Construction of an asymptotic expansion containing additional intermediate layers which approximates solution of a problem for nonlinear differential equation with partial derivatives of the four independent variables with a small parameter that includes the analysis of typical for this problem point of the “butterfly” catastrophe

Project: Award Project

Description

The project is aimed at asymptotic investigation of bisingularly perturbed (that is, not just singularly perturbed but additionally complicated by the presence of singularities in the coefficients of asymptotic expansions) problems for nonlinear partial differential equations with respect to four independent variables with a small parameter. The two problems are discussed in detail in the project separately for each of the two equations of mathematical physics with a small parameter describing the diffusion process and the wave process with smooth inhomogeneities in three spatial coordinates and with a slow variation in time. For each of these problems, the construction of the asymptotic expansion of the solution requires the introduction of two or more boundary layers near the typical critical point corresponding to the "butterfly" catastrophe, which allows us to investigate the rearrangements occurring in the vicinity of the indicated critical point. Bisingular problems are actual problems of mathematics, they are connected with various problems of mathematical physics that have applications in physics, chemistry, medicine, and biology. Catastrophes of higher orders than A3, such as the "swallowtail" and "butterfly", arising from the analysis of equations of mathematical physics with smooth inhomogeneities and changes in time in the case of a dependence on three spatial ordinates and time, have not yet been investigated. The novelty of the approach is related to the new type of matching condition obtained by the author for the power-logarithmic formal asymptotic expansions of the solution arising in such problems.
StatusFinished
Effective start/end date03/02/201631/12/2017

Keywords

  • 27.29.23
  • RFFI
  • Kuibyshev Research Division