A novel high-order approximation method for higher-dimensional time-fractional reaction-diffusion problems with weak initial singularity

The objective of this manuscript is to construct and analyze a fully discrete method to approximate one- and two-dimensional time-fractional reaction-diffusion equations defined in the Caputo sense. The current approach combines Alikhanov’s L۲-۱θ formula on a non-uniform graded mesh to discretize the time-fractional Caputo derivative and the discretization of the space variables using a cubic spline difference scheme. The two-dimensional problem is then separated into two one-dimensional problems using the alternating direction implicit (ADI) approach. The theoretical analysis, which covers both stability and convergence, has been provided for one- and two-dimensional problems. Further, to illustrate the accuracy and efficiency of the proposed method, numerical results for two test examples have been presented.

کلیدواژه ها:

Cubic spline difference scheme ، Caputo derivative ، L۲-۱θ formula ، Graded mesh ، ADI scheme ، Convergence analysis