A Characteristic Difference Scheme for Time‐Fractional Heat Equations Based on the Crank‐Nicholson Difference Schemes

Abstract

We consider the numerical solution of a time‐fractional heat equation, which is obtained from the standard diffusion equation by replacing the first‐order time derivative with Riemann‐Liouville fractional derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on Crank‐Nicholson method to the time‐fractional heat equations. We prove that the proposed method is unconditionally stable, and the numerical solution converges to the exact one with the order O(τ2 + h2). Numerical experiments are carried out to support the theoretical claims.