APPLICATION OF THE PARAMETRIZATION METHOD TO INTEGRO-DIFFERENTIAL EQUATIONS OF THE PANTOGRAPH TYPE

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Abstract

The first equations of the 1940 pantograph were considered in the works of Mahler. He used functional differential equations with compression arguments in number theory. He used functional differential equations  with compression arguments in number theory. In 1971, J. Ockendon used functional differential equations with transformed arguments of the form to describe the motion of an electric locomotive pantograph. Recently, pantograph type equations have been widely used. For example, a configurable (pantograph) phone device holder, a retractable (pantograph) microphone device, etc.

In this article, on the segment  discusses a two-point boundary value problem for systems of integro-differential equations of the pantograph type. To solve this problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we divide the segment in question into    parts. Let's denote the value of the desired function at the starting point of each segment through the parameters and replace it in the intervals . Then the problem under consideration is formally divided into two parts, i.e., a system of linear algebraic equations with respect to the introduced parameters and to the Cauchy problem for a system of integro-differential equations of the pantograph type. Thus, the problem is reduced to a closed system for determining the solution of a special Cauchy problem for systems of integro–differential equations of the pantograph type and a system of linear equations. Based on this, an algorithm for determining the solution of the initial problem is proposed.