On one algorithm for finding a solution to a two-point boundary value problem for loaded differential equations with impulse effect

Abstract

A linear two-point boundary value problem for a system of loaded differential equations with impulse effect is investigated. The parameterization method is used to solve the problem. The essence of parameterization method is that segment, where the loaded differential equation is considered, is divided into parts by loading points, and the initial problem is reduced to the boundary value problem with a parameter. The solution to boundary value problem with parameter is defined as a limit of systems sequence, consisting of the pairs of parameter and function. Parameters are defined by a system of linear algebraic equations. System of linear algebraic equations is determined by the matrices of boundary conditions, the system of loaded differential equations, and the conditions of impulse effect. An algorithm for finding the solution to linear two-point boundary value problem for the systems of loaded differential equations with impulse effect is offered. The convergence conditions of the algorithm providing the existence and uniqueness of solution to the considered problem are established. Sufficient conditions for unique solvability of the problem in the terms of initial data are received.