Speciality:
05.13.18 (Mathematical modeling, numerical methods, and the program systems)
Birth date:
13.11.1971
Keywords: cubature formulas; quadrature formulas; numerical methods of solving of partial differential equations; superradiance.
Subject:
The models of processes of a superradiance were studied. The task of mathematical modeling of superradiance processes was surveyed for multilevel mediums. Moreover the reflection problem of coherent impulse from a resonant medium was studied. Numerical methods of a solution of these models are constructed. In these models the tasks of optimal control and their finite-dimensional analogs are appeared. Other area of my study is a theory cubature formulas. Assume that $L$ is space of one-variable functions with respect to the norm $\left\| \varphi \right\| =\left( \int \sum\limits_{\alpha =1}^{m}a_{\alpha} \left(D^{\alpha }\varphi \right)^{2}dx\right)^{\frac{1}{2}}$. In space $L$ we construct the quadrature formulas with a uniform distribution of nodes. At these suppositions the explicit formulas for optimal coefficients are constructed with accuracy to exponentially small members. Other problem consists in approximation of a weight integral by the cubature formula such that the nodes of this formula can be out of the integration domain. The integrand function is a member of $\tilde {W}_2^\mu$ space of periodic functions. In these suppositions the asymptotic expansion of the optimal coefficients are constructed. This asymptotic expansion allows to prove a stability of the numerical algorithm and also to establish the Gibbs phenomenon. The Gibbs phenomenon illuminates the optimal coefficients fluctuations nearly the integration domain boundary. Besides the approximation problem of derivative of function is considered, where this function belong to space L_2^{\left( m \right)}. The explicit formulas for optimal coefficients are constructed. Denote by $A$ any functional. Let $A$ is defined on a given set of a functions. This functional is approximated by the special formula. The minimization problem of the functional miscalculation is studied. The explicit formulas for optimal coefficients are obtained. Suppose the thickness of the boundary layer is sufficiently wide; then the optimal coefficients is positive.
Main publications:
Andreev A. V., Sheetlin S. L. Superradiance and Raman scattering in three-level molecular system // Infrared Physics Technology, vol. 37, 7, p. 733–739, 1996.
