Numerical and exact solutions for time fractional Burgers' equation
Authors
Asıf Yokuş
 Department of Actuary, Firat University, Elazig, Turkey.
Doğan Kaya
 Department of Mathematics, Istanbul Commerce University, Istanbul, Turkey.
Abstract
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 ColeHopf 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 FourierVon 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.
Keywords
 Nonlinear time fractional Burgers’ equation
 an expansion method
 finite difference method
 Caputo formula
 linear stability
 ColeHopf transformation.
MSC
References

[1]
D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, Application of a fractional advectiondispersion equation, Water Resour. Res., 36 (2000), 1403–1412.

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

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

[4]
S. A. Elwakil, S. K. Ellabany, M. A. Zahran, R. Sabry, Modified extended tanhfunction method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179–188.

[5]
E.G. Fan, Extended tanhfunction method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212–218.

[6]
A. Gorguis, A comparison between ColeHopf transformation and the decomposition method for solving Burgers’ equations, Appl. Math. Comput., 173 (2006), 126–136.

[7]
S.M. Guo, Y.B. Zhou, The extended (\(\frac{G'}{G}\) )expansion method and its applications to the WhithamBroerKauplike equations and coupled HirotaSatsuma KdV equations, Appl. Math. Comput., 215 (2010), 3214–3221.

[8]
J.H. He, X.H. Wu, Expfunction method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700–708.

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

[10]
F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the spacetime fractional advectiondiffusion equation, Appl. Math. Comput., 191 (2007), 12–20.

[11]
F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuoustime finance, II, the waitingtime distribution, Phys. A, 287 (2000), 468–481.

[12]
M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advectiondispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.

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

[14]
Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286–293.

[15]
K. B. Oldham, J. Spanier, 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 YorkLondon (2006)

[16]
E. J. Parkes, B. R. Duffy, An automated tanhfunction method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun., 98 (1996), 288–300.

[17]
I. Podlubny, 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]
E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuoustime finance, Phys. A, 284 (2000), 376–384.

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

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

[21]
L.J. Su, W.Q. Wang, Z.X. Yang, Finite difference approximations for the fractional advectiondiffusion equation, Phys. Lett. A, 373 (2009), 4405–4408.

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

[23]
A. M. Wazwaz, The tanh method: solitons and periodic solutions for the DoddBulloughMikhailov and the TzitzeicaDodd Bullough equations, Chaos Solitons Fractals, 25 (2005), 55–63.

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

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

[26]
G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461–580.

[27]
X.D. Zheng, Y. Chen, H.Q. Zhang, Generalized extended tanhfunction method and its application to (1+1)dimensional dispersive long wave equation, Phys. Lett. A, 311 (2003), 145–157.

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