# A collocation method of lines for two-sided space-fractional advection-diffusion equations with variable coefficients

## TL;DR

This paper introduces a spectral collocation-based Method of Lines for solving space-fractional advection-diffusion equations with variable coefficients, demonstrating high accuracy and efficiency through numerical examples.

## Contribution

It develops a new basis function set and applies a spectral collocation Method of Lines to effectively solve complex space-fractional PDEs with variable coefficients.

## Key findings

- Achieves exponential accuracy in numerical solutions.
- Successfully handles equations with discontinuous coefficients.
- Demonstrates efficiency through benchmark problems.

## Abstract

We present the Method Of Lines (MOL), which is based on the spectral collocation method, to solve space-fractional advection-diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left- and right-sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.