INVERSE CONFORMABLE STURM-LIOUVILLE PROBLEM

Abstract

Bu çalışmada, Sturm-Liouville probleminden (SLP) ile ilgili olup; uyumlu türev yardımıyla hem direkt hem de teras problemin çözümü incelenmiştir. İlk olarak gerek mühendislikte gerek ise diğer alanlarda önemli bir yere sahip olan SLP probleminin temel özellikleri verilmiş, türevin sıfır ve bir değerleri arasındaki kesirli değerleri kullanılarak dönüşüm operatörünün varlığını ispatını verdik. Burada Hiperbolik tip kısmi türevli bir diferensiyel denklem ve bunula ilgili sınır şartları şeklinde elde edilen problem için bir Fredholm integral denklemi elde ettik. Literatürde ters nodal problem olarak bilinen problem için farklı sınır şartlarında özdeğerler, özfonksiyonlar ve nodal parametreler için asimptotik formüller verdik. Ve bunlar yardımıyla potansiyel fonksiyon için formüller verdik. Daha sonra bazı ek koşullar altında potansiyel fonksiyon için bazı teklik teoremlerini ispat edip özel durumda bu sonuçlardan yola çıkarak Ambarzumyan teoremine ulaştık. Son olarak gecikmeli SLP için ters nodal problemin çözümünü verip problemin dengeli olduğunu gösterdik. Genel olarak elde ettiğimiz sonuçların ? = 1 için klasik sonuçlar ile aynı olduğunu gösterdik.

This research work talks about Sturm-Liouville Problem (SLP); the direct and the inverse nodal forms of the problem (with / without delay) using a newly established non-integer value derivative called the conformable derivative. At first, looking in to the vitality of the SLP in the models of the real life problems addressing scientific, engineering, economic, etc, issues and the fact that most of these problems are better described using non-integer value derivative, we considered the conformable derivative method and proved the transformation operator for the SLP using this new differentiation method on a classical Sturm-Liouville Operator. There, we obtained a Hyperbolic partial differential equation and some suitable conditions for nucleus function N(x, ? ), then we finally obtained a Fredholm integral equation and the proof is validated by taking ? = 1 which returns the original/classical results. In addition, we gave various basic definitions, properties, lemmas and theorems used in this thesis, and they are mostly under this new derivative. Next, we considered the general conformable SLP and then established the spectral properties; the eigenvalues, the eigenfunctions, the nodal points, the nodal lengths, for some special boundary conditions and after that we calculated the potential functions and proved its uniqueness, in the direct case, using these spectral parameters as the direct and inverse of the conformable SLP. We considered and proved the Ambarzumyan's theorem for the conformable derivative SLP with additional condition. Lastly, we extended the conformable SLP and its inverse problems to the constant delay problem which is also new in the literature. There we similarly obtained the spectral properties for the problem in two different set of boundary conditions and we also expressed some important results like the proof for the specification of the spectrum in one case and proving the stability of the solution in the other case. New results, as an extension of the classical SLP (with and without delay) were established in this work in order to extend the concepts of the SLP to non-integer derivative phenomena, and the corresponding results in the classical cases of the problems discussed in this research documents can be obtained at ? = 1 and also the delay concept can be relax by taking ? = 0.