Section

Computer science

Title

Exact solution of optimization task generated by simplest heat conduction equation

Author(s)

Rodionov V.I.^{a},
Rodionova N.V.^{a}

Affiliations

Udmurt State University^{a}

Abstract

In the previous paper of the authors the parameter family of finitedimensional spaces of special quadratic splines of Lagrange's type has been defined. In each space, as a solution to the initialboundary 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 illsuited 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

UDC

519.651, 517.518.823

MSC

41A15

DOI

10.20537/vm120313

Received

24 May 2014

Language

Russian

Citation

Rodionov V.I., Rodionova N.V. Exact solution of optimization task generated by simplest heat conduction equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 3, pp. 141156.

References

 Rodionov V.I., Rodionova N.V. Exact formulas for coefficients and residual of optimal approximate spline of simplest heat conduction equation, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 4, pp. 154171.
 Fadeev D.K., Fadeeva V.N. Vychislitel'nye metody lineinoi algebry (Computing methods of linear algebra), St. Petersburg: Lan', 2002, 736 p.
 Suetin P.K. Klassicheskie ortogonal'nye mnogochleny (Classical orthogonal polynomials), Moscow: Nauka, 1976, 328 p.

Full text

