Analytical solutions and numerical schemes of certain generalized fractional diffusion models

Abstract

We give the analytical solutions of the fractional diffusion equations in one-, and two-dimensional space described by the Caputo left generalized fractional derivative. We introduce the forward Euler method for fractional diffusion equations represented by the Caputo left generalized fractional derivative. The contribution of this paper is to evaluate the impact of the second parameter of the Caputo left generalized fractional derivative in the behavior of the analytical solutions of the fractional diffusion equations, and to propose a numerical method for the generalized fractional diffusion equations. We will present the difference existing between the classical diffusion equation, the fractional diffusion equation described by Caputo fractional derivative and the fractional diffusion equation expressed by the Caputo left generalized fractional derivative. The Fourier-sine-Laplace-transform method is used to determine the analytical solutions of the fractional diffusion equations described by the Caputo left generalized fractional derivative. Some particular cases of diffusion equations are discussed, and the numerical simulations of their analytical solutions are presented and analyzed.