×

Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. (English) Zbl 1107.65124

Summary: Based on the symbolic computation system Maple, the Adomian decomposition method, developed for differential equations of integer order, is directly extended to derive explicit and numerical solutions of the fractional Korteweg-de Vries (KdV)-Burgers equation. The fractional derivatives are described in the Caputo sense. According to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional KdV-Burgers equation with time- and space-fractional derivatives. Finally, the solutions of our model equation are calculated in the form of convergent series with easily computable components.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
26A33 Fractional derivatives and integrals
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
68W30 Symbolic computation and algebraic computation

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, M. J.; Clarkson, P. A., Soliton, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press New York · Zbl 0762.35001
[2] Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical Korteweg-de Vries equation, J. Comput. Phys., 55, 2, 231 (1984) · Zbl 0541.65083
[3] Matveev, V. B.; Salle, M. A., Darboux Transformation and Solitons (1991), Springer: Springer Berlin · Zbl 0744.35045
[4] Clarkson, P. A.; Kruskal, M. D., New similarity reductions of Boussinesq equation, J. Math. Phys., 30, 2202 (1989)
[5] Conte, R.; Musette, M., Painleve analysis and Bäcklund transformation in the Kuramoto-Sivashinsky equation, J. Phys. A: Math. Gen., 22, 169 (1989) · Zbl 0687.35087
[6] Lou, S. Y.; Lu, J. Z., Special solutions from variable separation approach: Davey-Stewartson equation, J. Phys. A: Math. Gen., 29, 4029 (1996)
[7] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277, 212 (2000) · Zbl 1167.35331
[8] Yan, Z. Y., New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A, 292, 100 (2001) · Zbl 1092.35524
[9] Chen, Y.; Zheng, X. D.; Li, B.; Zhang, H. Q., New exact solutions for some nonlinear differential equations using symbolic computation, Appl. Math. Comput., 149, 277 (2004)
[10] Wang, Q.; Chen, Y.; Zhang, H. Q., A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation, Chaos, Solitons & Fractals, 23, 477 (2005) · Zbl 1072.35510
[11] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501 (1988) · Zbl 0671.34053
[12] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[13] Wazwaz, A. M., A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102, 77 (1999) · Zbl 0928.65083
[14] Wazwaz, A. M., Partial Differential Equations: Methods and Applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 0997.35083
[15] Yan, Z. Y., Numerical doubly periodic solution of the KdV equation with the initial condition via the decomposition method, Appl. Math. Comput., 168, 1065 (2005) · Zbl 1082.65587
[16] West, B. J.; Bolognab, M.; Grigolini, P., Physics of Fractal Operators (2003), Springer: Springer New York
[17] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[18] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[19] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[20] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent, Part II, J. Roy. Astr. Soc., 13, 529 (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.