Equiconvergence with Fourier Series for Non-Classical Sturm-Liouville Problems

Abstract

Sturm-Liouville type boundary-value problems arise in many engineering and scientific disciplinesas the mathematical modeling of systems and processes in the fields of chemistry, aerodynamics, electrodynamicsof complex medium or polymer rheology. For example the vibration of a homogeneous loaded strings, the earth’sfree oscillations, the interaction of atomic particles, sound, surface waves, heat transfer in a rod with heat capacityconcentrated at the ends, electromagnetic waves and gravitational waves can be solved using the Sturmian theory. Alarge class of physical problems require the investigation of the Sturm-Liouville type problems with discontinuities.Examples are vibration problems under various loads such as a vibrating string with a tip mass or heat conductionthrough a liquid solid interface.In this study we shall investigate some properties of the eigenfunctions of one discontinuous Sturm-Liouville Problem. We shall prove some preliminary results related to the basic solutions, Green’s function, resolvent operatorand selfadjointness of the considered problem. Particularly we shall present a new approach for constructing theGreen’s function which is not standard one generally found in textbooks. The obtained results are implemented tothe investigation of the basis properties of the system of eigenfunctions in modified Hilbert spaces. Finally, we shallshow that the eigenfunction expansion series regarding the convergence behaves in the same way as an ordinaryFourier series.