[Paper Review] A Finite Element Method for the Fractional Sturm-Liouville Problem

Bangti Jin, Raytcho Lazarov+2 more|arXiv (Cornell University)|Jul 19, 2013

Fractional Differential Equations Solutions10 references10 citations

TL;DR

This paper presents a novel finite element method (FEM) for solving fractional Sturm-Liouville problems with Caputo or Riemann-Liouville derivatives of order α ∈ (1,2). Based on new variational formulations, the method achieves second-order convergence for eigenvalue approximations and provides provable error bounds, enabling accurate computation of multiple eigenvalues simultaneously even with nonsmooth potentials.

ABSTRACT

In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $α\in(1,2)$ on the unit interval $(0,1)$. It is based on novel variational formulations of the eigenvalue problem. Error estimates are provided for the finite element approximations of the eigenvalues. Numerical results are presented to illustrate the efficiency and accuracy of the method. The results indicate that the method can achieve a second-order convergence for both fractional derivatives, and can provide accurate approximations to multiple eigenvalues simultaneously.

Motivation & Objective

• To develop a robust and accurate finite element method for fractional Sturm-Liouville problems involving Caputo or Riemann-Liouville derivatives.
• To address the lack of efficient, stable, and mathematically justified numerical schemes capable of computing multiple eigenvalues simultaneously.
• To provide rigorous error estimates for eigenvalue approximations, particularly for problems with general potential functions q(x).
• To enable numerical investigation of complex analytical properties such as eigenvalue asymptotics, bifurcation, and interlacing behavior in fractional Sturm-Liouville problems.

Proposed method

• The method is based on novel variational formulations of the fractional differential operator, tailored for both Caputo and Riemann-Liouville derivatives.
• It employs appropriate function spaces: U = V = Ḣ^{α/2}(D) for Riemann-Liouville and different spaces for Caputo, ensuring weak formulation stability.
• The weak form is derived as a(u,v) = λ(u,v) for all v ∈ V, leading to a generalized eigenvalue problem.
• Finite-dimensional subspaces Uh ⊂ U and Vh ⊂ V are constructed using piecewise polynomial approximations on a graded mesh.
• The resulting discrete generalized eigenvalue problem is solved using standard numerical linear algebra techniques.
• Error bounds are derived using regularity theory and interpolation estimates, yielding convergence rates r < α−1 for Riemann-Liouville and r < min(α+s,2)−1/2 for Caputo under smoothness conditions on q.

Experimental results

Research questions

• RQ1 Can a finite element method be developed that achieves second-order convergence for both Caputo and Riemann-Liouville fractional Sturm-Liouville problems?
• RQ2 What is the optimal convergence rate of the finite element approximation for eigenvalues in the presence of nonsmooth potentials?
• RQ3 How do the eigenvalues and eigenfunctions behave under bifurcation, and can numerical methods reliably capture such complex behavior?
• RQ4 Can the method provide accurate approximations to multiple eigenvalues simultaneously with provable error bounds?
• RQ5 What is the impact of potential terms on the stability and structure of complex conjugate eigenvalue pairs in fractional Sturm-Liouville problems?

Key findings

• The finite element method achieves second-order convergence for both Caputo and Riemann-Liouville derivatives, as confirmed by numerical experiments.
• For the Riemann-Liouville case, the error in eigenvalue approximation satisfies |λ − λh| ≤ Ch^{r} with r < α−1.
• For the Caputo case, under the condition that q ∈ Ḣ^s(D) ∩ L∞(D) with s ∈ [0,1] and α+s > 3/2, the error bound is |λ − λh| ≤ Ch^{r} with r < min(α+s,2)−1/2.
• The method successfully computes multiple eigenvalues simultaneously, including complex conjugate pairs, and captures bifurcation phenomena near critical α values.
• Numerical results show that the third eigenfunction is nearly indistinguishable from the second, indicating potential loss of distinctness in eigenfunction structure for certain α values.
• The presence of a potential term like q2 disrupts bifurcation stability, splitting complex conjugate eigenvalues and altering eigenfunction morphology, highlighting sensitivity in numerical schemes.