A Unified Spectral Method for FPDEs with Two-sided Derivatives; Stability, and Error Analysis
For researchers in numerical analysis, this work offers rigorous theoretical foundations for a spectral method applied to fractional PDEs, though it is an incremental extension of prior work.
The paper provides stability and error analysis for a unified Petrov-Galerkin spectral method for linear fractional PDEs with two-sided derivatives, proving existence, uniqueness, and convergence rates, validated by numerical simulations.
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.