О ЧИСЛЕННОМ РЕШЕНИИ СИСТЕМ СУЩЕСТВЕННО НАГРУЖЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ТРЕХТОЧЕЧНЫМ УСЛОВИЕМ

Abstract

A linear boundary value problem for systems of essentially loaded ordinary differential equations with a three-point condition is considered. The considering problem is reduced to a boundary value problem for loaded ordinary differential equations with a three-point condition. Based on D.S. Dzhumabaev's parameterization method, a numerical method for solving a boundary value problem for loaded ordinary differential equations with a three-point condition is developed and an algorithm for its implementation is proposed. By partitioning the interval and introducing additional parameters, the boundary value problem for loaded ordinary differential equations is reduced to an equivalent boundary value problem with a parameter. An equivalent boundary value problem with parameters consists of the Cauchy problem for a system of ordinary differential equations with parameters, a three-point condition, and a gluing condition. The solution of the Cauchy problem for a system of ordinary differential equations with parameters is constructed using the fundamental matrix of the differential equation. Substituting the values at the corresponding points of the constructed solution into the three-point condition and the gluing condition, a system of linear algebraic equations with respect to the parameters is compiled. A numerical method for solving a boundary value problem for essentially ordinary differential equations with a three-point condition based on the solving of the constructed system is proposed. The proposed numerical method is illustrated by an example.