Treffer: A highly efficient method for numerical solution of time-fractional partial differential equations

We propose a highly efficient method for the numerical solution of generalized diffusion equations with Caputo derivatives. A Galerkin finite element method is used for the space discretization whereas an improved non-classical method is used for the time integration. In practice, the computational implementation of this method presents challenges due to its arithmetic complexity and its high-memory requirements. The present study aims to overcome these drawbacks by introducing novel computational techniques to speed up the algorithm at low-memory storage. A detailed error analysis of the new method is also studied for time-fractional equations of diffusion type. Numerical results are also presented for several problems to validate the performance of the proposed techniques. The obtained results demonstrate that the method significantly reduces computational cost compared to the conventional method while maintaining the convergence order of the method in both space and time.
Moroccan Journal of Pure and Applied Analysis, Vol. 10 No. 1 (2024)