Our purpose in this paper was to determine the eigenvalues of the boundary value problem with separated and parameter dependent boundary conditions when the potential function is integrable and symmetric. We first provided the asymptotic estimates of the eigenvalues of the related problem. Second, we computed numerical results for the eigenvalues using the interpolating element free Galerkin method. Then, a few examples are presented to illustrate the power of the method and compare the asymptotical and numerical results for consistency and validity.

In this work, we asymptotically calculate the eigenvalue of a boundary value problem that involves an integrable and symmetric potential over a defined interval and includes eigenvalues in the boundary conditions. We apply the Riccati equation to derive the asymptotic solutions. To validate our findings, we compare them with numerical results obtained using the element–free Galerkin method. The comparison confirms the consistency and accuracy of the asymptotic approach.