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On a shooting algorithm for Sturm-Liouville eigenvalue problems with periodic and semi-periodic boundary conditions. (English) Zbl 0804.65085

The eigenvalues of a regular Sturm-Liouville system on (0,a) with periodic and semi-periodic boundary conditions satisfy an equation (i) ϕ 1 (a,λ)+ϕ 2 ' (a,λ)=D(λ)=±2, where ϕ 1 , ϕ 2 are basic solutions of the differential equation satisfying initial conditions ϕ 1 (0,λ)=1, ϕ 1 ' (0,λ)=0, ϕ 2 (0,λ)=0, ϕ 2 ' (0,λ)=1.

The author applies the shooting method to generate a sequence λ (k) , μ (k) of numbers and functions ϕ i (x,λ (k) ), ϕ i (x,μ (k) ), i=1,2, such that λ (k) , μ (k) converge to eigenvalues λ,μ of the periodic and semi-periodic boundary value problems respectively. Starting values for λ (k) , μ (k) are based on the known behavior of D(λ).

The new algorithm is applied to three test problems, one of which is the Mathieu equation. Calculations made on a CYBER 860 computer for the first ten pairs of eigenvalues are accurate to 10 -4 .

MSC:
65L15Eigenvalue problems for ODE (numerical methods)
34B24Sturm-Liouville theory
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators