A numerical treatment based on Haar wavelets for coupled KdV equation

Ömer Oruç; Fatih Bulut; Alaattin Esen

doi:10.11121/ijocta.01.2017.00396

Authors

Ömer Oruç
Fatih Bulut Inonu University
Alaattin Esen Inonu University

DOI:

https://doi.org/10.11121/ijocta.01.2017.00396

Keywords:

coupled kdv equation, Haar wavelet, numerical solution

Abstract

In this paper, numerical solutions of one dimensional coupled KdV equation has been investigated by Haar Wavelet method. Time derivatives given in this equation are discretized by finite differences and nonlinear terms appearing in the equations are linearized by some linearization techniques and space derivatives are discretized by Haar wavelets. For examining the performance of the proposed method, single soliton solution and conserved quantities of some test problems are used. Also, error analysis of numerical scheme is investigated and numerical results are compared with some results already existing in the literature.

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References

Hirota, R. and Satsuma, J., “Soliton solutions of a coupled Korteweg-de Vries equation,” Physics Letters A, 85 (8-9), 407–408 (1981).

Halim, A. A., Kshevetskii, S. P. and Leble, S. B., “Numerical integration of a coupled Korteweg-de Vries

system,” Computers & Mathematics with Applications, vol. 45 (4-5), 581–591 (2003).

Halim, A. A. and Leble, S. B., “Analytical and numerical solution of a coupled KdV-MKdV System,” Chaos, Solitons and Fractals, 19 (1), 99–108 (2004).

Zhu, S., A difference scheme for the coupled KdV equation. Communications in Nonlinear Science and Numerical Simulation, 4 (1), 60-63 (1999).

Kaya, D. and Inan, I. E., “Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation,” Applied Mathematics and Computation, 151(3), 775–787 (2004).

Fan, E. G., “Traveling wave solutions for nonlinear equations using symbolic computation,” Computers & Mathematics with Applications, 43 (6-7), 671–680 (2002).

Ma, Z. and Zhu, J., Jacobian elliptic function expansion solutions for the Wicktype stochastic coupled KdV equations. Chaos Solitons Fractals, 32, 1679- 1685 (2007)

Assas, L. M. B., “Variational iteration method for solving coupled-KdV equations,” Chaos, Solitons and Fractals, 38 (4), 1225–1228 (2008).

Abbasbandy, S., “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letters A: General, Atomic and Solid State Physics, 361 (6), 478–483 (2007).

Al-Khaled, K., Al-Refai, M. and Alawneh, A., Traveling wave solutions using the variational method and the tanh method for nonlinear coupled equations. Applied Mathematics and Computation, 202, 233-242 (2008).

Mokhtari, R. and Mohammadi, M., New exact solutions to a class of coupled nonlinear PDEs. The International Journal of Nonlinear Sciences and Numerical Simulation, 10 (6) , 779-796 (2009).

Ismail, M. S., “Numerical solution of coupled nonlinear Schrodinger equation by Galerkin method,” Mathematics and Computers in Simulation, 78 (4), 532–547 (2008).

Ismail, M. S. and Taha, T. R., “A linearly implicit conservative scheme for the coupled nonlinear Schrodinger equation,” Mathematics and Computers in Simulation, 74 (4-5), 302– 311 (2007).

Ismail, M. S. and Alamri, S. Z., “Highly accurate finite difference method for coupled nonlinear Schrodinger equation,” International Journal of Computer Mathematics,81 (3), 333– 351 (2004).

Wazwaz, A., “The KdV equation,” in Handbook of Differential Equations: Evolutionary Equations. VOL. IV, Handb. Differ. Equ., pp. 485–568, Elsevier/North- Holland, Amsterdam, The Netherlands, 2008.

Ismail, M. S., “Numerical solution of a coupled Korteweg-de Vries equations by collocation method,” Numerical Methods for Partial Differential Equations,25 (2), 275–291 (2009).

Kutluay, S. and Ucar,Y., “A quadratic BsplineGalerkin approach for solving a coupled KdV equation,” Mathematical Modelling and Analysis, 18

(1), 103–121 (2013).

Ismail, M. S. and Ashi, H. A., A Numerical Solution for Hirota-Satsuma Coupled KdV Equation, Abstract and Applied Analysis, Volume 2014, Article ID 819367, 9 pages http://dx.doi.org/10.1155/2014/819367

Chen, C. and Hsiao, C.H., Haar wavelet method for solving lumped and distributed parameter systems, IEE Proceedings - Control Theory and Applications, 144, 87–94 (1997).

Lepik, U., Numerical solution of differential equations using Haar wavelets, Mathematics and Computers in Simulation, 68, 127–143 (2005).

Lepik, U., Numerical solution of evolution equations by the Haar wavelet method, Applied Mathematics and Computation, 185,695–704 (2007).

Lepik, U., Solving PDEs with the aid of twodimensional Haar wavelets, Computers and Mathematics with Applications, 61, 1873–1879 (2011)

Celik, I., Haar wavelet method for solving generalized Burgers–Huxley equation, Arab Journal of Mathematical Sciences, 18 (1), 25–37(2012) .

Celik, I., Haar wavelet approximation for magnetohydrodynamic flow equations, Applied Mathematical Modelling, 37, 3894–3902(2013).

Jiwari, R., A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Computer Physics Communications, 183, 2413–2423 (2012).

Kaur, H., Mittal, R.C. and Mishra, V., Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics, Computer Physics Communications, 184, 2169–2177(2013).

Shi, Z., Cao, Y., and Chen, Q.J., Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method, Applied Mathematical Modelling, 36, 5143–5161 (2012).

Kumar, M. and Pandit, S., A composite numerical scheme for the numerical simulation of coupled Burgers’ equation, Computer Physics Communications, 185 (3), 809-817 (2014).

Mittal, R.C., Kaur, H. and Mishra, V., Haar wavelet based numerical investigation of coupled viscous Burgers’ equation, International Journal of Computer Mathematics, 92 (8), 1643-1659 (2014).

Oruç, Ö., Bulut, F. and Esen, A., Numerical Solutions of Regularized Long Wave Equation By Haar Wavelet Method, Mediterranean Journal of Mathematics, 13(5), 32353253 (2016)

Oruç, Ö., Bulut, F. and Esen, A., A Haar wavelet finite difference hybrid method for the numerical solution of the modified Burgers’ equation, Journal of Mathematical Chemistry, 53 (7) 1592-1607 (2015).

Oruç, Ö., Bulut, F. and Esen, A., Numerical Solution of the KdV Equation by Haar Wavelet Method, Pramana Journal of Physics, doi:10.1007/s12043-016-1286-7, 87: 94 (2016)

Bulut, F., Oruç, Ö. and Esen, A., Numerical Solutions of Fractional System of Partial Differential Equations by Haar Wavelets, Computer Modeling in Engineering & Sciences, 108 (4), 263-284 (2015).

Hein, H.and Feklistova, L., Free vibrations of nonuniform and axially functionally graded beams using Haar wavelets, Engineering Structures, 33 (12), 3696-3701 (2011).

Rubin, S. G. and Graves, R. A., Cubic spline approximation for problems in fluid mechanics, NASA TR R-436, Washington, DC, 1975.

Hunter, J. D., Matplotlib: A 2D graphics environment, Computing In Science & Engineering, 9(3), 90-95 (2007).

Ray, S.S., On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation, Applied Mathematics and Computation, 218, 5239–5248 (2012).