MODELING OF DYNAMIC PROCESSES BY THE METHOD OF HYBRID INTEGRAL TRANSFORM OF EULER-BESSEL TYPE ON THE SEGMENT

Abstract

At the present stage of scientific and technological progress, especially in connection with the widespread use of composite materials, there is an urgent need to study the physical and technical characteristics of such materials that are in different operating conditions, which mathematically leads to the problems of solving a separate system of partial differential equations of the second order on a piecewise homogeneous segment with the corresponding initial and boundary conditions, in particular, the dynamics problem mathematically leads to the construction of a solution of a separate system of partial differential equations of hyperbolic type. One of the effective methods for constructing of integral representations of analytic solutions of the algorithmic nature of the problems of mathematical physics is the method of hybrid integral transforms. In this paper we construct a solution of the dynamics problem on the two-component segment of polar axis r ∈[0;R2 ] with point of conjugation by the method of hybrid integral Euler-Bessel transform. The problem of dynamics on the two-component segment of polar axis mathematically leads to the construction of a limited solution of a separate system of two partial differential equations of hyperbolic type with corresponding initial conditions, conjugation conditions and boundary conditions. Applying to this boundary-value problem the hybrid integral Euler-Bessel transform, we obtain the Cauchy problem. Finding a solution to the Cauchy problem, we apply to it the inverse hybrid integral Euler-Bessel transform. A straight integral Euler-Bessel transform on the segment of polar axis with point of conjugation is written in the form of a matrix row. The output system and the initial conditions are written in a matrix form and we apply the operator matrix row to the given problem by the rule of multiplication of matrices. As a result we obtain the Cauchy problem for the ordinary differential equation of the second order. The inverse Euler-Bessel transform is written in the form of an operator matrix column and we apply it to the constructed solution of the Cauchy problem. After completing certain transformations, we obtain the unique solution of the original problem. The constructed solutions of boundary value problems have an algorithmic character, which allows us to use them both in theoretical studies and in numerical calculations.