title: Ð§Ð¸ÑÐ»ÐµÐ½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´, ÑÐ²ÑÐ·Ð°Ð½Ð½ÑÐ¹ Ñ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÐ¿Ð»Ð°Ð¹Ð½Ð°Ð¼Ð¸, Ð² Ð½Ð°ÑÐ°Ð»ÑÐ½Ð¾-ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐµ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð¿Ð»Ð¾ÑÐºÐ¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ñ ÐºÑÐ¸Ð²Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð³ÑÐ°Ð½Ð¸ÑÐµÐ¹
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: The method of separation of variables in thermal conductivity problem for region with curvilinear boundary
Creators: name: Ð¨Ð°Ð¿Ð¸ÑÐ¾, ÐÐºÐ°ÑÐµÑÐ¸Ð½Ð°
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: 2022
Subjects: Ð¡Ð¿Ð»Ð°Ð¹Ð½Ñ (Ð¼Ð°Ñ.)
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the method of separation of variables
parabolic initial-boundary value problem
region with curvilinear boundary
finite Fourier series

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«Ð§Ð¸ÑÐ»ÐµÐ½Ð½ÑÐ¹ Ð¼ÐµÑÐ¾Ð´, ÑÐ²ÑÐ·Ð°Ð½Ð½ÑÐ¹ Ñ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÐ¿Ð»Ð°Ð¹Ð½Ð°Ð¼Ð¸, Ð² Ð½Ð°ÑÐ°Ð»ÑÐ½Ð¾-ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐµ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð¿Ð»Ð¾ÑÐºÐ¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ñ ÐºÑÐ¸Ð²Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð³ÑÐ°Ð½Ð¸ÑÐµÐ¹Â».ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¼ÐµÑÐ¾Ð´Ð° ÑÐ°Ð·Ð´ÐµÐ»ÐµÐ½Ð¸Ñ Ð¿ÐµÑÐµÐ¼ÐµÐ½Ð½ÑÑ (ÐÐ Ð), ÑÐ²ÑÐ·Ð°Ð½Ð½Ð¾Ð³Ð¾ Ñ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÐ¼Ð¸ ÑÐ¿Ð»Ð°Ð¹Ð½Ð°Ð¼Ð¸. 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ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐµÐ¾ÑÐ¸Ð¸ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÑ ÑÐ¿Ð»Ð°Ð¹Ð½Ð¾Ð² Ð¸ ÑÐ¼ÐµÑÐ°Ð½Ð½ÑÑ Ð²Ð°ÑÐ¸Ð°ÑÐ¸Ð¾Ð½Ð½Ð¾-ÑÐµÑÐ¾ÑÐ½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð². ÐÐ·Ð»Ð¾Ð¶ÐµÐ½Ð¸Ðµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° Ð¼ÐµÑÐ¾Ð´Ð° ÑÐ°Ð·Ð´ÐµÐ»ÐµÐ½Ð¸Ñ Ð¿ÐµÑÐµÐ¼ÐµÐ½Ð½ÑÑ, Ð² ÐºÐ¾ÑÐ¾ÑÐ¾Ð¼ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÑÑÑÑ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÐµ ÑÐ¿Ð»Ð°Ð¹Ð½Ñ Ð¸ ÐºÐ¾ÑÐ¾ÑÑÐ¹ Ð¿ÑÐµÐ´Ð½Ð°Ð·Ð½Ð°ÑÐµÐ½ Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð½Ð°ÑÐ°Ð»ÑÐ½Ð¾-ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð¿Ð»Ð¾ÑÐºÐ¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸. Ð ÐµÑÐµÐ½Ð¸Ñ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐ¼ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¤ÑÑÑÐµ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ Ð¾Ð±Ð»Ð°ÑÑÐµÐ¹ Ñ ÐºÑÐ¸Ð²Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÐ¼Ð¸ Ð³ÑÐ°Ð½Ð¸ÑÐ°Ð¼Ð¸ Ð² ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÑÐ´Ð°Ñ Ð¤ÑÑÑÐµ, Ð¸ Ð¸Ñ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ. ÐÐ»Ð°ÑÑÐ¸ÑÐµÑÐºÐ¸Ð¹ Ð¼ÐµÑÐ¾Ð´ Ð¤ÑÑÑÐµ Ð½Ðµ Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÐµÑ Ð¿Ð¾Ð»ÑÑÐ°ÑÑ ÑÐµÑÐµÐ½Ð¸Ñ ÑÐ°ÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ Ð² ÑÑÐ´Ð°Ñ.Â ÐÐ¾Ð´Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ñ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ñ Ð´Ð»Ñ MatLab Ð¸ ÑÐµÑÐµÐ½Ð¸Ðµ Ñ ÐµÐµ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ Ð´Ð»Ñ ÐºÑÐ¸Ð²Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÑ Ð¾Ð±Ð»Ð°ÑÑÐµÐ¹. ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ ÑÐ°ÑÑÐµÑÑ, Ð¿Ð¾ÐºÐ°Ð·ÑÐ²Ð°ÑÑÐ¸Ðµ, ÑÑÐ¾ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÐµ c Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¼Ð¾Ð´Ð¸ÑÐ¸ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð³Ð¾ Ð¼ÐµÑÐ¾Ð´Ð° Ð¤ÑÑÑÐµ Ð¸ ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÑÐ´Ð¾Ð² Ð¤ÑÑÑÐµ Ð½Ð° ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ ÑÐµÑÐºÐ°Ñ ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ´Ð° Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ ÑÐµÐ¿Ð»Ð¾Ð¿ÑÐ¾Ð²Ð¾Ð´Ð½Ð¾ÑÑÐ¸ ÑÑÐ¾Ð´ÑÑÑÑ. ÐÐ½Ð°Ð»Ð¸Ð· Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÐµÑÐ¿ÐµÑÐµÐ½Ð¸Ñ MatLab. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð±ÑÐ»Ð¸ Ð¿Ð¾Ð»ÑÑÐµÐ½Ñ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¿Ð¾Ð´ÑÐ²ÐµÑÐ¶Ð´Ð°ÑÑÐ¸Ðµ, ÑÑÐ¾ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¾ÑÑÐ¾Ð³Ð¾Ð½Ð°Ð»ÑÐ½ÑÑ ÑÐ¿Ð»Ð°Ð¹Ð½Ð¾Ð² Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÐµÑ Ð¿Ð¾Ð»ÑÑÐ°ÑÑ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÐµ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸Ðµ ÑÐµÑÐµÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð±Ð¾Ð»Ð¸ÑÐµÑÐºÐ¸Ñ Ð½Ð°ÑÐ°Ð»ÑÐ½Ð¾-ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°Ñ Ð´Ð»Ñ Ð¾Ð±Ð»Ð°ÑÑÐµÐ¹ Ñ ÐºÑÐ¸Ð²Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÐ¼Ð¸ Ð³ÑÐ°Ð½Ð¸ÑÐ°Ð¼Ð¸, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð½Ðµ ÑÐ²Ð»ÑÑÑÑÑ Ð½Ð°Ð±Ð¾ÑÐ°Ð¼Ð¸ ÐºÐ¾Ð¾ÑÐ´Ð¸Ð½Ð°ÑÐ½ÑÑ Ð»Ð¸Ð½Ð¸Ð¹. ÐÐ½Ð°Ð»Ð¸Ð· Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹ Ð´Ð°ÐµÑ Ð¾ÑÐµÐ½ÐºÐ¸ Ð¸Ñ ÑÐ°ÐºÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸.
description: The subject of the graduate qualification work is âThe method of separation of variables in thermal conductivity problem for region with curvilinear boundaryâ. The given work is devoted to studying the method of separation of variables associated with orthogonal splines is considered. The research set the following goals:Using the theory of orthogonal splines and mixed variational grid methods. Description of the algorithm of the method of separation of variables, which uses orthogonal splines and which is designed to solve the initial boundary value problem of thermal conductivity for a flat region. Solutions by the modified Fourier method of dynamic heat conduction problems for regions with curvilinear boundaries in finite Fourier series, and their study. The classical Fourier method does not allow to obtain solutions to such problems in series. Modification of the program for MatLab and solution with its help of the dynamic problem of heat conduction for curvilinear regions. Calculations were carried out showing that the solutions of a number of dynamic thermal conductivity problems obtained using the modified Fourier method and finite Fourier series on various grids converge. The analysis was conducted by the method of mathematical modeling with the use of MatLabThe study resulted into analysis of the use of orthogonal splines makes it possible to obtain approximate analytical solutions of parabolic initial-boundary value problems for regions with curvilinear boundaries that are not sets of coordinate lines The analysis of approximate solutions gives estimates of their actual accuracy.