[Paper Review] Spectral Properties of Fractional Sturm-Liouville Problem for Diffusion Operator

Erdal Baş, Funda Metin|arXiv (Cornell University)|Dec 19, 2012

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper investigates a regular fractional Sturm-Liouville problem for a diffusion operator (FSLPDO), establishing that its eigenvalues are real and eigenfunctions are orthogonal. The study proves the fractional diffusion operator is self-adjoint, ensuring spectral properties essential for solving fractional differential equations in mathematical physics.

ABSTRACT

In this study, we give a regular fractional Sturm Liouville problem for diffusion operator (FSLPDO), research the spectral properties of the eigenfunctions and eigenvalues of the diffusion operator. We show that the eigenvalues and eigenfunctions of (FSLPDO) are real and orthogonal, respectively and fractional diffusion operator is self adjoint.

Motivation & Objective

• To analyze the spectral behavior of a fractional Sturm-Liouville problem involving a diffusion operator.
• To establish the reality of eigenvalues and orthogonality of eigenfunctions in the fractional setting.
• To prove that the fractional diffusion operator is self-adjoint, ensuring well-defined spectral theory.

Proposed method

• Formulation of a regular fractional Sturm-Liouville problem for a diffusion operator using Riemann-Liouville fractional derivatives.
• Application of spectral theory techniques adapted to fractional-order differential operators.
• Use of inner product spaces and functional analytic methods to examine self-adjointness.
• Derivation of eigenvalue and eigenfunction properties through boundary value analysis.
• Proof of real eigenvalues via symmetry and self-adjointness conditions.
• Demonstration of eigenfunction orthogonality using the inner product and self-adjoint operator properties.

Experimental results

Research questions

• RQ1 Are the eigenvalues of the fractional Sturm-Liouville diffusion operator real?
• RQ2 Are the eigenfunctions of the FSLPDO orthogonal with respect to the standard L2 inner product?
• RQ3 Does the fractional diffusion operator satisfy the conditions for self-adjointness in the Hilbert space setting?
• RQ4 How do the spectral properties of the fractional operator compare to classical Sturm-Liouville problems?
• RQ5 What functional analytic framework supports the spectral decomposition of the FSLPDO?

Key findings

• The eigenvalues of the fractional Sturm-Liouville diffusion operator are real.
• The eigenfunctions of the FSLPDO are orthogonal in the L2 inner product space.
• The fractional diffusion operator is self-adjoint under appropriate boundary conditions.
• The spectral properties of the FSLPDO mirror those of classical Sturm-Liouville problems, but in a fractional calculus context.
• The results validate the use of spectral methods for solving fractional differential equations involving diffusion.
• The framework supports future analysis of fractional partial differential equations through eigenfunction expansions.