Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

Abstract

Two alternating direction implicit difference schemes are derived for twodimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete L 1 (L ∞ ) norm with the convergence orders O(τ 2 | ln τ | + h 2 1 + h 2 2 + α 2 ) and O(τ 2 | ln τ | + h 4 1 + h 4 2 + α 4 ), respectively, where τ, h i (i = 1, 2) and α are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.

An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations

“…There has been some pioneer work on the numerical treatment of distributed order fractional differential equations . In Diethelm and Ford and Katsikadelis, one‐dimensional linear and nonlinear distributed order initial value problems have been investigated.…”

“…Note that the temporal convergence order remains low in the above mentioned papers and this will greatly affect the performance of the proposed numerical schemes in practice. Recently, Gao and Sun have successfully improved the temporal convergence order through the weighted shifted Grünwald‐Letnikov (WSGL) method. With composite trapezoid/Simpson formula for the integral and WSGL method for the fractional terms, coupled with finite difference/compact finite difference method in the spatial dimension, their best convergence order is .…”

“…A large K ensures high accuracy but it will also incur high computation cost. Motivated by all the above research, our objective in this paper is to improve the spatial convergence order to at least O ( h 4.5 ) while keeping the computation cost similar to that of the compact finite difference method which has a convergence of O ( h 4 ).…”

Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

“…Hu et al [33] considered a new time distributed-order and two-sided space-fractional advection-dispersion equation that was solved numerically using an implicit method for the solution the multi-term fractional equation. Gao et al published a series of papers [27,28,29] where: [27] two difference schemes were derived for both one-dimensional and two-dimensional distributed-order differential equations (he proved that the schemes are unconditionally stable and convergent); [28] the Grünwald formula was used to solve the one-dimensional distributed-order differential equations (two difference schemes were derived and the extrapolation method was applied to improve the approximate accuracy); [29] two alternating direction implicit difference schemes were derived for two-dimensional distributed-order fractional diffusion equations (he proved that the schemes are unconditionally stable and convergent). Wang et al [63] derived and analysed a second-order accurate implicit numerical method for the Riesz space distributedorder advection-dispersion equation.…”

Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

“…The difference schemes for solving the two-dimensional problem were also established in [22]. In [23] and [24], some effective ADI difference schemes for solving the two-dimensional time distributed-order fractional diffusion equations were presented and analyzed.…”