Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

Abstract

Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag AbstractThis paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.

Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients

“…The formulas (20) and (21) were obtained using the integration by parts as well as the representation for the generalized Green's function g n (x, ξ) by a series (15). Note that obtained in this Section absolute errors estimates of the FDmethod significantly improve the accuracy of the estimates obtained earlier in [5].…”

“…In this article a new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the fourth order Sturm-Liouville problem (see numerical algorithm from Section 7). We consider the eigenvalue problem on a finite interval [0, X] in the Hilbert space L 2 (0, X) for the fourth order ordinary differential equation (1) obtained with other existing numerical techniques [16,17,18,19,20,21].…”

“…Section 8 illustrates the theoretical results by numerical examples. In Examples 1 and 2 the numerical results obtained with the FD-method are compared with the numerical test results obtained with other existing numerical techniques [16,17,18,19,20,21].…”

Solving inverse Sturm-Liouville problems: theory and practice

“…In many important cases [4,8,9], the correction is known in closed form. Originally proposed [22] for centered finite difference solution of the direct problem (1-2) with c 1 = c 2 = 0 , asymptotic correction was soon extended to higher order methods [11,25] and to more general differential operators [4,6,25].…”

Stability of Thin-Walled Beams with Lateral Continuous Restraint

“…In the mathematical point of view it is an eigenvalue problem. In this case it is the so called Sturm-Liouville eigenvalue problem of differential equation of fourth order [4]. Solution of this complex problem is possible using combination of selected numerical methods, e.g.…”