Development and Analysis of Optimized Numerical Scheme for Three-Dimensional Linear Time-Fractional Diffusion Equations

In this paper, an optimized numerical method for solving time-fractional diffusion equations in three-dimensional space is proposed. Fractional-order differential equations can model complex physical processes with memory effects more accurately than differential equations with integer order derivatives. On the other hand, numerical solutions of this type of problem are usually challenging due to the existence of a singularity near the initial time. In the proposed method, in order to overcome this issue, the L۱ formula has been used on a graded mesh to maintain the accuracy of calculations near the initial time. Also, a fourth-order compact operator has been used to discretize the Laplace operator in three-dimensional space. Stability and convergence analyses show that this method has a convergence order proportional to the grading parameter in time and a fourth-order convergence order in space. The obtained results from numerical simulations confirm the high accuracy of the proposed method and the efficiency of the optimal selection of the scaling parameter in achieving the suitable convergence order, even in the presence of initial singularities.

کلیدواژه ها:

Time-fractional diffusion equation ، Graded mesh ، Compact finite difference scheme ، Stability and convergence analysis ، Three-dimensional diffusion equation