  RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  General information Latest issue Archive Guidelines for authors Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. IMI UdGU: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Izv. IMI UdGU, 2014, Issue 1(43), Pages 49–67 (Mi iimi290)  On solution of one optimization problem generated by simplest heat conduction equation

V. I. Rodionov

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: The solution of boundary value problem for the simplest heat conduction equation defined on a rectangle can be represented as the sum of two terms which are solutions of two boundary value problems: in the first case, the boundary functions are linear, while in the second case, the initial function is zero. This specificity allows us to apply two-dimensional splines for the numerical solution of both problems. The first problem was studied in previous papers where an economical algorithm was obtained for its numerical solution with linear computational complexity. This fact served as the basis for similar constructions in solving the second problem. Here we also define the finite-dimensional space of splines of Lagrangian type, and as a solution, we suggest the optimal spline giving the smallest residual. We have obtained exact formulas for the coefficients of this spline and its residual. The formula for the spline coefficients is a linear form of initial finite differences on the boundary. The formula for the residual is the sum of five simple terms and a negative definite quadratic form of new finite differences defined on the boundary. The entries of the matrix of the form are expressed through Chebyshev's polynomials, the matrix is invertible and is such that the inverse matrix has a tridiagonal form. This feature allows us to obtain upper and lower bounds for the spectrum of the matrix and to show that the residual is bounded by a constant independent of the dimension $N$. It is shown that the associated residual tends to zero with increasing $N$. Thus, the obtained optimal spline should be considered the pseudosolution of the second problem.

Keywords: heat conduction equation, interpolation, approximate spline, tridiagonal matrix, Chebyshev's polynomials. Full text: PDF file (292 kB) References: PDF file   HTML file
UDC: 519.651+517.518.823
MSC: 41A15

Citation: V. I. Rodionov, “On solution of one optimization problem generated by simplest heat conduction equation”, Izv. IMI UdGU, 2014, no. 1(43), 49–67 Citation in format AMSBIB
\Bibitem{Rod14} \by V.~I.~Rodionov \paper On solution of one optimization problem generated by simplest heat conduction equation \jour Izv. IMI UdGU \yr 2014 \issue 1(43) \pages 49--67 \mathnet{http://mi.mathnet.ru/iimi290} 

• http://mi.mathnet.ru/eng/iimi290
• http://mi.mathnet.ru/eng/iimi/y2014/i1/p49

 SHARE:      Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Rodionov, “O lineinom algoritme chislennogo resheniya kraevoi zadachi dlya prosteishego volnovogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:1 (2015), 126–144  •  Contact us: math-net2020_03 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020