A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

Abstract

In this paper, 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 the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2−α + h 2 ), conditionally. Numerical experiments are carried out to support the theoretical claims.MSC 2010 : 65M12, 65M06

Stability and convergence of a finite difference scheme for fractional partial differential equations by matrix method

“…Using the equations (3) and (4), we obtain the following difference scheme [7] which is accurate of order O(τ 2−α + h 2 );…”

A numerical solution for a class of time fractional diffusion equations with delay

“…In the work of Karatay et al (2013), an approximation to the time Caputo fractional derivative at t k+1/2 with 0 < α l < 1 was given as…”

“…Zhang and Sun (2013) introduced a linearized compact difference scheme for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. Karatay et al (2013) predicted an approximation for the time Caputo fractional derivative at time t k+1/2 with fractional order 0 < α < 1. They extended the idea of the Cranck-Nicholson method to time fractional heat equations with convergence order O τ 2−α + h 2 .…”

“…Based on the ideas of Zhang and Sun (2013) and Karatay et al (2013), we are interested in constructing a linearized difference scheme for (1) which is induced with fixed time delay in the source function as in the simulation of dynamical systems. Specifically, we consider t, u(x, t), u(x, t − s)), (2a) with the following initial and boundary conditions:…”

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

“…They do not provide any proof of convergence and stability of their method, and, they use a first order approximation for the discretisation of the Neumann boundary conditions. Another interesting work is the study proposed by Karatay et al [22] where they present a new numerical scheme, based on the Crank-Nicholson method, for the solution of the time-fractional heat equation. In this work, we consider the fractional version of the Pennes's bioheat transfer equation (1.1), with the thermal diffusivity coefficient assumed as a function of space.…”