Numerical method based on Galerkin approximation for the fractional advection-dispersion equation

TL;DR

This paper introduces a Galerkin-based numerical method using Legendre functions to approximate solutions of fractional advection-dispersion equations, demonstrating effectiveness through numerical examples.

Contribution

It develops a novel Galerkin approximation approach with Legendre functions for fractional advection-dispersion equations, including proof of weak asymptotic solutions.

Findings

The method produces accurate approximate solutions.

Numerical examples confirm the effectiveness of the approach.

The approach handles both homogeneous and non-homogeneous equations.

Abstract

We use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is proposed. This leads to a system of fractional ordinary differential equations whose solutions in turn give approximate solution for the advection-dispersion equations of fractional order. Under certain assumptions on the approximate solutions, it is shown that this sequence of approximate solutions forms a weak asymptotic solution. Numerical examples are given to show the effectiveness of the proposed method.