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This paper introduces a high-order numerical procedure to solve the two-dimensional distributed-order Riesz space-fractional diffusion equation. In the proposed technique, first, a second-order numerical integration rule is employed to estimate the integral of the distributed-order Riesz space-fractional derivative. Then, the time derivative is discretized by a second-order difference scheme. Finally, the spatial direction is approximated by a difference formulation with fourth-order accuracy. The stability of the semi-discrete scheme is analyzed. We conclude that the difference between two consecutive time steps i.e. U i , j n − U i , j n − 1 is nearly zero when n → ∞. So, a suitable term is added to the main difference scheme as by adding this term we could derive the main ADI scheme. Furthermore, to reduce the used CPU time, we combine the fourth-order ADI formulation with the proper orthogonal decomposition method and then we gain a POD based reduced-order compact ADI finite difference plane. In the next, the convergence order of the fully discrete formulation has been investigated. The numerical results show the efficiency of new technique. It must be noted that the finite difference method is an effective and robust numerical technique for solving nonlinear equations that the ADI approach can be combined with it to improve the numerical simulations. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6] . The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains. [ABSTRACT FROM AUTHOR]