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Presentation of the theses performed under the guidance of as. prof. Belov A.A. in the Moscow State University Khokhlachev Valentin Sergeevich. Improved error estimates for exponentially convergent quadratures For some important classes of integrand functions, the quadrature trapezoid formula has exponential convergence instead of power-law one. For this case, majorant Trefethen-Weidemann estimates of accuracy are known. We improve the majorant constant noticeably. We show that the Trefethen-Weidemann estimates are applicable only for integrand functions with first-order poles. Heuristic accuracy estimates for the poles of integer order are proposed. Tintul Maxim Alexandrovich. Multidimensional cubatures on the Sobol sequences The shifted Sobol points are proposed for calculating multidimensional cubatures by the Monte Carlo method. This modification increases the uniformity of the points distribution. A multigrid calculation strategy is proposed to find a posteriori statistical accuracy estimate. This significantly increases the robustness of the calculation. Zverev Alexey Andreevich. Bicompact difference schemes for one-dimensional quantum mechanical problems A bicompact difference scheme is constructed for quantum-mechanical problems with discrete spectrums. This is a two-point conservative scheme. If the special meshes are applied, the scheme provides the second order of accuracy for non-smooth and even discontinuous potentials. Therefore, the bicompact scheme is radically superior in accuracy to traditional three-point schemes. Vergazov Artem Sergeevich. Accuracy control for numerical integration of stiff systems Automatic step selection algorithms are widely used for numerical integration of stiff Cauchy problems for ODEs. The method of geometrically adaptive meshes is the most reliable technique. We construct a new test problem, in which the exact solution is presented in terms of elementary functions of both time and the arc length of the integral curve. A quantitative comparison of various difference schemes is carried out. A new strategy for the thickening of geometrically adaptive meshes is proposed, which increases the reliability of the calculation while preserving high accuracy. Topor Oleg Igorevich. Filling the database on reaction rates to control chemical and thermonuclear reactors The most reliable data on the rates of chemical and thermonuclear reactions are obtained by regression of the results of direct experimental measurements. Using the latest mathematical methods, we construct a database on the rates of a) thermonuclear reactions which are the most important in the problem of controlled thermonuclear fusion, and b) chemical reactions of thermal decomposition of ethane. Statistically reliable estimates of confidence intervals are found for the approximating curves.
