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ÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼Ñ Ð°Ð½ÑÐ»Ð°
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 research of the Cauchy problem for the Guyer-Krumhansl equation
Creators: name: ÐÐ°ÑÐµÐµÐ²Ð°, ÐÐ¸Ð»ÐµÐ½Ð°
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
Ð±Ð°Ð»Ð»Ð¸ÑÑÐ¸ÐºÐ¾-Ð´Ð¸ÑÑÑÐ·Ð¸Ð¾Ð½Ð½Ð°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ
ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÐÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼Ñ Ð°Ð½ÑÐ»Ð°
thermal conductivity
ballistic-diffusion model
Guyer-Krumhansl equation

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ÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼ÑÐ°Ð½ÑÐ»Ð° Ð¿ÑÐ¸ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÑ ÑÑÐ»Ð¾Ð²Ð¸ÑÑ. 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: ÐÐ·ÑÑÐµÐ½Ð¸Ðµ Ð±Ð°Ð»Ð»Ð¸ÑÑÐ¸ÐºÐ¾-Ð´Ð¸ÑÑÑÐ·Ð¸Ð¾Ð½Ð½Ð¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸, Ð²Ð²ÐµÐ´ÐµÐ½Ð½Ð¾Ð¹ Ð. Ð§ÐµÐ½Ð¾Ð¼ (2001). ÐÐ·ÑÑÐµÐ½Ð¸Ðµ Ð²ÑÐ²Ð¾Ð´Ð° ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼ÑÐ°Ð½ÑÐ»Ð°, Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÐµÑÑÑ Ð²ÑÑÐµÑÐ¿Ð¾Ð¼ÑÐ½ÑÑÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ. ÐÐ¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ°, Ð° ÑÐ°ÐºÐ¶Ðµ 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ÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼ÑÐ°Ð½ÑÐ»Ð° Ñ Ð½ÐµÑÐºÐ¾Ð»ÑÐºÐ¸Ð¼Ð¸ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÑÐ»Ð¾Ð²Ð¸ÑÐ¼Ð¸. Ð Ð°Ð±Ð¾ÑÐ° Ð¿ÑÐ¾Ð²ÐµÐ´Ð°Ð½Ð° Ð½Ð° Ð±Ð°Ð·Ðµ ÑÑÐ°ÑÑÐ¸ Â«Ð Ð°ÑÑÐ¸ÑÐµÐ½Ð½Ð°Ñ ÑÐµÑÐ¼Ð¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ Ð¿ÐµÑÐµÑÐ¾Ð´Ð½Ð¾Ð¹ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð² ÑÑÐ±ÐºÐ¾Ð½ÑÐ¸Ð½ÑÐ°Ð»ÑÐ½ÑÑ Ð¼Ð°ÑÑÑÐ°Ð±Ð°ÑÂ» Ð.ÐÐµÐ±Ð¾Ð½Ð°, Ð¥. ÐÐ°ÐºÑÐ°ÑÐ¸, Ð. ÐÑÐ¼ÐµÐ»Ð° Ð Ð§. ÐÑÐ±ÑÐ°, Ð² ÐºÐ¾ÑÐ¾ÑÐ¾Ð¹ Ð¿ÑÐ¸Ð²ÐµÐ´ÐµÐ½ Ð²ÑÐ²Ð¾Ð´ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ ÐÑÐ¹ÐµÑÐ°-ÐÑÑÐ¼ÑÐ°Ð½ÑÐ»Ð°. ÐÑÐ»Ð° Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð° Ð·Ð°Ð´Ð°ÑÐ° ÐÐ¾ÑÐ¸ Ñ Ð´Ð²ÑÐ¼Ñ Ð²Ð°ÑÐ¸Ð°Ð½ÑÐ°Ð¼Ð¸ Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÑ ÑÑÐ»Ð¾Ð²Ð¸Ð¹ â ÑÐ¸Ð·Ð¸ÑÐ½ÑÐ¼Ð¸ Ð¸ Ð½ÐµÑÐ¸Ð·Ð¸ÑÐ½ÑÐ¼Ð¸. ÐÑÐ±Ð¾Ñ Ð¿Ð¾ÑÐ»ÐµÐ´Ð½Ð¸Ñ Ð±ÑÐ» Ð¾Ð±ÑÑÐ»Ð¾Ð²Ð»ÐµÐ½ Ð¶ÐµÐ»Ð°ÐµÐ¼ ÑÑÐ°Ð²Ð½Ð¸ÑÑ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÐµ Ð½Ð°Ð¼Ð¸ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¸ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ, Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÐµ Ð. ÐÑÐºÐ¾Ð²ÑÐºÐ¸Ð¼ Ð² 2016 Ð¿ÑÐ¸ Ð°Ð½Ð°Ð»Ð¾Ð³Ð¸ÑÐ½Ð¾Ð¼ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ð¸ Ð´Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ. ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ ÐÐ¾ÑÐ¸ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»Ð¾ÑÑ ÑÐ°Ð¼Ð¾ÑÑÐ¾ÑÑÐµÐ»ÑÐ½Ð¾, ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð¸ ÐµÐ³Ð¾ Ð³ÑÐ°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð²Ð¸Ð·ÑÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð±ÑÐ»Ð¸ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÐµÐ½Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÑÐ·ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Python. 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¸ Ð½Ð°Ð¹Ð´ÐµÐ½Ñ Ð¾Ð±Ð° ÑÐµÑÐµÐ½Ð¸Ñ Ð·Ð°Ð´Ð°ÑÐ¸ ÐÐ¾ÑÐ¸, ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð¾ Ð¸Ñ Ð°ÑÐ¸Ð¼Ð¿ÑÐ¾ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¿Ð¾Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ, ÑÐ´Ð°Ð»ÐµÐ½Ñ Ð½ÐµÐ¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð½Ð¾ÑÑÐ¸ Ð²Ð¸Ð´Ð° 0/0 Ð¿ÑÐ¸ Ð½ÐµÐºÐ¾ÑÐ¾ÑÑÑ Ð·Ð½Ð°ÑÐµÐ½Ð¸ÑÑ Ð¿ÐµÑÐµÐ¼ÐµÐ½Ð½ÑÑ.
description: The given work is devoted to the study of the Cauchy problem for the Guyer-Krumhansl equation under various initial conditions. The research set the following goals: 1. Study of the ballistics-diffusion model introduced by G. Chen (2001). 2. Study of the derivation of the Guyer-Krumhansl equation, which describes the above-mentioned model. 3. Formulation as well as analytical and numerical study of the Cauchy problem for the Guyer-Krumhansl equation with several initial conditions. The work is based on the article "An extended thermodynamic model of transient heat conduction at sub-continuum scales" by G. Lebon, H. Machrafi, M. Grmela, and Ch. Dubois, which gives the derivation of the Guyer-Krumhansl equation. The Cauchy problem was posed with two variants of the initial conditions â physical and non-physical. The choice of the latter was determined by the desire to compare the results obtained by us and the results obtained by K. Zhukovsky in 2016 with a similar research of this equation. The research of the Cauchy problem was conducted independently, the numerical solution and its graphical visualization were presented using the Python programming language. As a result of the work all the goals were achieved. Both solutions of the Cauchy problem were found, their asymptotic behavior was considered, and uncertainties of the form 0/0 were removed for some values of the variables.