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

Singh, Richa; Singh, Anshima; Kumar, Sunil; Vigo-Aguiar, Jesus

doi:10.22034/cmde.2025.61994.2712

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

Document Type : Research Paper

Authors

¹ Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India.

² Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, India.

³ Department of Applied Mathematics, University of Salamanca, Salamanca, Spain.

10.22034/cmde.2025.61994.2712

Abstract

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 L2-1θ 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.

Keywords

Main Subjects

Fractional partial differential equations

References

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