Spectral theory for various derivative operators on time scales andsome applications

Time scale theory is a mathematical framework that allows the unification of continuous and discrete structures under a single framework using dynamical systems. The time scale approach, which allows problems defined on different structures to be solved with a single theory, offers significant advantages in both theoretical and applied problems. Fractional calculus, one of the key concepts in this study, is a mathematical discipline that generalizes the classical concepts of derivatives and integrals to define non-integer derivatives and integrals. In cases where the classical derivative operator is inadequate, systems constructed using fractional derivative operators offer more flexible and realistic solutions. In many fields such as economics, psychology, physics, and applied mathematics, systems can be more effectively represented using fractional derivatives. The proportional derivative operator on time scale has the potential to successfully model the behavior of systems defined on both discrete and continuous structures and to unify these dynamic processes. In this thesis, the two fundamental theories of classical spectral analysis, the Sturm–Liouville and Dirac problems, are revisited on time scale and investigated using different derivative operators. These problems, defined using the proportional derivative operator on time scale, are examined in terms of their solution structures, eigenvalue behaviors, and some important spectral properties. Generalizations made on time scale are also supported by fractional calculus approaches, enabling the solution of related problems within a common framework for both continuous and discrete cases. Furthermore, Ambarzumian theorems of classical spectral theory are revisited on time scale for both Sturm–Liouville and Dirac problems, and are expressed and proven by generalizing them to the delta, nabla, proportional delta, proportional nabla, and diamond alpha derivatives. The proportional derivative used in this study combines the concepts of fractional calculus with time scale theory, enabling the analysis of dynamical systems on different structures. Mathematical modeling, another topic in this study, aims to represent real-world processes with mathematical structures. In this context, the use of time-scale and proportional derivatives in modeling studies offers more comprehensive and flexible solutions for models. The models chosen in this study were restructured using delta derivatives and proportional derivatives on time scale instead of classical derivatives, in line with the nature of dynamic systems, to more realistically represent the processes. The resulting solutions were examined in detail for different time scales and different proportional orders.