On a nonlocal Sturm–Liouville problem with composite fractional derivatives
30 Jan 2021-Mathematical Methods in The Applied Sciences (John Wiley & Sons, Ltd)-Vol. 44, Iss: 2, pp 1931-1941
About: This article is published in Mathematical Methods in The Applied Sciences.The article was published on 2021-01-30. It has received 3 citations till now. The article focuses on the topics: Sturm–Liouville theory & Fractional calculus.
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TL;DR: In this paper , the authors considered the fractional Sturm-Liouville problem with homogeneous Neumann boundary conditions and presented the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule.
Abstract: 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.
6 citations
TL;DR: The spectral properties of Sturm-Liouville operators on function spaces restricted by homogeneous Dirichlet boundary conditions have been studied in this paper for regular eigenvalue problems.
Abstract: 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.
4 citations
TL;DR: In this article , the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm-Liouville equation with truncated Marchaud fractional derivative term by two-parameter method is discussed.
Abstract: In this paper, we discuss the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm–Liouville equation with truncated Marchaud fractional derivative term by two-parameter method. To this end, we first study the properties of eigenvalues for a two-parameter nonlocal Sturm–Liouville eigenvalue problem. We obtain the two-parameter nonlocal Sturm–Liouville eigenvalue problem that has countable number of simple real eigenvalues. Meanwhile, the asymptotic behavior of eigenvalues is studied by aid of the analytic perturbation theory.
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02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
01 Jan 2015
3,828 citations
"On a nonlocal Sturm–Liouville probl..." refers background in this paper
...1) with y(0) = 0 = y(1) or y(0) = 0 = y′(1) − μ(I1−α 0+ ( D1−y))(1) is simple....
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...and (D1−y)(x) = ( Dα1− [ y(t) − n−1 ∑ k=0 (−1)k y (k)(1) k! (1 − t)k ]) (x), x ∈ (0, 1], (2....
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...1) with y(0) = 0 = y(1) or y(0) = 0 = y′(1) − μ(I1−α 0+ ( D1−y))(1) by means of operator theory and the results about the initial value problems (1....
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...1) with y(0) = 0 = y(1) or y(0) = 0 = y′(1) − μ(I1−α 0+ ( D1−y))(1) in this paper....
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TL;DR: In this paper, the link between the treatments of constrained systems with fractional derivatives by using both Hamiltonian and Lagrangian formulations is studied, and it is shown that both treatments for systems with linear velocities are equivalent.
180 citations
TL;DR: 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.
Abstract: 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.
130 citations
28 May 2012
TL;DR: In this paper, a fractional Sturm-Liouville operator (FSLO) and a regular FSLP are introduced, and the properties of the eigenfunctions and eigenvalues of the operator are investigated.
Abstract: In this paper, we define a Fractional Sturm-Liouville Operator (FSLO), introduce a regular Fractional Sturm-Liouville Problem (FSLP), and investigate the properties of the eigenfunctions and the eigenvalues of the operator. We demonstrate that these properties are similar and in some cases identical to those for Integer Sturm-Liouville Operator. We briefly introduce a Reflected Fractional Sturm-Liouville Operator (RFSLO) and demonstrate that neither the FSLO nor the RFSLO are symmetric. We shall consider the topic of reflection symmetry in a subsequent paper.
65 citations