A novel spectral method for the subdiffusion equation

TL;DR

This paper introduces a new spectral method for the subdiffusion equation that effectively handles initial-time singularities through variable transformation and spectral approximation, achieving exponential convergence.

Contribution

It presents a novel spectral method combining variable transformation and $$-fractional Sobolev spaces, with theoretical analysis and numerical validation for subdiffusion equations.

Findings

Exponential convergence of the proposed spectral method.

Effective handling of solutions with limited regularity.

Numerical examples demonstrating efficiency and accuracy.

Abstract

In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional high-order numerical methods is inefficient. We try to overcome this difficulty in a novel approach by combining variable transformation techniques with spectral methods. The idea is to first use suitable variable transformation to re-scale the underlying equation, then construct spectral methods for the re-scaled equation. We establish a new variational framework based on the $\psi$-fractional Sobolev spaces. This allows us to prove the well-posedness of the associated variational problem. The proposed spectral method is based on the variational problem and generalized Jacobi polynomials to approximate the re-scaled fractional differential equation. Our theoretical and numerical investigation show that the proposed method is exponentially convergent for general right hand side functions, even though the exact solution has very limited regularity. Implementation details are also provided, along with a series of numerical examples to show the efficiency of the proposed method.