Wavelet Galerkin method for fractional elliptic differential equations

TL;DR

This paper develops a wavelet Galerkin method for fractional elliptic differential equations, leveraging Riesz bases of wavelets to improve computational efficiency and stability over traditional methods.

Contribution

It introduces a novel wavelet Galerkin approach for FEDEs with small, bounded condition numbers and Toeplitz matrix structures, enhancing numerical efficiency.

Findings

Reduced condition numbers of stiffness matrices

Toeplitz structure enables faster computations

Numerical results show improved efficiency and stability

Abstract

Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically solving fractional elliptic differential equations (FEDEs). The technique issues are solved and the detailed algorithm descriptions are provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin method we propose for FEDEs has the striking benefit of efficiency, since the condition numbers of the corresponding stiffness matrixes are small and uniformly bounded; and the Toeplitz structure of the matrix still can be used to reduce cost. Numerical results and comparison with the ordinary Galerkin methods are presented to demonstrate the advantages of the wavelet Galerkin method we provide.