Fractional Sturm-Liouville Problem Research Articles

Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem

Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem

Approximation and convergence of generalized fractional Sturm-Liouville problem via integral form

Approximation and convergence of generalized fractional Sturm-Liouville problem via integral form

Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems

Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems

Spectral Expansion for Conformable Fractional Sturm- Liouville Problem on the Whole Line

In this article, we discuss a conformable fractional Sturm-Liouville boundary-value problem on the whole line. We prove the existence of a spectral function for the singular conformable fractional Sturm-Lioville problem. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function for conformable fractional Sturm-Liouville problem on the whole line.

A numerical method for fractional Sturm–Liouville problems involving the Cauchy–Euler operators

The objective of this paper is to establish a numerical scheme for the solutions of fractional Sturm–Liouville problems involving Cauchy–Euler operators. The proposed scheme is based on the normalized Φ-Legendre functions. The Sturm–Liouville operators in this work are chosen to be a combination of the left Caputo and right Riemann–Liouville type fractional Cauchy–Euler operators. The choice of fractional operators is inspired by the structure of the classical Sturm–Liouville operator. Some important properties of eigenvalues and eigenfunctions corresponding to a class of generalized fractional Sturm–Liouville operators are also investigated. In addition to the fractional Sturm–Liouville problems, we propose schemes for solutions of some terminal and boundary value problems. Furthermore, we derive the upper bounds of the errors in approximations of derivatives of the unknown functions in terms of normalized Φ-Legendre functions. The accuracy of the proposed schemes is illustrated by numerical examples.

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

SummaryThis paper develops a Müntz–Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm–Liouville problem. We first derive the necessary optimality conditions as a two‐point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz–Legendre scaling functions have been obtained, and by utilizing it, the two‐point boundary value problem has been converted into a system of algebraic equations. The ‐error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz–Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method.

A numerical approximation for generalized fractional Sturm–Liouville problem with application

In this paper, we present a numerical scheme for the generalized fractional Sturm–Liouville problem (GFSLP) with mixed boundary conditions. The GFSLP is defined in terms of the B-operator consisting of an integral operator with a kernel and a differential operator. One of the main features of the B-operator is that for different kernels, it leads to different Sturm–Liouville Problems (SLPs), and thus the same formulation can be used to discuss different SLPs. We prove the well-posedness of the proposed GFSLP. Further, the approximated eigenvalues of the GFSLP are obtained for two different kernels namely a modified power kernel and the Prabhakar kernel in the B-operator. We obtain real eigenvalues and corresponding orthogonal eigenfunctions. Theoretical and numerical convergence orders of eigenvalues and eigenvectors are also discussed. Further, the numerically obtained eigenvalues and eigenfunctions are used to construct an approximate solution of the one-dimensional fractional diffusion equation defined in a bounded domain.

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

Inverse conformable Sturm-Liouville problems by three spectra with discontinuities and boundary conditions

In this manuscript, we consider the conformable fractional Sturm-Liouville problem (CFSLP) with finite numbers of transmission conditions at an interior point in [0, ?]. Also, we study the uniqueness theorem for inverse second order of fractional differential operators by applying three spectra with a finite number of discontinuities at interior points. For this aim, we investigate the CFSLP in three intervals [0, ?], [0, p], and [p, ?] such that p ? (0, ?) is an interior point.

Legendre approximation method for computing eigenvalues of fourth order fractional Sturm–Liouville problem

In this paper, a special class of fractional Sturm–Liouville problems of order α (3<α≤4) is discussed. The method is based on utilizing the shifted Legendre polynomials on the interval [0,L] to find numerical approximations for eigenvalues and corresponding eigenfunctions. The proposed method is implemented on several numerical examples. The numerical results indicate the high performance of the method, its effectiveness, and its simplicity in use.

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Exact and Numerical Solution of the Fractional Sturm-Liouville Problem with Neumann Boundary Conditions.

In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. Finally, we present the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule.

Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

&lt;abstract&gt;&lt;p&gt;This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.&lt;/p&gt;&lt;/abstract&gt;

Inverse nodal problem for conformable Sturm-Liouville operator with jump conditions

We consider a conformable fractional Sturm-Liouville problem with the discontinuous(or jump) condition inside the interval. The asymptotic formulas for the eigenvalues, nodal parameters (nodal points and nodal lengths) of this problem are calculated by the modified Pr?fer substitutions. Also, using these asymptotic formulas, an explicit formula for the potential functions is given. After all, we discuss Lipschitz stability for the considered problem.

Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

This work deals with the pseudospectral method to solve the Sturm-Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integrodifferential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we find the roots of the characteristics polynomial of the coefficients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

Conformable fractional Sturm–Liouville problems on time scales

We consider a conformable fractional Sturm–Liouville problem on bounded time scales. Firstly, the existence and uniqueness of solutions of the conformable fractional Sturm–Liouville problem are proved. Maximal and minimal symmetric operators are introduced and some results are given. Finally, the Green's function of this problem is constructed, and an eigenfunction expansion is given.

Eigenvalues of fractional Sturm-Liouville problems by successive method

In this paper, we consider a fractional Sturm-Liouville equation of the form, − cDα 0+ ◦ Dα 0+ y(t) + q(t)y(t) = λy(t), 0 < α < 1, t ∈ [0, 1], with Dirichlet boundary conditions I 1−α 0+ y(t)|t=0 = 0, and I 1−α 0+ y(t)|t=1 = 0, where, the sign ◦ is composition of two operators and q ∈ L2 (0, 1), is a real valued potential function. We use a recursive method based on Picard’s successive method to find the solution of this problem. We prove the method is convergent and show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.

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.