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

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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 τ 2 h 2 . Numerical experiments are carried out to support the theoretical claims.

We consider the nonlocal boundary value problem for difference equations (uk−uk−1)/τ+Auk=φk, 1≤k≤N, Nτ=1, and u0=u[λ/τ]+φ, 0<λ≤1, in an arbitrary Banach space E with the strongly positive operator A. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given

In this paper, a new difference scheme is constructed based on Crank Nicholson difference scheme. It can be used for solving Time Fractional Advection Dispersion Equations involving Caputo fractional derivative. We prove that the proposed method is unconditionally stable by using spectral stability technique. Numerical experiments are presented.