On spectral Petrov-Galerkin method for solving fractional initial value problems in weighted Sobolev space

TL;DR

This paper develops and analyzes a spectral Petrov-Galerkin method for fractional initial value problems, providing optimal error estimates and an efficient iterative solver, validated through numerical experiments.

Contribution

The paper introduces a spectral Petrov-Galerkin approach tailored for fractional problems with singularities, including an optimal error estimate and a quasi-linear complexity solver.

Findings

Convergence order in weighted L2-norm is 3α+1 for smooth sources.

The proposed method achieves high accuracy and efficiency.

Numerical experiments confirm theoretical error estimates and solver performance.

Abstract

In this paper, we investigate a spectral Petrov-Galerkin method for fractional initial value problems. Singularities of the solution at the origin inherited from the weakly singular kernel of the fractional derivative are considered, and the regularity is constructed for the solution in weighted Sobolev space. We present an optimal error estimate of the spectral Petrov-Galerkin method, and prove that the convergence order of the method in the weighted $L^2$-norm is $3\alpha+1$ for smooth source term, where $\alpha$ is the order of the fractional derivative. An iteration algorithm with a quasi-linear complexity is considered to solve the produced linear system. Numerical experiments verify the theoretical findings and show the efficiency of the proposed algorithm, and exhibit that the presented numerical method works well for some time-fractional diffusion equations after suitable temporal semi-discrete.