Nonstandard finite difference method for solving complex-order fractional Burgers' equations

Abstract

The aim of this work is to present numerical treatments to a complex order fractional nonlinear one-dimensional problem of Burgers' equations. A new parameter ${\sigma }_t$ is presented in order to be consistent with the physical model problem. This parameter characterizes the existence of fractional structures in the equations. A relation between the parameter ${\sigma }_t$ and the time derivative complex order is derived. An unconditionally stable numerical scheme using a kind of weighted average nonstandard finite-difference discretization is presented. Stability analysis of this method is studied. Numerical simulations are given to confirm the reliability of the proposed method.

Keywords: Burgers’ equations; Complex order fractional derivative; Nonstandard weighted average finite difference method; Stability analysis.