A fractional spline collocation-Galerkin method for the time-fractional diffusion equation

TL;DR

This paper introduces a novel collocation-Galerkin numerical method using fractional splines to efficiently solve time-fractional diffusion equations, with demonstrated accuracy and effectiveness through numerical tests.

Contribution

It develops a new fractional spline-based collocation-Galerkin method for time-fractional diffusion equations, enabling closed-form derivatives and improved computational efficiency.

Findings

Method achieves high accuracy in numerical tests

Derivatives of fractional splines expressed in closed form

Numerical results confirm method's effectiveness

Abstract

The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.