A Second-Order Accurate Numerical Approximation for Two-Sided Fractional Boundary Value Advection-Diffusion Problem

Abstract

Fractional order partial differential equations are generalization of classical partial<br />differential equations. Increasingly, these models are used in applications such as fluid flow, finance<br />and others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and convergence of the method are examined. It is shown that the fractional Crank-Nicolson method based on the shifted Gr"{u}nwald<br />formula is unconditionally stable. Some numerical examples are presented and compared with the exact analytical solution<br />for its order of convergence.<br /><br /><br />Fractional order partial differential equations are generalization of classical partial<br />differential equations. Increasingly, these models are used in applications such as fluid flow, finance<br />and others. In this paper we examine some practical numerical methods to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicolson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and convergence of the method are examined. It is shown that the fractional Crank-Nicolson method based on the shifted Gr"{u}nwald<br />formula is unconditionally stable. Some numerical examples are presented and compared with the exact analytical solution<br />for its order of convergence.