Lazurchak I. Functionally discrete methods for boundary value problems and their implementation by computational mathematics

In this thesis a two-sided functional-discrete (FD-) method is developed for linear and quasilinear ordinary differential equations and their systems with generalized boundary conditions (including quasi-periodic, Bitsadze-Samarsky boundary conditions). FD-method is applied to singular differential equations with boundary conditions defined on an semiaxis and for the Helmholtz equation for multi-connected domains on a plane. FD-methods for Sturm-Liouville problems containing an eigenparameter in boundary conditions, as well as multipoint and integral boundary conditions, are developed and justified. For matrix Sturm-Liouville problems with inseparable boundary conditions an efficient algorithmic implementation on the basis of the proposed approach is developed. Based on the developed set of programs a number of numerical experiments are conducted using well-known testing examples. These tests show full consistency with the theory and provide new insight to enhance the functionality of computer mathematics research in theoretical and numerical studies.