%0 Journal Article %T Numerical and exact solutions for time fractional Burgers' equation %A Yokuş, Asıf %A Kaya, Doğan %J Journal of Nonlinear Sciences and Applications %D 2017 %V 10 %N 7 %@ ISSN 2008-1901 %F Yokuş2017 %X 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. %9 journal article %R 10.22436/jnsa.010.07.06 %U http://dx.doi.org/10.22436/jnsa.010.07.06 %P 3419--3428 %0 Journal Article %T Application of a fractional advection-dispersion equation %A D. A. Benson %A S. W. Wheatcraft %A M. M. Meerschaert %J Water Resour. Res. %D 2000 %V 36 %F Benson2000 %0 Journal Article %T Fractional diffusion equations by the Kansa method %A W. Chen %A L.-J. Ye %A H.-G. Sun %J Comput. Math. Appl. %D 2010 %V 59 %F Chen2010 %0 Journal Article %T New similarity solutions for the modified Boussinesq equation %A P. A. Clarkson %J J. Phys. A %D 1989 %V 22 %F Clarkson1989 %0 Journal Article %T Modified extended tanh-function method for solving nonlinear partial differential equations %A S. A. Elwakil %A S. K. El-labany %A M. A. Zahran %A R. Sabry %J Phys. Lett. A %D 2002 %V 299 %F Elwakil2002 %0 Journal Article %T Extended tanh-function method and its applications to nonlinear equations %A E.-G. Fan %J Phys. Lett. A %D 2000 %V 277 %F Fan2000 %0 Journal Article %T A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations %A A. Gorguis %J Appl. Math. Comput. %D 2006 %V 173 %F Gorguis2006 %0 Journal Article %T The extended (\(\frac{G'}{G}\) )-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations %A S.-M. Guo %A Y.-B. Zhou %J Appl. Math. Comput. %D 2010 %V 215 %F Guo2010 %0 Journal Article %T Exp-function method for nonlinear wave equations %A J.-H. He %A X.-H. Wu %J Chaos Solitons Fractals %D 2006 %V 30 %F He2006 %0 Journal Article %T A meshfree method for numerical solution of KdV equation %A S. U. Islam %A A. J. Khattakand %A I. A. Tirmizi %J Eng. Anal. Bound. Elem. %D 2008 %V 32 %F Islam2008 %0 Journal Article %T Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation %A F. Liu %A P. Zhuang %A V. Anh %A I. Turner %A K. Burrage %J Appl. Math. Comput. %D 2007 %V 191 %F Liu2007 %0 Journal Article %T Fractional calculus and continuous-time finance, II, the waiting-time distribution %A F. Mainardi %A M. Raberto %A R. Gorenflo %A E. Scalas %J Phys. A %D 2000 %V 287 %F Mainardi2000 %0 Journal Article %T Finite difference approximations for fractional advection-dispersion flow equations %A M. M. Meerschaert %A C. Tadjeran %J J. Comput. Appl. Math. %D 2004 %V 172 %F Meerschaert2004 %0 Book %T An introduction to the fractional calculus and fractional differential equations %A K. S. Miller %A B. Ross %D 1993 %I A Wiley-Interscience Publication, John Wiley & Sons, Inc. %C New York %F Miller1993 %0 Journal Article %T Generalized Taylor’s formula %A Z. M. Odibat %A N. T. Shawagfeh %J Appl. Math. Comput. %D 2007 %V 186 %F Odibat2007 %0 Book %T The fractional calculus %A K. B. Oldham %A J. Spanier %D 2006 %I 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] %C New York-London %F Oldham2006 %0 Journal Article %T An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations %A E. J. Parkes %A B. R. Duffy %J Comput. Phys. Commun. %D 1996 %V 98 %F Parkes1996 %0 Book %T Fractional differential equations %A I. Podlubny %D 1999 %I 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. %C San Diego, CA %F Podlubny1999 %0 Journal Article %T Fractional calculus and continuous-time finance %A E. Scalas %A R. Gorenflo %A F. Mainardi %J Phys. A %D 2000 %V 284 %F Scalas2000 %0 Journal Article %T Finite difference approximations for a fractional advection diffusion problem %A E. Sousa %J J. Comput. Phys. %D 2009 %V 228 %F Sousa2009 %0 Journal Article %T Finite difference methods for fractional dispersion equations %A L.-J. Su %A W.-Q. Wang %A Q.-Y. Xu %J Appl. Math. Comput. %D 2010 %V 216 %F Su2010 %0 Journal Article %T Finite difference approximations for the fractional advection-diffusion equation %A L.-J. Su %A W.-Q. Wang %A Z.-X. Yang %J Phys. Lett. A %D 2009 %V 373 %F Su2009 %0 Journal Article %T The (\(\frac{G'}{G}\) )-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics %A M.-L. Wang %A X.-Z. Li %A J.-L. Zhang %J Phys. Lett. A %D 2008 %V 372 %F Wang2008 %0 Journal Article %T The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd- Bullough equations %A A. M. Wazwaz %J Chaos Solitons Fractals %D 2005 %V 25 %F Wazwaz2005 %0 Book %T Solutions of some nonlinear partial differential equations and comparison of their solutions %A A. Yokus %D 2011 %I Ph.D. Thesis, Fırat University, Elazig %C Turkey %F Yokus2011 %0 Journal Article %T Weighted average finite difference methods for fractional diffusion equations %A S. B. Yuste %J J. Comput. Phys. %D 2006 %V 216 %F Yuste2006 %0 Journal Article %T Chaos, fractional kinetics, and anomalous transport %A G. M. Zaslavsky %J Phys. Rep. %D 2002 %V 371 %F Zaslavsky2002 %0 Journal Article %T Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation %A X.-D. Zheng %A Y. Chen %A H.-Q. Zhang %J Phys. Lett. A %D 2003 %V 311 %F Zheng2003 %0 Journal Article %T New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation %A P. Zhuang %A F. Liu %A V. Anh %A I. Turner %J SIAM J. Numer. Anal. %D 2008 %V 46 %F Zhuang2008