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Journal ArticleDOI

A fractional approach to the Sturm-Liouville problem

20 Apr 2013-Central European Journal of Physics (Springer Vienna)-Vol. 11, Iss: 10, pp 1246-1254
TL;DR: In this paper, an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order, is presented.
Abstract: The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory.
Citations
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Journal ArticleDOI
TL;DR: This study introduces two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TF SLP-II) of order $ u \in (0,2)$.
Abstract: Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Levy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Levy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm--Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order $ u \in (0,2)$. In the regular case, the corresponding tempered differential operators are associated with tempering functions $p_I(x) = \exp(2\tau) $ and $p_{II}(x) = \exp(-2\tau)$, $\tau \geq 0$, respectively, in the regular TFSLP-I...

75 citations


Cites background from "A fractional approach to the Sturm-..."

  • ...Bas and Metin [6], Klimek and Agrawal [13], Zayernouri and Karniadakis [30], and Rivero, Trujillo, and Velasco [25] defined different classes of fractional Sturm–Liouville operators and investigated the properties of the corresponding eigenfunctions and the eigenvalues....

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  • ...[25] M. Rivero, J. J. Trujillo, and M. P. Velasco, A fractional approach to the Sturm–Liouville problem, Central European J. Phys., 11 (2013), pp. 1246–1254....

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Journal ArticleDOI
TL;DR: It is proved that these fractional Sturm-Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to F SLOs-1 and FSLOs-2 and form two sets of non-polynomial bases.

73 citations


Cites background or methods from "A fractional approach to the Sturm-..."

  • ...Moreover, in [68], a general approach of (4) not only with both the left and right sided Riemann-Liouville and Caputo fractional derivatives on a compact interval [a, b], but the Liouville fractional derivatives on R have been examined and some spectral properties, such as orthogonality of the eigenfunctions and the fact that the eigenvalues are all real valued, have been proved, but in these papers, the authors did not obtain any explicit form of eigenfunctions for (4)....

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  • ...Due to the orthogonality of the eigenfunctions of (4) [68, 44], the explicit form of the eigenfunctions has very important role to establish the theories of the spectral methods such as Galerkin, Tau, Petrov-Galerkin, collocation and pseudo-spectral methods based on the eigenfunctions obtained from equation (4)....

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  • ...It is worth observing that in [68], the authors have used some other fractional operators to construct FSLPs....

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Journal ArticleDOI
TL;DR: In this article, the authors apply the methods of fractional variational analysis to the regular fractional Sturm-Liouville eigenvalue problem and prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues.

73 citations


Cites background from "A fractional approach to the Sturm-..."

  • ..., [1,2,9,10,11,12,13,15,16,18,20]) which are integrals and derivatives of arbitrary real or complex order....

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Posted Content
TL;DR: In this article, the authors apply methods of fractional variational analysis to the regular fractional Sturm-Liouville eigenvalue problem and prove existence of countable set of orthogonal solutions and corresponding eigenvalues.
Abstract: This article is devoted to the regular fractional Sturm--Liouville eigenvalue problem. Applying methods of fractional variational analysis we prove existence of 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.

63 citations

Journal ArticleDOI
TL;DR: In this paper, a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered, and a numerical scheme is presented.
Abstract: In this paper a fractional differential equation of the Euler-Lagrange/Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.

48 citations

References
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Book
01 Jan 1999

15,898 citations


Additional excerpts

  • ...Let λ > 0 and <(α) ≥ 0, then: Dα −e = λαe−λx (27) Dα +eλx = λαeλx (28) The following rules for fractional integration by parts for the Riemann-Liouville, Caputo and Liouville fractional derivatives hold: Lemma 1....

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Book
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


Additional excerpts

  • ..., n (25) Dα b−(b− x)α−j = 0, j = 1, 2, ....

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Book
19 May 1993
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Abstract: Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations Further Results Associated with Fractional Differential Equations The Weyl Fractional Calculus Some Historical Arguments.

7,643 citations


"A fractional approach to the Sturm-..." refers background in this paper

  • ...In this section, we introduce the fractional derivatives and integrals used in this work and some their properties (see also [11], [17], [23], [25], [26], [28], [30])....

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Book
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.

7,096 citations

Book
12 Jul 1974
TL;DR: In the beginning, when having significantly cash, why don't you attempt to acquire something basic in the beginning? That's something that will guide you to understand even more in the region of the globe, experience, some places, history, amusement, and a lot more as discussed by the authors.
Abstract: Eventually, you will unquestionably discover a additional experience and realization by spending more cash. nevertheless when? realize you acknowledge that you require to acquire those every needs when having significantly cash? Why don't you attempt to acquire something basic in the beginning? That's something that will guide you to understand even more in the region of the globe, experience, some places, gone history, amusement, and a lot more?

3,379 citations