Fractional Sturm–Liouville problem

Klimek, Małgorzata; Agrawal, Om P.

doi:10.1016/j.camwa.2012.12.011

Cited by 140 publications

(78 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bas and Metin [6], Klimek and Agrawal [13], Zayernouri and Karniadakis [30], and Rivero, Trujillo, and Velasco [25] defined different classes of fractional Sturm-Liouville operators and investigated the properties of the corresponding eigenfunctions and the eigenvalues. In [12], the exact eigensolutions are obtained in terms of the standard Legendre polynomials. In addition, in [14], variational methods for the FSLP are investigated.…”

Section: Introductionmentioning

confidence: 99%

Tempered Fractional Sturm--Liouville EigenProblems

Zayernouri

Ainsworth

Karniadakis

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Lévy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm-Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order ν ∈ (0, 2). In the regular case, the corresponding tempered differential operators are associated with tempering functions p I (x) = exp(2τ ) and p II (x) = exp(−2τ ), τ ≥ 0, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in [−1, 1]. In contrast, the corresponding differential operators in the singular setting are associated with different forms of p I (x) = exp(2τ )(1 − x) 1+α (1 + x) 1+β and p II (x) = exp(−2τ )(1 − x) 1+α (1 + x) 1+β , vanishing at x = ±1 in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann-Liouville and tempered Caputo type of fractional order μ = ν/2 ∈ (0, 1). We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov-Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations.

show abstract

Section: Introductionmentioning

confidence: 99%

Tempered Fractional Sturm--Liouville EigenProblems

Zayernouri

Ainsworth

Karniadakis

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Recently, there are several analytical studies devoted to fractional Sturm-Liouville eigenvalue problems; see [23][24][25][26][27]. In these studies some of the well-known results of the Sturm-Liouville problems are extended to the fractional ones with left-and right-sided fractional derivatives of Riemann-Liouville and Caputo and Riesz derivatives.…”

Section: Introductionmentioning

confidence: 99%

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Al-Refai

Abdeljawad

2017

Complexity

View full text Add to dashboard Cite

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

show abstract

“…However, the fractional variational SLEs have only recently been investigated for local and nonlocal fractional operators [34,35,36].…”

Section: Introductionmentioning

confidence: 99%

A Variational Approach of the Sturm-Liouville Problem in Fractional Difference Calculus

Mert¹,

Erbe²,

Abdeljawad³

2018

DSA

View full text Add to dashboard Cite

In this article, we formulate and analyze a nabla fractional difference Sturm Liouville problem (SLP ) with the nabla left Caputo fractional difference and the nabla right Riemann-Liouville fractional difference. The discrete fractional variational calculus is used to study the eigenvalues and eigenfunctions of the formulated SLP by presenting a new nabla fractional difference isoperimetric variational problem.

show abstract

Fractional Sturm–Liouville problem

Cited by 140 publications

References 12 publications

Tempered Fractional Sturm--Liouville EigenProblems

Tempered Fractional Sturm--Liouville EigenProblems

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

A Variational Approach of the Sturm-Liouville Problem in Fractional Difference Calculus

Contact Info

Product

Resources

About