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Application of new operator-functional approaches to development of effective numerical methods for solving modern problems in natural sciences

Work number - M 72

Presented bythe Institute of Mathematics of NAS of Ukraine.

Authors:Lysenko L. O.,  Romaniuk N. M., Sytnyk D. O.

The goal of the work is to construct efficient numerical methods for solving mathematical problems which described by differential operator equations and are important from the practical point of view.

New existence conditions and representation of the solution nonlocal-in-time Cauchy problem for first-order differential equation in Banach space with a strongly positive operator coefficient and multipoint non-local condition were obtained. Authors developed exponentially convergent  method without accuracy saturation to approximate the solution of mentioned non-local problem. The general scheme of functionally discrete (FD-) method for solving nonlinear functional equations in abstract setting were proposed and applied to obtain the  superexponentially--convergent numerical method for the nonlinear boundary value problem on a finite interval.

Authors developed and justified the FD-methods for spectral problems for the Schrödinger operator on a finite interval. In the polynomial potential case  new symbolic-algorithmic implementation of FD-method, relying on usual algebraic operations only, were developed. The proposed method was extended to the problem in which  potential is the derivative of a function of bounded variation. New abstract scheme of FD-method for the approximations of eigen-pairs of linear operators with discrete spectrum, with the capability to treat eigenvlaues of multiplicity more than one, were developed.  The sufficient conditions for its superexponential convergence have been established.

The V. K. Dzyadyk method of moment representations were generalized to the case of multidimensional numerical sequences. New theorems on construction of rational approximations for power series of two or more variables were established. Authors constructed and investigated Pade approximants for broad classes of special functions of two and more variables, in particular, for the Appell, Humbert and Lauricella hypergeometric series. Convergence of Pade-type approximants for Humbert series was proved, and new asymptotic formulas were developed.

The level of obtained results is no lower than international analogues.

Number of publications: 45, including 1 monograph, 27 articles (9 - in foreign journals). According to the Scopus database the total number of the authors’ citations is 4, h-index (for work) = 1; Google Scholar citations: 32, h-index (for the work) = 3.