A Unified Spectral Method for FPDEs with Two-sided Derivatives; Stability, and Error Analysis

TL;DR

This paper develops a spectral Petrov-Galerkin method for linear fractional PDEs with two-sided derivatives, providing stability, error analysis, and numerical validation across multiple dimensions.

Contribution

It offers a unified spectral approach with proven stability and error bounds for fractional PDEs in multi-dimensional spaces, extending previous methods.

Findings

Proved existence and uniqueness of the weak form.

Established stability and error estimates.

Numerical results confirm theoretical convergence rates.

Abstract

We present the stability and error analysis of the unified Petrov-Galerkin spectral method, developed in \cite{samiee2017Unified}, for linear fractional partial differential equations with two-sided derivatives and constant coefficients in any ($1+d$)-dimensional space-time hypercube, $d = 1, 2, 3, \cdots$, subject to homogeneous Dirichlet initial/boundary conditions. Specifically, we prove the existence and uniqueness of the weak form and perform the corresponding stability and error analysis of the proposed method. Finally, we perform several numerical simulations to compare the theoretical and computational rates of convergence.