Gateaux and Frechet derivatives and their applications

In this thesis, the use of Newton Method in the approximate solution of the nonlinear operator equations is investigated.In the first chapter, a brief history of Newton method is given. Some basic mathematical concepts are given in the second chapter. Some of them can be listed as linear and nonlinear operators, Lipshcitz condition, Banach space, operator equation and its solution.In the third chapter, detailed information about the Frechet and Gateaux derivatives which are necessary for differentiating the functionals that used in Newton method is presented.In the following chapter, the application of Newton method to the nonlinear differential equations and the nonlinear equation systems is explained. Additionally, by giving some examples, theoretical and practical results are displayed in the last chapter.Also, the packet programs of the solutions obtained by Newton method are taken place in the appendices.Keywords: Approximate solution, Newton method, Frechet derivative, Gateaux derivative.