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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
1
Pages
141-152
 Section Computer science Title Exact solution of optimization task generated by simplest wave equation Author(-s) Rodionova N.V.a Affiliations Udmurt State Universitya Abstract In the previous paper of the author 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 wave 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 positive definite quadratic form of the new finite differences defined on the boundary. The elements of the matrix of form 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/vm140112 Received 21 June 2013 Language Russian Citation Rodionova N.V. Exact solution of optimization task generated by simplest wave equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 141-152. References Rodionova N.V. Exact formulas for coefficients and residual of optimal approximate spline of simplest wave equation, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 1, pp. 144-154 (in Russian). 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. 154-171 (in Russian). Suetin P.K. Klassicheskie ortogonal'nye mnogochleny (Classical orthogonal polynomials), Moscow: Nauka, 1976, 328 p. Rodionov V.I., Rodionova N.V. Exact solution of optimization task generated by simplest heat conduction equation, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, no. 3, pp. 141-156 (in Russian). Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoi fiziki (Equations of mathematical physics), Moscow: Nauka, 1977, 736 p. Full text