Sember D. A functional discrete method for solving a nonlinear Klein-Gordon equation.

The dissertation is devoted to the development and generalization of the numerical-analytical method for solving a nonlinear Klein-Gordon equation. Based on the main idea of the FD-method for solving operator equations of general type, a numerical-analytical method for solving nonlinear Klein-Gordon equation with bounded and unbounded nonlinearities was constructed. It was proved that for quite a wide range of input data (nonlinearity, initial conditions etc.) the FD-method has a superexponential convergence rate. As an auxiliary statements it was also proved a few theorems about existence and uniqueness of local solutions for both Cauchy and Goursat problems. The thesis contains quite a deep investigation of algorithmic implementation approaches for the proposed methods. It was shown that to achieve the best performance of the algorithms their analytical parts have to be combined with well known techniques of numerical integration.