Solvability of Multipoint-Integral Boundary Value Problem for a Third-Order Differential Equation and Parametrization Method

Abstract

A multipoint-integral boundary value problem for a third order differential equation with variable coefficients is considered. The questions of the existence of a unique solution of the considered problem and ways of its construction are investigated. The multipoint-integral boundary value problem for the differential equation of third order with variable coefficients is reduced to a multipoint-integral boundary value problem for a system of three differential equations by introducing new functions. To solve the resulting multipoint-integral boundary value problem, a parametrization method is applied. Algorithms of finding the approximate solution to the multipoint-integral boundary value problem for the system of three differential equations are constructed and their convergence is proved. The conditions of the unique solvability of the multipoint-integral boundary value problem for the system of three differential equations are established in the terms of initial data. The results are also formulated relative to the original of the multipoint-integral boundary value problem for the differential equation of third order with variable coefficients. The obtained results are applied to a two-point boundary value problem for the third order ordinary differential equation.