Qualitative and quantitative properties of eigenvalue problems with eigenparameter in the boundary conditions

We have examined the general differential equation

y" + q(x)y - -λy

with boundary conditions

(cosγ)y(L) - (sinγ)y(L) - 0

and

-[β₁y(0) + β₂ y'(0)] = λ[β₁’y(0) + β₂’y'(0)].

The qualitative study of this problem Involves the Investigation of the number of zeros for each elgenfunctlon and the separation of the zeros of the elgenfunctlons.

The quantitative study Involves numerical solutions of the above differential equation. The IMSL computer programs ZPOLR and EQRT2S were used In this study and can be found In the program library at The University of Tennessee. We used ZPOLR to solve for the eigenvalues of a trldlagonal symmetric matrix and EQRT2S to solve for the roots of a polynomial. Other numerical methods Included Newton's method for solving differential equations and an algorithm to solve for the angle 8 using a bisection approach.

The qualitative findings show that the zeros of the elgenfunctlons separate with a minor discrepancy when a certain angle crosses a multiple of w. The nth elgenfunctlon has n zeros until this angle crosses a multiple of JT and then the nth elgenfunctlon has (n-1) zeros.

The qualitative findings show that the finite difference and bisection methods work well in solving for the eigenvalues. The series method, while solving for the first few zeros successfully, also generates negative and extraneous roots and is therefore considered to be unreliable.