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

Abstract

At the current stage of scientific and technical progress, there is a need to study the physical and technical characteristics of composite materials, which are increasingly used for the production of various parts. Modeling physical processes in such materials, in particular the diffusion process, mathematically leads to the problem of solving a separate system of partial differential equations of the second order of the parabolic type on a piecewise homogeneous interval with certain initial and boundary conditions, since for different materials physical processes are described by different differential operators. One of the most effective methods of obtaining integral images of analytical solutions of the algorithmic nature of such mathematical physics problems is the method of hybrid integral transforms, which arose in the second half of the 20th century. In this work, the solution of the diffusion problem on the two-component segment [0;R2] with one point of conjugation is obtained using the Euler-Bessel hybrid integral transform. Mathematical modeling of diffusion processes in two-component materials mathematically means constructing a limited solution of a separate system of two partial differential equations of the parabolic type with certain boundary conditions, initial conditions, and conjugation conditions. Applying to such a boundary-value problem the previously constructed Euler-Bessel hybrid integral transform on a segment, we obtain the Cauchy problem for an ordinary differential equation. Having found the solution of the Cauchy problem, we apply to it the inverse Euler-Bessel hybrid integral transform. The direct Euler-Bessel hybrid integral transform on a segment with one point of conjugation can be written in the form of a row matrix. If at the same time the original system and initial conditions are written in matrix form, then, applying the row operator matrix to such a problem according to the rule of matrix multiplication, we get the Cauchy problem for an ordinary differential equation of the first order, which is easily solved. If we write the inverse Euler-Bessel hybrid integral transform in the form of a column operator matrix, then, applying it to the resulting solution of the Cauchy problem, after performing elementary transformations, we obtain a unique solution of the original problem in analytical form. Constructed solutions of boundary value problems are algorithmic in nature, which allows them to be used both in theoretical studies and in numerical calculations.