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

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 ] χ 0 ( y ) = 0 , χ 1 ( y ) = 0 , where q ∈ L 1 and, χ 0 and χ 1 are continuous linear functionals defined by χ 0 ( y ) = ∫ 0 1 [ y ( t ) d ψ 1 ( t ) + y ′ ( t ) d ψ 2 ( t ) ] , χ 1 ( y ) = ∫ 0 1 [ y ( t ) d ϕ 1 ( t ) + y ′ ( t ) d ϕ 2 ( t ) ] , where χ 0 and χ 1 are independent, and ψ 1 , ψ 2 , ϕ 1 and ϕ 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.