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## Archive of Issues

Russia Izhevsk
Year
2015
Volume
25
Issue
1
Pages
126-144
 Section Computer science Title On the linear algorithm of numerical solution of a boundary value problem for a simple wave equation Author(-s) Rodionov V.I.a Affiliations Udmurt State Universitya Abstract The solution of a boundary value problem for a simple wave equation defined on a rectangle can be represented as a sum of two terms. They are solutions of two boundary value problems: in the first case, the boundary functions are constant, while in the second the initial functions have a special form. Such decomposition allows to apply two-dimensional splines for the numerical solution of both problems. The first problem was studied previously, and an economical algorithm of its numerical solution was developed. To solve the second problem we define a finite-dimensional space of splines of Lagrangian type, and recommend an optimal spline giving the smallest residual as a solution. We obtain exact formulas for the coefficients of this spline and its residual. The formula for the coefficients of this spline is a linear form of initial finite differences defined on the boundary. The formula for the residual is a sum of two simple terms and two positive definite quadratic forms of new finite differences defined on the boundary. Elements of matrices of forms are expressed through Chebyshev polynomials, both matrices are invertible and have the property that their inverses matrices are of tridiagonal form. This feature allows us to obtain upper and lower bounds for the spectrum of matrices, and to show that the residual tends to zero when the numerical problem dimension increases. This fact ensures the correctness of the proposed algorithm of numerical solution of the second problem which has linear computational complexity. Keywords wave equation, interpolation, approximate spline, tridiagonal matrix, Chebyshev polynomials UDC 519.651, 517.518.823 MSC 41A15 DOI 10.20537/vm150114 Received 20 September 2014 Language Russian Citation Rodionov V.I. On the linear algorithm of numerical solution of a boundary value problem for a simple wave equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 126-144. References Rodionov V.I. On application of special multivariate splines of any degree in the numerical analysis, Vestn. Udmurt. Univ. Mat. Mekh. Komp. Nauki, 2010, no. 4, pp. 146-153 (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). Rodionov V.I., Rodionova N.V. Exact solution of optimization task generated by simplest heat conduction equation, Vestn. Udmurt. 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