Numerical solutions of fractional convection-diffusion equation using finite-difference and finite-volume schemes

Abstract

Many natural phenomena in physics and engineering can be modeled by linear and nonlinear partial differential equations, which are constructed using derivatives of fractional order. The main purpose of this work is to facilitate the implementation of space finite-volume and finite-difference schemes to solve fractional convection-diffusion equation of order  without source term along with appropriate initial conditions. The fractional derivative is described in Riemann-Liouville sense. The highlight of the proposed methods is to introduce an alternative way to discretize the space-fractional derivative utilizing the fractional Grünwald formula. Numerical results are provided to examine the accuracy of the proposed scheme and to compare it in different conditions. The obtained results show that the proposed techniques are simple, accurate, and applicable to a wide range of space-fractional models that arise in the natural sciences.