Stability Analysis of a Finite Difference Scheme for a Nonlinear Time Fractional Convection Diffusion Equation

The nonlinear time fractional convection diffusion equation (TFCDE) is obtained from a standard nonlinear convection diffusion equation by replacing the first-order time derivative with a fractional derivative (in Caputo sense) of order a. (0, 1). Developing numerical methods for solving fractional partial differential equations is of increasing interest in many areas of science and engineering. In this chapter, an explicit conservative finite difference scheme for TFCDE is introduced. We find its Courant-Friedrichs-Lewy (CFL) condition and prove encouraging results regarding stability, namely, monotonicity, the total variation diminishing (TVD) property and several bounds. Illustrative numerical examples are included in order to evaluate potential uses of the new method.