Abstract
The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.
Highlights
IntroductionSturm–Liouville problems that arise when modeling many real problems appearing in physics, engineering and other branches of natural science
In this study we are interested in the eigenvalues and eigenfunctions of two-intervalSturm–Liouville problems that arise when modeling many real problems appearing in physics, engineering and other branches of natural science
In this paper we have investigated a new type singular Sturm–Liouville problem
Summary
Sturm–Liouville problems that arise when modeling many real problems appearing in physics, engineering and other branches of natural science. They arise when considering Kirchoff’s law in electrical circuits, the balance of tension in elastic, the steady-state temperature in a heated rod, the vibrations of a string or the energy eigenfunctions of a quantum mechanical oscillator, in which eigenvalues correspond to the resonant frequencies or energy levels (See, [1,2]). Chen and Ho [3] used the differential transformation method (DTM) to calculate the eigenvalues of the linear Sturm–Liouville problem d dy( x ) [ p( x ). Golmankhaneh et al [4] used the homotop perturbation method (HPM), the variational iteration method (VIM) and the new iteration method (NIM) for finding approximation solutions of nonlinear
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