Q-differential transform method and its applications

The differential transformation method is an effective method for solving many linear or non-linear ordinary or partial differential equations. The fact that it is easy to apply compared to other methods and that it can be applied effectively to non- linear equations is one of the reasons why the method is preferred. One of the current issues of recent years is q-calculus. Since the q-derivative here is defined only by the difference ratio without limit, q-calculus is also known as unlimited calculus. Investigations on the q-differential transformation method as q-analogue of the differential transformation method have been made and continue to be made. In this thesis, the one-dimensional q-differential transformation method and its basic properties are examined and its application on some q-initial value problems is given. In the first part of the thesis, literature summary about q-calculus, differential transformation method and q-differential transformation method is given. In the second part of the thesis, basic definitions, theorems and properties related to q- calculus and differential transformation method are given. In the third part of the thesis, the q-differential transformation method is defined and the basic theorems and proofs about the method are given. The fourth chapter of the thesis is devoted to the application of the q-differential transformation method to q-differential equations. In this context, initial value problems involving linear and non-linear q-differential equations such as q-Lane-Emden differential equation, q-Riccati differential equation, q-integro-differential equation are solved with this method, and exact solutions or serial solutions are obtained. By the definition of the q-derivative, when the limit for q→1 is taken, the findings and results in the q-calculus are reduced to their classical counterparts. In the problems examined within the scope of the thesis, when the limit for q→1 is taken, the classical equivalents and solutions are obtained. In the last part of the thesis, other conclusions and suggestions are given.