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PresentationsUsing Padé approximants in combinations of asymptotic expansions of nonlinear problems in mathematical physicsPeter the Great St. Petersburg Polytechnic University pavelbakaev2006@gmail.com The report presents the results of approximating asymptotic expansions of a number of mathematical physics problems using rational functions. Nonlinear dynamical systems (DS) are a generalization of a large class of real physical problems arising in mathematical modeling. Examples of nonlinear initial and initial-boundary value problems are considered as model DS [1]. One of the promising approaches is the use of Padé approximations, which significantly improve the accuracy of expansions in small parameters compared to classical Taylor series. The specifics of solving boundary value problems are demonstrated for Airy equations [1-3] and Schrödinger equations [4]. It is shown that for the one-dimensional stationary Schrödinger equation, which reduces to an Airy equation, one can use the obtained Padé approximation of the asymptotic solution to the Airy boundary value problem [3]. The differences between solving initial value problems and boundary value problems in mathematical physics are noted in the context of constructing asymptotics and combining asymptotic expansions using Padé approximants (PA). One of the principles of asymptotic methods for solving boundary value problems is the hypothesis of the existence of asymptotics for two limiting values of the parameter. To connect non-overlapping asymptotics, methods based on two-point Padé approximations (TPPA) have been developed [3]. The use of TPPA in many cases allows overcoming the local nature of solutions inherent to asymptotic methods and perturbation theory. When it is necessary to determine TPPA parameters corresponding to the specifics of a given problem, additional conditions obtained from integral relations of the Bubnov-Galerkin type are used [3].
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