Ambarzumyan theorems on time scale and multiplicative analysis

Non-Newtonian analysis such as both time scale and multiplicative analysis offer a new perspective on mathematical problems encountered in science and engineering. In this thesis, basic concepts such as limit, continuity, derivative and integral on time scale and multiplicative analysis are introduced and the basic properties of these concepts are given. Some of the given features of these concepts, which are expressed on time scale and multiplicative analysis, are compared with classical analysis and examples are given. Differential equations are studied on time scale and multiplicative analysis. The method of uncertain coefficients and variation of parameters in classical case are defined on time scale and multiplicative analysis, and examples are solved. The meaning of direct and inverse problems is expressed through the Sturm-Liouville equation and Dirac equation system. After briefly mentioning the inverse eigenvalue problems, the Ambarzumian theorem and its proof are given for the Sturm-Liouville equation, which is the first example of the inverse problems. Then, this theorem is given for Dirac system. In the last part of the thesis, the Ambarzumyan theorem on time scale and multiplicative analysis is expressed and proven for the Sturm-Liouville equation and Dirac equation systems, which are redefined in these analysis.