A FINITE VOLUME–ADAMS–BASHFORTH–MOULTON SCHEME FOR TIME-FRACTIONAL DIFFUSION

Abstract

Fractional differential equations (FDEs) are widely used to model physical processes which contain memory effects and anomalous diffusion. The current study presents a numerical method for the time-fractional diffusion equation by combining the Adams–Bashforth–Moulton predictor–corrector (ABM–PC) approach for time discretization with the finite volume method (FVM) in space discretization, thus preserving the mass conservation property. Numerical experiments performed on different fractional orders α = 0.3, 0.5, 0.7, 0.9, and α = 1.0 show that increasing values of α introduce smoothness and stability in the diffusion profiles. However, decreasing values of α introduce stronger memory effects which clearly manifest as mild oscillations. In the classical limit of α = 1.0, the numerical solution agrees very well with the analytical solution obtained from. In general, the proposed FVM–ABM–PC scheme is robust and can easily be extended to higher dimensional and nonlinear fractional diffusion problems.