Investigation of solutions of some equations in mathematical physics using Nikiforov-Uvarov method

In this thesis, Nikiforov-Uvarov NU method which is used to solve second order linear differential equations, is studied. A particular solution of Heun equation which is encountered by physicists in many fields, is obtained by the NU method. The NU method is based on reducing the differential equation to a generalized hypergoemetric-type differential equation. In this context, the NU method is succeed only differential equations which have at most three singular points. In order to extend the range of equations which are solvable with the NU method, the boundary conditions of the method are changed and its extended form is obtained. It is presented that eigenvalue and eigenfunction solutions of Heun equation and its three confluent forms are attained by the extended NU method. The eigenstates of a physical system are determined from the solutions of the Schrödinger equation. The NU method can be used in order to solve Schrödinger equation for some given potential functions. In the remaining part of this thesis, it is searched whether the validity of the achievement of NU method in the integer calculus is conserved in the non-integer calculus. For this purpose, fractional NU method is derived by means of "conformable fractional derivative" definition and three different physical applications are handled in order to present the accuracy of this fractional method.