Numerical and exact solutions for time fractional Burgers' equation

Yokuş, Asıf¹ ; Kaya, Doğan²

¹ Department of Actuary, Firat University, Elazig, Turkey
² Department of Mathematics, Istanbul Commerce University, Istanbul, Turkey

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3419-3428.

Résumé

The main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers’ equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers’ equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers’ equation is linearized by using the Cole- Hopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in $L_2$ and $L_\infty$. All of obtained results are discussed with an example of the Burgers’ equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study.

Détail

DOI : 10.22436/jnsa.010.07.06

Classification : 65M06, 35R11
Keywords: Nonlinear time fractional Burgers’ equation, an expansion method, finite difference method, Caputo formula, linear stability, Cole-Hopf transformation.

Bibliographie

[1] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M. Application of a fractional advection-dispersion equation, Water Resour. Res., Volume 36 (2000), pp. 1403-1412 | DOI

[2] Chen, W.; Ye, L.-J.; Sun, H.-G. Fractional diffusion equations by the Kansa method, Comput. Math. Appl., Volume 59 (2010), pp. 1614-1620 | DOI

[3] Clarkson, P. A. New similarity solutions for the modified Boussinesq equation, J. Phys. A, Volume 22 (1989), pp. 2355-2367 | DOI

[4] Elwakil, S. A.; El-labany, S. K.; Zahran, M. A.; Sabry, R. Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A, Volume 299 (2002), pp. 179-188 | DOI

[5] Fan, E.-G. Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, Volume 277 (2000), pp. 212-218 | Zbl | DOI

[6] Gorguis, A. A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations, Appl. Math. Comput., Volume 173 (2006), pp. 126-136 | Zbl | DOI

[7] Guo, S.-M.; Zhou, Y.-B. The extended (\(\frac{G'}{G}\) )-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., Volume 215 (2010), pp. 3214-3221 | DOI | Zbl

[8] He, J.-H.; Wu, X.-H. Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, Volume 30 (2006), pp. 700-708 | DOI

[9] Islam, S. U.; Khattakand, A. J.; Tirmizi, I. A. A meshfree method for numerical solution of KdV equation, Eng. Anal. Bound. Elem., Volume 32 (2008), pp. 849-855 | DOI

[10] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., Volume 191 (2007), pp. 12-20 | DOI

[11] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E. Fractional calculus and continuous-time finance, II, the waiting-time distribution, Phys. A, Volume 287 (2000), pp. 468-481 | DOI

[12] Meerschaert, M. M.; Tadjeran, C. Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., Volume 172 (2004), pp. 65-77 | DOI | Zbl

[13] Miller, K. S.; Ross, B. An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993

[14] Odibat, Z. M.; Shawagfeh, N. T. Generalized Taylor’s formula, Appl. Math. Comput., Volume 186 (2007), pp. 286-293 | DOI

[15] Oldham, K. B.; Spanier, J. The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 2006

[16] Parkes, E. J.; Duffy, B. R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun., Volume 98 (1996), pp. 288-300 | Zbl | DOI

[17] Podlubny, I. Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999

[18] Scalas, E.; Gorenflo, R.; Mainardi, F. Fractional calculus and continuous-time finance, Phys. A, Volume 284 (2000), pp. 376-384 | DOI

[19] Sousa, E. Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., Volume 228 (2009), pp. 4038-4054 | DOI

[20] Su, L.-J.; Wang, W.-Q.; Xu, Q.-Y. Finite difference methods for fractional dispersion equations, Appl. Math. Comput., Volume 216 (2010), pp. 3329-3334 | DOI

[21] Su, L.-J.; Wang, W.-Q.; Yang, Z.-X. Finite difference approximations for the fractional advection-diffusion equation, Phys. Lett. A, Volume 373 (2009), pp. 4405-4408 | DOI

[22] Wang, M.-L.; Li, X.-Z.; Zhang, J.-L. The (\(\frac{G'}{G}\) )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, Volume 372 (2008), pp. 417-423 | DOI

[23] Wazwaz, A. M. The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd- Bullough equations, Chaos Solitons Fractals, Volume 25 (2005), pp. 55-63 | DOI | Zbl

[24] Yokus, A. Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph.D. Thesis, Fırat University, Elazig, Turkey, 2011

[25] Yuste, S. B. Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. , Volume 216 (2006), pp. 264-274 | DOI

[26] Zaslavsky, G. M. Chaos, fractional kinetics, and anomalous transport, Phys. Rep., Volume 371 (2002), pp. 461-580 | Zbl | DOI

[27] Zheng, X.-D.; Chen, Y.; Zhang, H.-Q. Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation, Phys. Lett. A, Volume 311 (2003), pp. 145-157 | Zbl | DOI

[28] Zhuang, P.; Liu, F.; Anh, V.; Turner, I. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., Volume 46 (2008), pp. 1079-1095 | DOI | Zbl