title: ÐÐ½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ ÐÐ¾ÑÐ¸ Ð´Ð»Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð°
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Analytical and numerical solution of the Cauchy problem for the Kovacs-Van equation
Creators: name: ÐÐ¾ÑÐ¾Ð»ÐµÐ²ÑÐºÐ°Ñ, ÐÑÐµÐ½Ð¸Ñ
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2021
Subjects: ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÑ
ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð°
Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¤ÑÑÑÐµ
thermal conductivity
Kovacs-Van equation
Fourier transform

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÐÐ¾ÑÐ¸ Ð´Ð»Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð°. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. ÐÑÐ¸Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð° Ðº Ð¾Ð´Ð½Ð¾Ð¼Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ. 2. ÐÐ¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ°, Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ ÐÐ¾ÑÐ¸ Ð´Ð»Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð° Ñ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¸Ð¼Ð¸ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÑÐ»Ð¾Ð²Ð¸ÑÐ¼Ð¸. 3. ÐÑÐ¾Ð²ÐµÑÐºÐ° Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð². ÐÑÐ»Ð° Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð° Ð·Ð°Ð´Ð°ÑÐ° ÐÐ¾ÑÐ¸ Ð´Ð»Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÐ¾Ð²Ð°ÑÐ°-ÐÐ°Ð½Ð°. ÐÐ½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸ Ð±ÑÐ»Ð¾ Ð½Ð°Ð¹Ð´ÐµÐ½Ð¾ ÐµÐµ ÑÐµÑÐµÐ½Ð¸Ðµ, Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐ»Ð¸ÑÑ: Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¤ÑÑÑÐµ, Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ ÐÐ°Ð¿Ð»Ð°ÑÐ°, ÑÐ¾ÑÐ¼ÑÐ»Ñ ÐÐ°ÑÐ´Ð°Ð½Ð¾, Ð¼ÐµÑÐ¾Ð´ ÐÑÐ°Ð¼ÐµÑÐ° Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð¡ÐÐÐ£. Ð§Ð¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¸Ð½ÑÐµÐ³ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ Ð²Ð¸Ð·ÑÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð² Ð±ÑÐ»Ð¸ Ð¿Ð¾Ð»ÑÑÐµÐ½Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÑÐ·ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ R. Ð¢Ð°ÐºÐ¶Ðµ Ð±ÑÐ» ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½ ÑÐ»ÑÑÐ°Ð¹ Ð½ÐµÑÐ¸Ð·Ð¸ÑÐ½ÑÑ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÑ ÑÑÐ»Ð¾Ð²Ð¸Ð¹. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð¸ Ð´Ð¾ÑÑÐ¸Ð³Ð½ÑÑÑ Ð²ÑÐµ Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð½ÑÐµ ÑÐµÐ»Ð¸. ÐÑÐ»Ð¸ Ð³ÑÐ°ÑÐ¸ÑÐµÑÐºÐ¸ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÐµÐ½Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÑÐ°Ð·Ð½ÑÑ Ð²ÑÐ¾Ð´Ð½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð², Ð´Ð²ÑÑ Ð²Ð¸Ð´Ð¾Ð² Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÑ ÑÑÐ»Ð¾Ð²Ð¸Ð¹. Ð¢Ð°ÐºÐ¶Ðµ Ð±ÑÐ»Ð° Ð²ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð° Ð¿ÑÐ¾Ð²ÐµÑÐºÐ° Ð¿ÑÐ°Ð²Ð¸Ð»ÑÐ½Ð¾ÑÑÐ¸ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ.
description: The given work is devoted to the study of the Cauchy problem for the Kovacs-Van equation. The research set the following goals: 1. Reduction of the Kovacs-Van model to a single equation. 2. Formulation, analytical and numerical study of the Cauchy problem for the Kovacs-Van equation with several initial conditions. 3. Checking the results obtained The Cauchy problem for the Kovacs-Van equation was posed. Analytically, its solution was found, in the course of the study, the Fourier transform, the Laplace transform, Cardano formulas, and the Kramer method for solving System of linear equations were used. Numerical integration and visualization of the results were obtained using the programming language R. The case of nonphysical initial conditions was also considered. As a result of the work, all the set goals were achieved. Solutions for different input parameters and two types of initial conditions were graphically presented. The correctness of the study was also checked.