A Numerical Study of Eigenvalues and Eigenfunctions of Fractional Sturm-Liouville Problems via Laplace Transform
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DOI: 10.1007/s13226-020-0436-2
A NUMERICAL STUDY OF EIGENVALUES AND EIGENFUNCTIONS OF FRACTIONAL STURM-LIOUVILLE PROBLEMS VIA LAPLACE TRANSFORM Mahnaz Kashfi Sadabad, Aliasghar Jodayree Akbarfam and Babak Shiri University of Tabriz, Faculty of Mathematical Science, Tabriz, Iran e-mails: [email protected]; [email protected]; [email protected] (Received 28 April 2018; after final revision 14 April 2019; accepted 29 April 2019) 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. Key words : Fractional Sturm-Liouville problem; Caputo fractional derivative; Laplace transform. 2010 Mathematics Subject Classification : 34B24, 65N25, 35R11.
1. I NTRODUCTION Sturm-Liouville problems have been known since 1836. The concept of Sturm-Liouville problems plays an important role in mathematics and physics [4]. In recent years, researchers have focused on a certain generalization type of the classical Sturm-Liouville problem to the fractional one. Many authors have studied the Sturm-Liouville problem in which the second order derivative is replaced by a Caputo fractional derivative. Study of this type of Sturm-Liouville problem is important for solving diffusion equations and oscillator problems [11]. We consider the Caputo fractional Sturm-Liouville problem (CFSLP) of the form C α 0 Dt y(t)
+ (λr(t) − q(t)) y(t) = 0, t ∈ [0, 1],
(1)
858
M. K. SADABAD, A. J. AKBARFAM AND B. SHIRI
where q and r are real-valued constant functions, 1 < α ≤ 2 and
C α 0 Dt
is the Caputo fractional
derivative. The boundary conditions are ay(0) + by 0 (0) = 0, cy(1) + dy 0 (1) = 0,
(2)
where a, b, c, d ∈ R, a2 + b2 6= 0 and c2 + d2 6= 0. The problems (1)-(2) are solved using some numerical schemes such as Adomian decomposition method [2], Predictor-corrector algorithm with Newton method [8], Homotopy analysis method [1], Homotopy Perturbation Method [3, 11], and Augmented-RBF Method [5]. The aim is to solve the problems (1)-(2) with constant coefficients. As a comparison with other methods, we use the Laplace transform to obtain algebraic equations and introduce a simple and effective method for approximating the eigenvalues and the eigenfunctions of the CFSLPs with a high convergence rate. The structure of the paper is as follows: In Section 2, we recall some definitions and results related to fractional calculus and Laplace transform. In Section 3, we construct the method using the Laplace transform. In Section 4, a convergence analysis is obtained. In Section 5, some examples is provided to show simplicity and efficiency of the method. 2. P RELIMINARIES In this section we recall some of the basic definit
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