Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems

Abstract

In this paper, we shall use the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems with nonlocal conditions {(- y″ + q (x) y = λ y, x ∈ [0, 1]; χ0 (y) = 0, χ1 (y) = 0,) where q ∈ L1 and, χ0 and χ1 are continuous linear functionals defined by χ0 (y) = ∫01 [y (t) d ψ1 (t) + y′ (t) d ψ2 (t)], χ1 (y) = ∫01 [y (t) d φ{symbol}1 (t) + y′ (t) d φ{symbol}2 (t)], where χ0 and χ1 are independent, and ψ1,ψ2, φ{symbol}1 and φ{symbol}2 are functions of bounded variations. Integration is in the sense of Riemann-Stieltjes. A few numerical examples will be presented to illustrate the merits of the method, and comparisons will be made with the exact eigenvalues when they are available. © 2009 Elsevier Ltd. All rights reserved.