An appropriate method for eigenvalues approximation of sixth-order Sturm-Liouville problems by using integral operation matrix over the Chebyshev polynomials

In this study, we suggest an efficient and useful method by using Chebyshev polynomials to approximate eigenvalues of a non-singular sixth-order Sturm-Liouville equation. This method uses Chebyshev polynomials and integration operation matrix for approximating of function and its derivatives. We convert the original problem for finding the eigenvalues of Sturm-Liouville problem to a problem of finding the eigenvalues matrix. The obtained results demonstrate appropriate reliability and efficiency of the proposed method. INTRODUCTION In recent years numerical methods for solving of eigenvalue problems have been considered by researchers because these problems have numerous applications in physics and engineering scope, such as pendulums, rotating and vibrating shaft, viscous flow between rotating cylinders, thermal instability of fluid spheres and spherical shells, earthquake behavior and ring structures (see [6, 7, 8]). In this paper, we present a numerical method to approximate eigenvalues of sixth-order Sturm-Liouville problem. ( p(x)y′′′(x) )′′′ = (s(x)y′′(x))′′ − (r(x)y′(x))′ − λ(w(x) − q(x))y(x), a x b, (1) with boundary condition: α jy j)(a) + β jy j)(b) = 0, 0 ≤ j ≤ 5, (2) where p, s, r, w and q are piecewise continuous functions with p(x) > 0, and w(x) > 0 for all a < x < b, and α j, β j are constant. Eq. (1) is often referred to the circular ring structure with parabolic variable thickness as constraints, [6]. Note that eigenvalues as studied in [2, 3], for Eq. (1) and boundary condition (2) form a non-decreasing sequence λ0 λ1 λ2 · · · λn, so that lim n→∞ λn = ∞, and maximal degree of each eigenvalue is 3. The solution of these problems always has real form and often analytical answers are not accessible. Therefore developing and implementing of a newmethod to obtain an approximate solution has received considerable interest of researchers in recent years. For instance, Lesnic and Attily [8] used the ADM method, Greenberg and Marletta [4, 5] used Theta Matrix (SLEUTH). Recently, Siyyam and Syam [13] implemented iterated variation method and [1] used conjugating the Chebyshev collocation-path following method. The present work is motivated by desire to approximate eigenvalues and eigenfunctions of Eqs. (1) and (2) using an efficient and useful numerical algorithm based on integration of truncated Chebyshev polynomials series. Also we divided original interval, [a, b] to some subintervals. The results obtained are good and satisfactory. We demonstrate the reliability and efficiency of our method by two numerical examples. Application of Mathematics in Technical and Natural Sciences AIP Conf. Proc. 1684, 090001-1–090001-7; doi: 10.1063/1.4934326 © 2015 AIP Publishing LLC 978-0-7354-1331-3/$30.00 090001-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 188.136.149.34 On: Mon, 02 Nov 2015 09:37:50 This paper is organized as follow: A brief description of Chebyshev polynomials properties and also integration operation matrix of discrete Chebyshev polynomials series is presented in Section 2. A description of used method for discrete Sturm-Liouville problem (1) with boundary condition (2) and also the numerical method of finding eigenvalues using special technique are considered in Section 3. Finally, numerical examples with their results and comparison with other similar articles are discussed in Section 4. SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS Now, we mention some important properties of Chebyshev polynomials applied in this research [14]: a) Chebyshev polynomials with general term are Tn(x) = {cos(arccos nx)}n=0; −1 x 1, and with recursive terms T0(x) = 1, T1(x) = x, Tn+1(x) = 2xTn(x) − Tn−1(x); b) Chebyshev polynomials with weighted function w(x) = (1 − x2)− 2 in interval [−1, 1] are orthogonal and