Fractional Sturm-Liouville Problem Research Articles

On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions

The present work deals with some spectral properties of the problem \medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$, \vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip \noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter. First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space $L^{2}_q[a,b]$. Then, we investigate some asymptotic properties of the spectrum as $\alpha \underset {\lt }{\rightarrow } 1$. We give, in particular, the asymptotic expansion of the first eigenvalue.

Applications of the Fractional Sturm-Liouville Problem to the Space-Time Fractional Diffusion in a Finite Domain

Abstract The space–time fractional diffusion equations on finite domain model anomalous diffusion behavior with large particle jumps combined with long waiting times. In this work we prove existence of strong solutions for such equations. Our proofs strongly depend on the fractional Sturm–Liouville theory, precisely on the problem of finding eigenvalues and corresponding eigenfunctions to the certain fractional differential equation. Using the method of separating variables and applying theorem ensuring existence of solutions to the fractional Sturm–Liouville problem we solve several types of fractional diffusion equations.

Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

ABSTRACTIn this article, we first introduce a singular fractional Sturm–Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results.

A simple finite element method for boundary value problems with a Riemann–Liouville derivative

We consider a boundary value problem involving a Riemann–Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα−1 in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and L2(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm–Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

Spectral Analysis for Fractional Hydrogen Atom Equation

In this paper, spectral analysis of fractional Sturm Liouville problem defined on (0,1], having the singularity of type at zero and research the fundamental properties of the eigenfunctions and eigenvalues for the operator. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore,we give some important theorems and lemmas for fractional hydrogen atom equation.

A unified Petrov–Galerkin spectral method for fractional PDEs

Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov–Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form 0Dt2τu+∑j=1dcj[ajDxj2μju]+γu=f, where 2τ, μj∈(0,1), in a (1+d)-dimensional space–time domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1-D) and the corresponding convergence study of the scheme (in multi-D). The unified PG spectral method applies to the entire family of linear hyperbolic-, parabolic- and elliptic-like equations. We develop the PG method based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi poly-fractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from poly-fractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Besides the high-order spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in time-integration schemes for solving time-dependent FPDEs as well, rather than employing algebraically accurate traditional methods, especially when 2τ=1. Finally, we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices, which reduces the computational cost drastically. We demonstrate that this framework can reduce to hyperbolic FPDEs such as time- and space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost. Several numerical tests confirm the efficiency and spectral convergence of the unified PG spectral method for the aforementioned families of FPDEs. Moreover, we demonstrate the computational efficiency of the new approach in higher-dimensions e.g., (1+3), (1+5) and (1+9)-dimensional problems.

Variational methods for the fractional Sturm–Liouville problem

This article is devoted to the regular fractional Sturm–Liouville eigenvalue problem. By applying the methods of fractional variational analysis, we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.

Fractional Sturm-Liouville eigen-problems

We first consider a regular fractional Sturm-Liouville problem of two kinds RFSLP-I and RFSLP-II of order ( 0 , 2 ) . The corresponding fractional differential operators in these problems are both ...

Exponentially accurate spectral and spectral element methods for fractional ODEs

Current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy. Here, we present different methods for fractional ODEs that lead to exponentially fast decay of the error. First, we develop a Petrov–Galerkin (PG) spectral method for Fractional Initial-Value Problems (FIVPs) of the form Dtν0u(t)=f(t) and Fractional Final-Value Problems (FFVPs) DTνtu(t)=g(t), where ν∈(0,1), subject to Dirichlet initial/final conditions. These schemes are developed based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), which has been recently developed in [1]. Specifically, we obtain solutions to FIVPs and FFVPs in terms of the new fractional (non-polynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Subsequently, we develop a Discontinuous Spectral Method (DSM) of Petrov–Galerkin sense for the aforementioned FIVPs and FFVPs, where the basis functions do not satisfy the initial/final conditions. Finally, we extend the DSM scheme to a Discontinuous Spectral Element Method (DSEM) for efficient longer time-integration and adaptive refinement. In these discontinuous schemes, we employ the asymptotic eigensolutions to FSLP-I & -II, which are of Jacobi polynomial forms, as basis and test functions. Our numerical tests confirm the exponential/algebraic convergence, respectively, in p- and h-refinements, for various test cases with integer- and fractional-order solutions.

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 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.

Regular Fractional Sturm-Liouville Problem with Generalized Derivatives of Order in (0,1)

In this paper, we define a Generalized Fractional Sturm-Liouville Operator (GFSLO) and introduce a regular Generalized Fractional Sturm-Liouville Problem. In the construction of the operator and the problem, we apply a generalized fractional derivatives built using a general kernel. We investigate the properties of the eigenfunctions and the eigenvalues of the GFSLO, and demonstrate that these properties are analogous to those for classical Sturm-Liouville Operator dependent on first-order derivatives. As an example, we study the case when the integrals and the resulting derivatives are built using the generalized Mittag-Leffler function as a kernel.

Fractional Sturm–Liouville problem

In this paper, we define some Fractional Sturm–Liouville Operators (FSLOs) and introduce two classes of Fractional Sturm–Liouville Problems (FSLPs) namely regular and singular FSLP. The operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right Riemann–Liouville and left and right Caputo fractional derivatives. For both classes we investigate the eigenvalue and eigenfunction properties of the FSLOs. In the class of regular FSLPs, we discuss two types of FSLPs. As an operator for the class of singular FSLPs, we introduce a Fractional Legendre Equation (FLE) and discuss its solution. It is shown that the Legendre Polynomials resulting from an FLE are the same as those obtained from the integer order Legendre equation; however, the eigenvalues of the two equations differ. Using the Legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations, one ordinary and the other partial. It is our hope that this paper will initiate new research in the area of FSLPs and many of its variations.

An inverse Sturm–Liouville problem with a fractional derivative

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm–Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order α∈(1,2) of fractional derivative is sufficiently away from 2.

On the numerical solution of fractional Sturm–Liouville problems

The differential equation of Sturm-Liouville problems is generalized into fractional form by replacing the first-order derivative by a fractional derivative of order α, 0<α≤1. We showed briefly that this class of eigenvalue could be very promising to the solution of linear fractional partial differential equations. The homotopy perturbation method is considered for computing the eigenelements of the present problem. Based on our simulations some theoretical conjectures are reported.

Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator \(\mathcal{L}\). It can be seen in this paper that the auxiliary parameter \(\hbar,\) which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.

An efficient method for solving fractional Sturm–Liouville problems

The numerical approximation of the eigenvalues and the eigenfunctions of the fractional Sturm–Liouville problems, in which the second order derivative is replaced by a fractional derivative, is considered. The present results can be implemented on the numerical solution of the fractional diffusion-wave equation. The results show the simplicity and efficiency of the numerical method.