RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki:
Year:
Volume:
Issue:
Page:
Find






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


Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 3, Pages 141–156 (Mi vuu343)  

This article is cited in 4 scientific papers (total in 4 papers)

COMPUTER SCIENCE

Exact solution of optimization task generated by simplest heat conduction equation

V. I. Rodionov, N. V. Rodionova

Udmurt State University, Izhevsk, Russia

Abstract: In the previous paper of the authors the parameter family of finite-dimensional spaces of special quadratic splines of Lagrange's type has been defined. In each space, as a solution to the initial-boundary problem for the simplest heat conduction equation, we have proposed the optimal spline, which gives the smallest residual. We have obtained exact formulas for coefficients of this spline and its residual. The formula for coefficients of this spline is a linear form of initial finite differences. The formula for the residual is a positive definite quadratic form of these quantities, but because of its bulkiness it is ill-suited for analyzing of the approximation quality of the input problem at the variation with the parameters.
For the purposes of the present paper, we have obtained an alternative representation for the residual, which is the sum of two positive definite quadratic forms of the new finite differences defined on the boundary. The matrix of the first form has second order and the apparent spectrum. The elements of the second matrix of order $N+1$ are expressed in terms of Chebyshev's polynomials, the matrix is invertible and the inverse matrix has a tridiagonal form. This feature allows us to obtain, for the spectrum of the matrix, upper and lower bounds that are independent of the dimension $N$. Said fact allows us to make a study of the quality of approximation for different dimensions $N$ and weights $\omega\in[-1,1]$. It is shown that the parameter $\omega=0$ gives the best approximation and the residual tends to zero as $N$ increasing.

Keywords: interpolation, approximate spline, Chebyshev's polynomials.

Full text: PDF file (249 kB)
References: PDF file   HTML file
UDC: 519.651+517.518.823
MSC: 41A15
Received: 24.05.2012

Citation: V. I. Rodionov, N. V. Rodionova, “Exact solution of optimization task generated by simplest heat conduction equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 3, 141–156

Citation in format AMSBIB
\Bibitem{RodRod12}
\by V.~I.~Rodionov, N.~V.~Rodionova
\paper Exact solution of optimization task generated by simplest heat conduction equation
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2012
\issue 3
\pages 141--156
\mathnet{http://mi.mathnet.ru/vuu343}


Linking options:
  • http://mi.mathnet.ru/eng/vuu343
  • http://mi.mathnet.ru/eng/vuu/y2012/i3/p141

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. N. V. Rodionova, “Tochnoe reshenie odnoi zadachi optimizatsii, porozhdennoi prosteishim volnovym uravneniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 1, 141–152  mathnet
    2. V. I. Rodionov, “O reshenii odnoi zadachi optimizatsii, porozhdennoi prosteishim uravneniem teploprovodnosti”, Izv. IMI UdGU, 2014, no. 1(43), 49–67  mathnet
    3. 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  mathnet  elib
    4. A. N. Mzedavee, V. I. Rodionov, “Tochnoe reshenie odnoi zadachi optimizatsii, porozhdennoi trekhmernym uravneniem Laplasa”, Izv. IMI UdGU, 51 (2018), 52–78  mathnet  crossref  elib
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
    Number of views:
    This page:232
    Full text:89
    References:28
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020