# Differential quadrature method for space-fractional diffusion equations on 2D irregular domains

## TL;DR

This paper introduces a flexible differential quadrature method using radial basis functions for solving 2D space-fractional diffusion equations on irregular domains, enabling efficient modeling of super-diffusion phenomena.

## Contribution

It develops a novel DQ approach with RBFs for fractional derivatives on irregular domains, extending the applicability of numerical methods to complex geometries.

## Key findings

- Method effectively solves 2D fractional diffusion equations on irregular domains.
- Numerical examples demonstrate high accuracy and flexibility.
- Approach reduces PDEs to ODEs discretized by Crank-Nicolson scheme.

## Abstract

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L\'{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.