Abstract
Solving time-fractional diffusion equation using a numerical method has become a research trend nowadays since analytical approaches are quite limited. There is increasing usage of the finite difference method, but the efficiency of the scheme still needs to be explored. A half-sweep finite difference scheme is well-known as a computational complexity reduction approach. Therefore, the present paper applied an unconditionally stable half-sweep finite difference scheme to solve the time-fractional diffusion equation in a one-dimensional model. Throughout this paper, a Caputo fractional operator is used to substitute the time-fractional derivative term approximately. Then, the stability of the difference scheme combining the half-sweep finite difference for spatial discretization and Caputo time-fractional derivative is analyzed for its compatibility. From the formulated half-sweep Caputo approximation to the time-fractional diffusion equation, a linear system corresponds to the equation contains a large and sparse coefficient matrix that needs to be solved efficiently. We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated over relaxation (PAOR) algorithm to achieve a high convergence solution. The convergence of the developed method is analyzed. Finally, some numerical experiments from our research are given to illustrate the efficiency of our computational approach to solve the proposed problems of time-fractional diffusion. The combination of a half-sweep finite difference scheme and PAOR algorithm can be a good alternative computational approach to solve the time-fractional diffusion equation-based mathematical physics model.
Highlights
The increasing interest in the application of fractional-order partial differential equations (FPDE) to replace the classical integer-order partial differential equations can be seen obviously in recent years
We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated over relaxation (PAOR) algorithm to achieve a high convergence solution
The contribution of this paper is to present a preconditioned accelerated over relaxation (PAOR) algorithm via an half-sweep finite difference (HSFD) scheme to solve a one-dimensional time-fractional diffusion equation (TFDE) efficiently
Summary
The increasing interest in the application of fractional-order partial differential equations (FPDE) to replace the classical integer-order partial differential equations can be seen obviously in recent years. This work is motivated by the recent active development of an accurate and efficient numerical approach to solve FPDE. There are three types of FPDE, namely time-fractional, space-fractional, and time-space-fractional differential equations. These FPDEs can be distinguished based on the presence of fractional derivative, which either on time derivative, space derivative or both time and space derivative. Time-fractional type is the subject of interest and an investigation on developing an efficient computational approach for timefractional diffusion equation is conducted
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