Numerical and exact solutions for time fractional Burgers' equation

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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
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