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

Abstract

Two alternating direction implicit difference schemes are derived for two-dimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete $$L^1(L^\infty )$$L1(L?) norm with the convergence orders $$O(\tau ^2|\ln \tau |+h_1^2+h_2^2+\Delta \alpha ^2)$$O(?2|ln?|+h12+h22+Δ?2) and $$O(\tau ^2|\ln \tau |+h_1^4+h_2^4+\Delta \alpha ^4),$$O(?2|ln?|+h14+h24+Δ?4), respectively, where $$ \tau , h_i \;(i=1,2)$$?,hi(i=1,2) and $$\Delta \alpha $$Δ? are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.