An exponential B-spline collocation method for fractional sub-diffusion equation

TL;DR

This paper introduces an exponential B-spline collocation method for efficiently solving fractional sub-diffusion equations of Caputo type, demonstrating its stability, solvability, and superior accuracy through numerical experiments.

Contribution

It develops a novel exponential B-spline collocation approach combined with GMMP scheme for fractional PDEs, with rigorous analysis and improved computational efficiency.

Findings

Method produces tri-diagonal algebraic systems for fast solutions.

Numerical results confirm the method's accuracy and stability.

Outperforms existing algorithms in the literature.

Abstract

In this article, we propose an exponential B-spline collocation method to approximate the solution of the fractional sub-diffusion equation of Caputo type. The present method is generated by use of the Gorenflo-Mainardi-Moretti-Paradisi (GMMP) scheme in time and an efficient exponential B-spline based method in space. The unique solvability is rigorously discussed. Its stability is well illustrated via a procedure closely resembling the classic von Neumann approach. The resulting algebraic system is tri-diagonal that can rapidly be solved by the known algebraic solver with low cost and storage. A series of numerical examples are finally carried out and by contrast to the other algorithms available in the literature, numerical results confirm the validity and superiority of our method.