to Wavelets. Academic Press, San Diego, Calif De Boor C (1978) A practical guide to spline. Springer-Verlag, New York Dehghan M (2005) On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer Meth Part D E 21:24–40 Dehghan M (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simulat 71:16–30 Dehghan M (2007) The one-dimensional heat equation subject to a boundary integral specification. Chaos Soliton Fract 32:661–675 Dehghan M, Shokri A (2007) A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dynam 50(2007):111–120 Dehghan M, Saray BN, Lakestani M (2012) Three methods based on the interpolation scaling functions and the mixed collocation finite difference schemes for the numerical solution of the nonlinear generalized Burgers–Huxley equation. Math Comput Model 55(3–4):1129–1142 Fornberg B, Whitham GB (1978) A numerical and theoretical study of certain nonlinear wave phenomena. Philos Trans Roy Soc 289:373–404 Gardner LRT, Ali AHA (1988) A numerical solution for the Korteweg–de Vries equation using Galerkins method with hermite polynomial shape functions. Proc Int Conf Model Simul 1:81–93 Gardner LRT, Gardner GA, Ali AHA (1991) Simulation of solitons using quadratic spline finite elements. Comput Meth Appl Mechan Eng 92(1991):231–243 Goda K (1975) On stability of some finite difference schemes for the Korteweg–de Vries equation. J Phys Soc Jpn 39:229–236 Goswami JC, Chan AK (1999) Fundamentals of wavelets: theory, algorithms, and applications. Wiley, Hoboken Inc M (2007) Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method. Chaos Soliton Fract 34(4):1075–1081 Kutluay S, Bahadír AR, Özedes A (2000) A small time solutions for the Korteweg–de Vries equation. Appl Math Comput 107:203–210 Lakestani M, Dehghan M (2009) Numerical solution of Fokker–Planck equation using the cubic B-spline scaling functions. Numer Meth Part D E 25(2):418–429 Lakestani M, Dehghan M (2010a) Collocation and finite difference-collocation methods for the solution of nonlinear Klein–Gordon equation. Comput Phys Commun 181(8):1392–1401 Lakestani M, Dehghan M (2010b) The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement. J Comput Appl Math 235:669–678 Lakestani M, Dehghan M (2012) Numerical solutions of the generalized Kuramoto–Sivashinsky equation using B-spline functions. Appl Math Model 36:605–617 Lakestani M, Dehghan M, Irandoust-pakchin S (2012) The construction of operational matrix of fractional derivatives using B-spline functions. Commun Nonlinear Sci Numer Simul 17(3):1149–1162 Lu D, Hong B, Tian L (2009) New Solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers’ equations. Commun Nonlinear Sci Numer Simulat 14(1):77–84 Özer A, Kutluay S (2005) An analytical-numerical method for solving the Korteweg–de Vries equation. Appl Math Comput 164:789–797 Rathish Kumar BV, Mehra M (2005) A wavelet Taylor Galerkin method for parabolic and hyperbolic partial differential equations. Int J Comput Method 2(1):75–97 Rubin SG, Graves RA (1975) A cubic spline approximation for problems in fluid mechanics. NASA TR R-436, Washington, DC Siraj-ul-Islam, Khattak AJ, Tirmizi IA (2008) A meshfree method for numerical solution of KdV equation. Eng Anal Bound Elem 32:849–855 Soliman AA (2004) Collocation solution of the Korteweg–de Vries equation using septic splines. Int J Comp Math 81:325–331 Soliman AA (2006) Numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos Soliton Fract 29(2):294–302 Soliman AA, Abdou MA (2007) Exact travelling wave solutions of nonlinear partial differential equations. Chaos Soliton Fract 32(2):808–815 Vliengenthart AC (1971) On finite difference methods for Korteweg–de Vries equation. J Eng Math 5:137–155 Wazwaz AM (2006) Two reliable methods for solving variants of the KdV equation with compact and noncompact structures. Chaos Soliton Fract 28:457–462 Wazwaz AM (2008) New sets of solitary wave solutions to the KdV, mKdV, and the generalized KdV equations. Commun Nonlinear Sci Numer Simulat 13(2):331–339 Zabusky NJ (1976) A synergetic approach to problem of nonlinear dispersive wave propagation and interaction. In: Ames W (ed) Proceeding of Symposium. Non-linear Partial Differential Equations. Academic Press, New York, pp 223–258