A numerical algorithm for the solution of the sine gordon equation in a class of generalized functions

In this three-part thesis, a simple and effective numerical algorithm is proposed to solve sine Gordon equation in a class of generalized functions. As it is known, to discretize any differential equation into finite differences, the solution must have higher order derivatives. However, for a nonlinear equation, the order of the differentiability of the solution is generally less than that of the equation itself. This makes it impossible to use the method of finite differences. Consequently, in the first part of the thesis, we construct the necessary infrastructure. Indeed, the general theory of the generalized functions and the concepts of the generalized function, generalized derivative and weak solution are given. Secondly, in order to discuss the solution of sine Gordon equation, some methods such as seperation of variables, Bäcklund transformation and inverse scattering are used. These methods work in the Schwartz class of functions. Finally, a numerical algorithm for solving sine Gordon equation are presented. To do so, a specially constructed auxiliary problem, which is equivalent to the main problem, is suggested. The differentiable property of the auxiliary problem is one degree higher than that of the main problem. Key Words: Generalized Functions, Weak Solution, Solitons, Sine Gordon Equation, Finite Differences Method