SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR FUNCTIONAL-DIFFERENTIAL EQUATIONS OF PANTOGRAPH TYPE
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Abstract
The first equations of the pantograph type were considered in the works of M. Mahler in 1940, which laid the foundation for the use of functional-differential equations with argument compression in number theory. In 1971, J. Ockendon used such equations to model the movement of the current collector of an electric locomotive. In recent decades, pantograph-type equations have been widely used in engineering and construction: adjustable (pantograph) telephone holders, retractable (pantograph) microphone systems, etc.
This article studies a two-point boundary value problem for a system of functional-differential equations of the pantograph type. The parameterization method proposed by Professor D. Jumabayev is used to find a solution. The interval is divided into parts of equal length; the values of the desired function are determined by parameters at the initial point characteristic of each part, and substitutions of a specific form are introduced in the inner intervals.
As a result, the original boundary value problem is formally decomposed into two interrelated subproblems:
- a special Cauchy problem for the functional-differential system of the pantograph type;
- a system of linear algebraic equations with respect to the input parameters.
The special Cauchy problem for the functional-differential system of the pantograph type and the system of linear algebraic equations with respect to the input parameters obtained in this way form a closed system, and this closed system completely determines the solution of the original boundary value problem. The article presents a computational algorithm based on the above structure and discusses its effectiveness in the numerical analysis of pantograph-type functional-differential models.
