On the discretization of solutions of the wave equation with initial conditions from generalized Sobolev classes. (English. Russian original) Zbl 1283.65093
Math. Notes 91, No. 3, 430-434 (2012); translation from Mat. Zametki 91, No. 3, 459-463 (2012).
From the introduction: The general statement of the problem is to derive upper and lower bounds (coinciding up to constants, if possible) for the quantity
\[ \begin{split} \delta_N (A; B, D_N; F)_Y = \min_{N_1+N_2=N (l^{N_1,N_2)},\varphi_N)\in D_{N_1,N_2}} \inf \\ \sup_{f=(f_1,f_2)\in F^{(1)} \times F{(2)}} \|u(\cdot;f) - \varphi_N(l_1^{(1)}(f_1),\dots,l_1^{(N_1)}(f_1),l_2^{(1)}(f_2),\dots,l_2^{(N_2)}(f_2);\cdot) \|_Y\end{split} \] and to find a computational aggregate implementing the upper bound.
\[ \begin{split} \delta_N (A; B, D_N; F)_Y = \min_{N_1+N_2=N (l^{N_1,N_2)},\varphi_N)\in D_{N_1,N_2}} \inf \\ \sup_{f=(f_1,f_2)\in F^{(1)} \times F{(2)}} \|u(\cdot;f) - \varphi_N(l_1^{(1)}(f_1),\dots,l_1^{(N_1)}(f_1),l_2^{(1)}(f_2),\dots,l_2^{(N_2)}(f_2);\cdot) \|_Y\end{split} \] and to find a computational aggregate implementing the upper bound.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35L05 | Wave equation |
References:
| [1] | N. Temirgaliev, Mathematics: Selected Works (Izd. ENU, Astana, 2009) [in Russian]. |
| [2] | N. Temirgaliev, Computational Width. Algebraic Number Theory and Harmonic Analysis in Reconstruction Problems (Quasi-Monte-Carlo method). Embedding and Approximation Theory. Fourier Series, Special issue containing the prominent works of mathematicians of Gumilev Eurasian National University, Astana, Vestnik ENU, 1 (2010). |
| [3] | Sh. U. Azhgaliev, Mat. Zametki 82(2), 177 (2007) [Math. Notes 82 (2), 153 (2007)]. · doi:10.4213/mzm3789 |
| [4] | A. B. Utesov, The Problem of Reconstructing Functions and Integrals for Generalized Classes and the Solutions of the Heat Equation, Cand. Sci. (Phys.-Math.) Dissertation (Almaaty, 2001) [in Russian]. |
| [5] | E. I. Shangireev, On the Reconstruction of the Solutions of the Wave Equation, Cand. Sci. (Phys.-Math.) Dissertation (Karaganda, 2002) [in Russian]. |
| [6] | I. Zh. Ibatulin and N. Temirgaliev, Differ. Uravn. 44(4), 491 (2008) [Differ. Equations 44 (4), 510 (2008)]. |
| [7] | V. I. Kolyada, Sibirsk.Mat. Zh. 14(4), 766 (1973). |
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