Fractional Sturm-Liouville Problem Research Articles

Theory and numerical approaches of high order fractional Sturm–Liouville problems

Theory and numerical approaches of high order fractional Sturm–Liouville problems

On partial fractional Sturm\u2013Liouville equation and inclusion

The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

Sharp asymptotics in a fractional Sturm-Liouville problem

Abstract The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.

Müntz sturm-liouville problems: Theory and numerical experiments

Abstract This paper presents two new classes of Müntz functions which are called Jacobi-Müntz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new Müntz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given.

Variational Approach for Tempered Fractional Sturm–Liouville Problem

In this paper, we study the properties of eigenvalue for the regular tempered fractional Sturm–Liouville problem (TFSLP) of order $$ \mu $$ . Using a fractional variational approach, we show that the set of eigenvalues for TFSLP are infinite, and correspond to unique eigenfunctions. We establish that the eigenvalues of the problem are distinct, and the corresponding eigenfunctions are orthogonal to each other. We also show that the minimum value of the functional corresponding to TFSLP is the lowest eigenvalue.

An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville problems

In this paper, we construct numerical schemes based on the Lagrange polynomial interpolation to solve Fractional Sturm–Liouville problems (FSLPs) in which the fractional derivatives are considered in the Caputo sense. First, we convert the differential equation with boundary conditions into integral form and discretize the fractional integral to generate a system of algebraic equations in the matrix form. Next, we calculate the set of approximate eigenvalues and corresponding eigenvectors. The eigenfunctions are approximated and some of their properties are investigated. The experimental rate of convergence of numerical calculations for the eigenvalues is reported and the order convergence of the numerical method is obtained. Finally, some examples are presented to illustrate the efficiency and accuracy of the numerical method.

Fast preconditioned iterative methods for fractional Sturm–Liouville equations

AbstractIn this paper, we have considered fast solutions of the linear system arising from the fractional Sturm–Liouville problem, whose coefficient matrix contains the product of Toeplitz‐like matrices. Based on suitable circulant approximations of the related coefficient matrix, we establish a matching preconditioner of matrix‐free form. In theory, the spectrum of the preconditioned matrix is shown to cluster around [1/2, 1), which suggests the fast convergence speed of the proposed preconditioner within Krylov subspace acceleration. In addition, to reduce the computation time and storage, we consider an all‐at‐once discretized system and explore its low‐rank tensor structure and alternating iterative tensor algorithms. Numerical experiments are given to show the effectiveness of the proposed solution techniques compared with some existing methods.

Existence of solutions for infinite systems of regular fractional Sturm-Liouville problems in the spaces of tempered sequences

In this paper we investigate the existence of solutions for infinite systems of regular fractional Sturm-Liouville problems by an application of Meir-Keeler fixed point theorem in the tempered sequence spaces. We provide examples to check validity of our obtained results.

Fully spectral‐Galerkin method for the one‐ and two‐dimensional fourth‐order time‐fractional partial integro‐differential equations with a weakly singular kernel

AbstractIn the current paper, a space–time spectral‐Galerkin method is presented for the one‐ and two‐dimensional (1D & 2D) fourth‐order time‐fractional partial integro‐differential equation (TFPIDE) with a weakly singular kernel. In temporal direction, a Petrov‐Galerkin approach is used for discretization. Indeed, eigenfunctions of the first and second kind fractional Sturm‐Liouville problem (FSLP), which called Jacobi polyfractonomials, are used as temporal basis for the trial and test spaces, respectively. Also, spatial discretization is based on the Galerkin approximation with a combination of Legendre polynomials as basis. Fully discrete scheme, give rises to obtaining approximate solution of desired problem via solving a system of linear algebraic equations. Spectral accuracy and efficiency of the considered method are numerically demonstrated by some test problems with smooth and non‐smooth exact solutions in one‐ and two‐dimensional cases.

A Numerical Study of Eigenvalues and Eigenfunctions of Fractional Sturm-Liouville Problems via Laplace Transform

In this paper, we consider a class of fractional Sturm-Liouville problems, in which the second order derivative is replaced by the Caputo fractional derivative. The Laplace transform method is applied to obtain algebraic equations. Then, the eigenvalues and the eigenfunctions of the fractional Sturm-Liouville problems are obtained numerically. We provide a convergence analysis for given method. Finally, the simplicity and efficiency of the numerical method is shown by some examples.

Variational Approximation for Fractional Sturm–Liouville Problem

Abstract In this paper, we consider a regular Fractional Sturm–Liouville Problem (FSLP) of order μ (0 < μ < 1). We approximate the eigenvalues and eigenfunctions of the problem using a fractional variational approach. Recently, Klimek et al. [16] presented the variational approach for FSLPs defined in terms of Caputo derivatives and obtained eigenvalues, eigenfunctions for a special range of fractional order 1/2 < μ < 1. Here, we extend the variational approach for the FSLPs and approximate the eigenvalues and eigenfunctions of the FSLP for fractional-order μ (0 < μ < 1). We also prove that the FSLP has countably infinite eigenvalues and corresponding eigenfunctions.

Adomian decomposition method for solving nonlinear fractional sturm-liouville problem

In the present paper, the Adomian decomposition method is employed for solving nonlinear fractional Sturm-Liouville equation. The numerical results for the eigenfunctions and the eigenvalues are obtained. Also, the present results are demonstrated by the tables and the graphs for different values of considered problem.

Representation of solutions for Sturm-Liouville eigenvalue problems with generalized fractional derivative.

We analyze fractional Sturm-Liouville problems with a new generalized fractional derivative in five different forms. We investigate the representation of solutions by means of ρ-Laplace transform for generalized fractional Sturm-Liouville initial value problems. Finally, we examine eigenfunctions and eigenvalues for generalized fractional Sturm-Liouville boundary value problems. All results obtained are compared with simulations in detail under different α fractional orders and real ρ values.

Optimal control of a fractional Sturm–Liouville problem on a star graph

ABSTRACT This paper is devoted to elliptic fractional boundary value problems of Sturm–Liouville type on an interval and on a general star graph. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls.

Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation

Distributed fractional derivative operators can be used for modeling of complex multiscaling anomalous transport, where derivative orders are distributed over a range of values rather than being just a fixed integer number. In this paper, we consider the space-time Petrov–Galerkin spectral method for a two-dimensional distributed-order time-fractional fourth-order partial differential equation. By applying a proper Gauss-quadrature rule to discretize the distributed integral operator, the problem is converted to a multi-term time-fractional equation. Then, the proposed method for solving the obtained equation is based on using Jacobi polyfractonomial, which are eigenfunctions of the first kind fractional Sturm–Liouville problem (FSLP), as temporal basis and Legendre polynomials for the spatial discretization. The eigenfunctions of the second kind FSLP are used as temporal basis in test space. This approach leads to finding the numerical solution of the problem through solving a system of linear algebraic equations. Finally, we provide some examples with smooth solutions and finite regular solutions to numerically demonstrate the efficiency, accuracy, and exponential convergence of the proposed method.

Existence results of solution for fractional Sturm–Liouville inclusion involving composition with multi-maps

ABSTRACT This paper is introduced as complementary studies based on fractional Sturm–Liouville problems in a Banach space. We explore the existence results for new considered problems which can be considered as mixture of equations and inclusions. For the sake of that, we use jointly continuous composed functions with multi-valued maps and denote this form by eq-inclusion problems. The form of the solutions is calculated by the rules of Caputo derivative and the corresponding integral. The concept “continuous image of multi-valued maps” is useful to show that the strong results will be under inclusion hypothesis. The argument and fit technicals used here consider both Lipschitz and non-Lipschitz cases with using nonlinear alternative Leray Schauder type and Covitiz and Nadler theorems.

A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators

<p style='text-indent:20px;'>In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems (<inline-formula><tex-math id="M1">\begin{document}$ FSLPs $\end{document}</tex-math></inline-formula>). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators (<inline-formula><tex-math id="M2">\begin{document}$ FSLOs $\end{document}</tex-math></inline-formula>). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.

Analysis of Spectral Approximations Using Eigenfunctions of Fractional Sturm–Liouville Problems

The eigenfunctions of fractional Sturm–Liouville problems (FSLPs) serve as basis functions for the construction of efficient spectral methods for fractional differential equations. In this work, we provide a rigorous error analysis of spectral approximations using eigenfunctions of FSLPs in the uniform norm. Specifically, we first establish bounds on the coefficients in spectral expansions in terms of eigenfunctions of regular and singular FSLPs and then apply these bounds to derive error estimates for spectral approximations in the uniform norm. We also clarify the effect of the parameters in these eigenfunctions on the decay rate of the expansion coefficients. Numerical examples are provided to confirm our analysis.

Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem

ABSTRACTIn this paper, we have developed the spectral theory for a conformable fractional Sturm-Liouville problem with boundary conditions which include conformable fractional derivatives of order α, , and prove a completeness theorem and an expansion theorem. We obtain the canonical infinite product constructed by the zero set of the characteristic function of . Also, we calculate the regularized trace formula of the eigenvalues and investigate the inverse nodal problem for this problem with real-valued coefficients on a finite interval. The oscillation of the eigenfunctions for sufficiently large n is established, and an asymptotic formula for elements constructed by of nodal points is obtained. The uniqueness theorem is proved, and an effective procedure for solving the inverse problem is given. Finally, we present two examples to illustrate our theoretical findings.

Numerical approximation to Prabhakar fractional Sturm–Liouville problem

In this paper, we treat a numerical scheme for the regular fractional Sturm–Liouville problem containing the Prabhakar fractional derivatives with the mixed boundary conditions. We show that the eigenfunctions corresponding to distinct numerical eigenvalues are orthogonal in the Hilbert spaces. The numerical errors and convergence rates are also investigated. Further, we consider a space-fractional diffusion equation and study the associated fractional Sturm–Liouville problem along with the convergence analysis.