Fractional Sturm-Liouville Problem Research Articles

Homogeneous Robin Boundary Conditions and Discrete Spectrum of Fractional Eigenvalue Problem

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

Discontinuous fractional Sturm–Liouville problems with transmission conditions

The main purpose of this study is the investigation of discontinuous Sturm–Liouville problem with fractional derivatives. We give an operator theoretic framework of the problem under consideration. Namely, we define an operator A in the Hilbert space L2[−1,1], the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem, respectively. Then, we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function.

Theory of discrete fractional Sturm–Liouville equations and visual results

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grunwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.

Existence of positive solutions for regular fractional Sturm-Liouville problems

Existence of positive solutions for regular fractional Sturm-Liouville problems

On fractional-Legendre spectral Galerkin method for fractional Sturm–Liouville problems

In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples.

A Reliable Method for Solving Fractional Sturm–Liouville Problems

In this paper, a reliable method for solving fractional Sturm–Liouville problem based on the operational matrix method is presented. Some of our numerical examples are presented.

Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions

In this article, we study Sturm-Liouville Equations (SLEs) in the frame of fractional operators with exponential kernels. We formulate some Fractional Sturm-Liouville Problems (FSLPs) with the differential part containing the left and right sided derivatives with exponential kernels. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators (FSLOs) by using fractional integration by parts formulas. The nabla discrete version of our results are also established. Finally, an example is analyzed to illustrate the method of solution.

Homogeneous Robin Boundary Conditions and Discrete Spectrum of Fractional Eigenvalue Problem

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subjected to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space.

Exact and numerical solutions of the fractional Sturm–Liouville problem

Abstract In the paper, we discuss the regular fractional Sturm-Liouville problem in a bounded domain, subjected to the homogeneous mixed boundary conditions. The results on exact and numerical solutions are based on transformation of the differential fractional Sturm-Liouville problem into the integral one. First, we prove the existence of a purely discrete, countable spectrum and the orthogonal system of eigenfunctions by using the tools of Hilbert-Schmidt operators theory. Then, we construct a new variant of the numerical method which produces eigenvalues and approximate eigenfunctions. The convergence of the procedure is controlled by using the experimental rate of convergence approach and the orthogonality of eigenfunctions is preserved at each step of approximation. In the final part, the illustrative examples of calculations and estimation of the experimental rate of convergence are presented.

Fractional Sturm–Liouville problems for Weber fractional derivatives

In this paper, we introduce the regular and singular fractional Sturm–Liouville problem (SLP) where the operator is the Weber fractional derivative of order α. We show then the eigenvalues of fractional SLP are real and the eigenfunctions corresponding to distinct eigenvalues are orthogonal. We also consider a Jacobi type singular fractional SLP and treat the behaviours of its eigenvalues. We further introduce the finite fractional Fourier transform and use this transform for solving the space-fractional diffusion equation problem.

An application of comparison criteria to fractional spectral problem having Coloumb potential

In this study, the zeros of eigen functions of spectral theory are considered in fractional Sturm-Liouville problem. The 1st and 2nd comparison theorems for fractional Sturm-Liouville equation with boundary condition and their proofs are given. In this way, our new approximation will contribute to construct fractional Sturm-Liouville theory. Also, its an application is given in case of Coulomb potential and the results are presented by a symbolic graph.

On the fundamental solutions of a discontinuous fractional boundary value problem

The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the constructed operator in Hilbert spaces.

Existence of solutions for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator

This paper is concerned with the solvability for fractional Sturm-Liouville boundary value problems with p(t)-Laplacian operator at resonance using Mawhin’s continuation theorem. Sufficient conditions for the existence of solutions have been acquired, and they would extend the existing results. Furthermore, an example is provided to illustrate the main result.

Finite Fractional Sturm–Liouville Transforms For Generalized Fractional Derivatives

In this article, we introduce the regular and singular fractional Sturm–Liouville problems with the Hilfer and Hilfer–Prabhakar derivatives. We show that these problems with the corresponding boundary conditions have real eigenvalues and their eigenfunctions are orthogonal. Also, the finite fractional Sturm–Liouville transforms and their inversion formulas are established, and as an application, the formal solution of the fractional Laplace equation in prolate spheroidal coordinates is obtained using the finite Legendre transform.

Regular Fractional Differential Equations in the Sobolev Space

Abstract In this paper regular fractional Sturm-Liouville boundary value problems are considered. In particular, new inner products are described in the Sobolev space and a symmetric operator is established in this space.

Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm–Liouville problem

ABSTRACTIn this paper, we discuss a class of eigenvalue problems of fractional differential equations of order with variable coefficients. The method of solution is based on utilizing the fractional series solution to find theoretical eigenfunctions. Then, the eigenvalues are determined by applying the associated boundary conditions. A notable result, for certain cases, is that the eigenfunctions are characterized in terms of the Mittag-Leffler or semi Mittag-Leffler functions. The present findings demonstrate, for certain cases, the existence of a critical value at which the problem has no eigenvalue (for ), only one eigenvalue (at ), a finite or infinitely many eigenvalues (for ). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples.

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems (SLPs) on bounded domains is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.

Numerical Computation of Fractional Second-Order Sturm-Liouville Problems

In this article, we investigate the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. The fractional derivative in this paper is in the conformable fractional derivative sense. We implement the reproducing kernel Hilbert space method to approximate the eigenvalues. Convergence of the proposed method is discussed. The main properties of the Sturm-Liouville problem are investigated. Numerical results demonstrate the accuracy of the present algorithm. Comparisons with other methods are presented.

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

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.

The fractional Sturm–Liouville problem—Numerical approximation and application in fractional diffusion

The numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional Sturm–Liouville equation is subjected to the mixed boundary conditions. This non-integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right Caputo derivative. The numerical eigenvalues are thus real, and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite-dimensional Hilbert space. The advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite-dimensional weighted Lebesgue integrable function space. The developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1D space-time fractional diffusion problem in a bounded domain.