## THERMAL SCIENCE

International Scientific Journal

### NUMERICAL SOLUTIONS OF THE FRACTIONAL KDV-BURGERS-KURAMOTO EQUATION

**ABSTRACT**

Non-linear terms of the time-fractional KdV-Burgers-Kuramoto equation are linearized using by some linearization techniques. Numerical solutions of this equation are obtained with the help of the finite difference methods. Numerical solutions and corresponding analytical solutions are compared. The L2 and L∞ error norms are computed. Stability of given method is investigated by using the Von Neumann stability analysis.

**KEYWORDS**

PAPER SUBMITTED: 2017-06-13

PAPER REVISED: 2017-11-19

PAPER ACCEPTED: 2017-11-21

PUBLISHED ONLINE: 2018-01-07

**THERMAL SCIENCE** YEAR

**2018**, VOLUME

**22**, ISSUE

**Supplement 1**, PAGES [S153 - S158]

- Cohen, B., Krommes, J., Tang, W. and Rosenbluth, M., Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nuclear Fusion, 16 (1976), pp. 971-992.
- Guo, B. , Xueke, P. and Huang, F., Fractional partial differential equations and their numerical solutions, 2015.
- Das, S., Functional fractional calculus. Springer-Verlag Berlin, Heidelberg, 2011.
- Huang, F. and Liu, S., Physical mechanism and model of turbulent cascades in a barotropic atmosphere, Advances in Atmospheric Sciences, 21 (2004), pp. 34-40.
- Safari, M., Ganji, D.D. and Moslemi, M., Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation, Computers&Mathematics with Applications, 58 (2009), pp. 2091-2097.
- Hashemi, M. S.and Dumitru, B., Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line, Journal of Computational Physics, 316 (2016), pp. 10-20.
- Hashemi, M. S., Baleanu and Parto-Haghighi, D., A lie group approach to solve the fractional poisson equation, Romanian Journal of Physics, 60 (2015), pp. 1289-1297.
- Hashemi, M. S., Baleanu, D., Parto-Haghighi, M. and Darvishi, E., Solving the time-fractional diffusion equation using a Lie group integrator, Thermal Science, 19 (2015), pp. 77-83.
- Oldham, K.B. and Spanier, J. , The Fractional Calculus. Academic, New York, 1974.
- Song, L. and Zhang, H., Application of homotopy analysis method to fractional KdV-Burgers- Kuramoto equation, Physics Letter A, 367 (2007), pp. 88-94.
- Wei, L. and He, Y., Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation, Cornell University Library, arXiv: 1201.1156v1 2012.
- Pashayi, S., Hashemi, M. S. and Shahmorad, S., Analytical lie group approach for solving fractional integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 51 (2017), pp. 66-77.
- Topper, J. and Kawahara, T., Approximate equations for long nonlinear waves on a viscous fluid, Journal of the Physical Society of Japan, 44 (1978), pp. 663-666.
- Ucar, Y., Yagmurlu, N. M. and Tasbozan, O., Numerical Solutions of the Modified Burgers' Equation by Finite Difference Methods, Journal of Applied Mathematics, Statistics and Informatics, 13 (1) (2017)., pp. 19-30.
- Yokus, A., Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, In press, corrected proof, 2017.
- Yokus, A. and Kaya, D., Numerical and exact solutions for time fractional burgers equation, Journal of Nonlinear Sciences and Applications, 10 (2017), pp. 3419-3428.
- Kawahara, T., Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation, Physical Review Letters, 51 (1983), pp. 381-383.