Eigenvalues of Sturm-Liouville Problem Containing an Eigenparameter in Seperated Boundary Conditions

The present paper is concerned with the asymptotic behavior and numerical computation of the eigenvalues of Sturm–Liouville problems in which the eigenvalue parameter appears both in the differential equation and in the separated boundary conditions. The potential function is assumed to be absolutely continuous and symmetric with a single-well structure. As the eigenparameter becomes sufficiently large, asymptotic approximations are obtained using the relationship between the first-order Riccati equation and the second-order Sturm–Liouville equation. Numerical results for the eigenvalues of the related problem are then derived using the element-free Galerkin method. Numerical simulations of the eigenvalues support the validity of the obtained asymptotic estimates.