[Paper Review] Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations

Lizhen Chen, Zhiping Mao+1 more|arXiv (Cornell University)|Mar 9, 2018

Fractional Differential Equations Solutions17 references3 citations

TL;DR

This paper proposes a Jacobi-Galerkin spectral method for solving Riesz fractional differential eigenvalue problems with homogeneous Dirichlet boundary conditions. By formulating the problem in a symmetric weak form within specially defined Sobolev spaces, the method preserves matrix symmetry and positive definiteness, enabling rigorous error estimates and proving exponential convergence. The key contribution is the numerical validation of the Weyl-type asymptotic law $\lambda_n = \mathcal{O}(n^{2\alpha})$ and condition number scaling $\chi_N = \mathcal{O}(N^{4\alpha})$. The method achieves spectral accuracy and confirms theoretical convergence rates through extensive numerical experiments.

ABSTRACT

An efficient Jacobi-Galerkin spectral method for calculating eigenvalues of Riesz fractional partial differential equations with homogeneous Dirichlet boundary values is proposed in this paper. In order to retain the symmetry and positive definiteness of the discrete linear system, we introduce some properly defined Sobolev spaces and approximate the eigenvalue problem in a standard Galerkin weak formulation instead of the Petrov-Galerkin one as in literature. Poincaré and inverse inequalities are proved for the proposed Galerkin formulation which finally help us establishing a sharp estimate on the algebraic system's condition number. Rigorous error estimates of the eigenvalues and eigenvectors are then readily obtained by using Babuška and Osborn's approximation theory on self-adjoint and positive-definite eigenvalue problems. Numerical results are presented to demonstrate the accuracy and efficiency, and to validate the asymptotically exponential oder of convergence. Moreover, the Weyl-type asymptotic law $ λ_n=\mathcal{O}(n^{2α})$ for the $n$-th eigenvalue $λ_n$ of the Riesz fractional differential operator of order $2α$, and the condition number $N^{4α}$ of its algebraic system with respect to the polynomial degree $N$ are observed.

Motivation & Objective

• To develop a high-order, stable numerical method for eigenvalue problems of Riesz fractional differential equations with Dirichlet boundary conditions.
• To preserve symmetry and positive definiteness of the discrete system, ensuring robust conditioning and efficient solution of the algebraic eigenvalue problem.
• To rigorously establish error estimates for eigenvalues and eigenvectors using Babuška and Osborn's approximation theory for self-adjoint, positive-definite eigenvalue problems.
• To numerically validate the Weyl-type asymptotic law $\lambda_n = \mathcal{O}(n^{2\alpha})$ and the condition number scaling $\chi_N = \mathcal{O}(N^{4\alpha})$.
• To demonstrate the method's exponential convergence rate and robustness across varying fractional orders $2\alpha \in (0,2)$ and higher orders up to $5.6$.

Proposed method

• The method employs a Galerkin weak formulation in specially constructed Sobolev spaces that accommodate the singular behavior of eigenfunctions near the domain boundaries.
• Jacobi basis functions $\{(1-x^2)^\alpha J^{\alpha,\alpha}_n(x)\}$ are used to approximate eigenfunctions, capturing the $\sim (1-x^2)^\alpha$ singularity at $x = \pm 1$.
• The formulation ensures symmetry and positive definiteness of the resulting algebraic system, enabling efficient and stable solution of the generalized eigenvalue problem.
• Poincaré and inverse inequalities are derived for the proposed Galerkin space, which are essential for bounding the condition number of the discrete system.
• Error estimates for eigenvalues and eigenvectors are derived using Babuška and Osborn's theory for self-adjoint, positive-definite eigenvalue problems.
• Numerical experiments are conducted with varying polynomial degrees $N$ and fractional orders $2\alpha$, using high-precision solvers to verify convergence and asymptotic behavior.

Experimental results

Research questions

• RQ1 Does the proposed Jacobi-Galerkin spectral method achieve exponential convergence for eigenvalues of Riesz fractional differential equations?
• RQ2 Is the condition number of the discrete system accurately predicted by $\mathcal{O}(N^{4\alpha})$ as $N$ increases?
• RQ3 Do the computed eigenvalues follow the Weyl-type asymptotic law $\lambda_n = \mathcal{O}(n^{2\alpha})$ for large $n$?
• RQ4 Can the method maintain high accuracy and stability for higher fractional orders $2\alpha > 2$, such as $3.6$ and $5.6$?
• RQ5 How well do the numerical eigenfunctions capture the expected singular behavior $\sim (1-x^2)^\alpha$ near the endpoints?

Key findings

• The method achieves asymptotically exponential convergence for eigenvalues, confirmed by numerical results showing rapid error decay with increasing polynomial degree $N$.
• The eigenvalues scale as $\lambda_n = \mathcal{O}(n^{2\alpha})$, with numerical results showing a strong linear dependence of $\lambda_n$ on $n^{2\alpha}$ across $2\alpha \in (1.2, 2.0)$.
• The condition number of the discrete system scales as $\chi_N = \mathcal{O}(N^{4\alpha})$, confirmed by log-log plots of $\chi_N$ versus $N$ matching the predicted power law.
• For $2\alpha = 2$, the first eigenvalue $\lambda_1$ converges to the exact value $\left(\frac{\pi}{2}\right)^2$, and $\rho_n = \lambda_n / \left(\frac{\pi n}{2}\right)^{2\alpha}$ converges to 1 as $n$ increases.
• The method remains accurate and efficient for higher fractional orders, including $2\alpha = 3.6$ and $5.6$, with consistent convergence behavior observed.
• The numerical results validate the theoretical prediction that the first eigenfunction $\psi_1(x)$ behaves as $\sim (1-x^2)^\alpha$ near the boundaries, which the Jacobi basis captures effectively.