About one approach for solving bounary-value problems of differential equation of pantograph type

Abstract

Pantograph equations have been studied for a long time. In 1940, K. Mahler introduced functional differential equations of this type into number theory. In 1971, J. Ockendon used a functional differential equation with a transformed argument, , to describe the dynamics of the pantograph of an electric locomotive. Subsequently, Pantograph-type equations were studied in the works of many authors. This article considers a boundary value problem for a differential equation of the pantograph type. To solve the posed boundary value problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we denote the value of the function at the initial point of the segment under consideration by a parameter and perform a change of variable Then the solvability of the original boundary value problem is reduced to studying the solvability of the resulting Cauchy problem for the original equation and to a linear algebraic equation to determine the introduced parameter. Next, using the method of successive approximations, we find solutions to the Cauchy problem for the Pantograph type equation. The convergence of the resulting sequence and the convergence of its solution to the solution of the Cauchy problem of Pantograph type are proved. By requiring continuity of the free term, we establish its unique solvability. Substituting the resulting solution into a linear algebraic equation to determine the entered parameter, we calculate the values of the entered parameter using the original data. Substituting the resulting expressions into we find the solution to the original problem . And assuming the unique solvability of a linear algebraic equation, we establish the solvability of the boundary value problem for an equation of Pantograph type.