# Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

## TL;DR

This paper develops and analyzes rational spectral methods for solving PDEs with fractional Laplacians in unbounded domains, providing optimal error estimates and demonstrating superior numerical performance over existing Hermite-based approaches.

## Contribution

The paper introduces rational basis spectral methods for fractional PDEs in unbounded domains, including explicit formulas, error analysis, and numerical validation.

## Key findings

- Rational spectral methods achieve optimal error estimates.
- The proposed methods outperform Hermite function approaches.
- Numerical results confirm high accuracy and efficiency.

## Abstract

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.