Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation

We first consider a regular fractional Sturm–Liouville problem of two kinds RFSLP-I and RFSLP-II of order ν∈(0,2). The corresponding fractional differential operators in these problems are both of Riemann–Liouville and Caputo type, of the same fractional order μ=ν/2∈(0,1). We obtain the analytical e...

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Bibliographic Details
Published in:Journal of computational physics Vol. 252; pp. 495 - 517
Main Authors: Zayernouri, Mohsen, Karniadakis, George Em
Format: Journal Article
Language:English
Published: Elsevier Inc 01.11.2013
Subjects:
Approximation
Asymptotic properties
Convergence
Eigenfunctions
Eigenvalues
Exponential convergence
Fractal expansion set
Mathematical analysis
Mathematical models
Operators
Poly-fractonomial eigenfunctions
Regular/singular fractional Sturm–Liouville operators
Exponential convergence
Poly-fractonomial eigenfunctions
Regular/singular fractional Sturm–Liouville operators
Fractal expansion set
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:We first consider a regular fractional Sturm–Liouville problem of two kinds RFSLP-I and RFSLP-II of order ν∈(0,2). The corresponding fractional differential operators in these problems are both of Riemann–Liouville and Caputo type, of the same fractional order μ=ν/2∈(0,1). We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm–Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2013.06.031