Computation of the eigenvalues of Sturm-Liouville problems with parameter dependent boundary conditions using the regularized sampling method

Abstract

Abstract. The purpose in this paper is to compute the eigenvalues of SturmLiouville problems with quite general separated boundary conditions nonlinear in the eigenvalue parameter using the regularized sampling method, an improvement on the method based on Shannon sampling theory, which does not involve any multiple integration and provides higher order estimates of the eigenvalues at a very low cost. A few examples shall be presented to illustrate the power of the method and a comparison made with the the exact eigenvalues obtained as squares of the zeros of the exact characteristic functions.

Eigenvalues of Vectorial Sturm-Liouville Problems with Parameter Dependent Boundary Conditions

“…We had even to subtract terms involving multiple integrals to get sharper results when it comes to computing of the eigenvalues. The regularized sampling method has been introduced recently [1] to overcome this problem; we do not have to subtract any term involving any (multiple) integration. In fact we multiplied ( , ) − ( , ) and ( , ) − ( , ) by an appropriate function of and got the eigenvalues with much greater precision at a reduced cost.…”

“…Example 1 (Chanane [1], 1D-version taken from fom Binding and Browne [19]). Consider from the exact eigenvalues.…”

“…In [1] we introduced the regularized sampling method, a method to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) with parameter dependent boundary conditions. We subsequently used this method to compute the eigenvalues of singular and non-self-adjoint Sturm-Liouville problems.…”

A sinc-method computation for eigenvalues of Schrödinger operators with eigenparameter-dependent boundary conditions

“…In an earlier work [10] analyticity as well as growth conditions are assumed on the coefficient matrix. Then it is stated that, [9, p. 230] the problem with its general setting has a discrete spectrum.…”

On Computing of Eigenvalues of Differential Equations Q = λP with Eigenparameter in Boundary Conditions

“…Illustrative examples and tables are included in the last section. It is worthy to mention that the sampling scheme is used to approximate eigenvalues for different types of boundary value problems in (Boumenir, 1999;2000a;2000b;Chanane, 1999;2005).…”