Abstract
This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.
Highlights
One of the topical problems in the singular perturbation theory is the study of nonlinear singularly perturbed partial differential equations, the solutions of which have boundary and interior layers. Such equations are of great interest both in the qualitative theory of differential equations and in many applied problems. Such equations arise in mathematical models of processes of the reaction–diffusion and reaction–diffusion–advection type in chemical kinetics, synergetics, astrophysics, biology, and other fields, where the processes under study are characterized by narrow boundary regions of rapid changes in process parameters or by sharp interior layers of various types, stationary or moving
The results presented in this paper form the basis for further development of methods for asymptotic analysis of new classes of nonlinear singularly perturbed problems for partial differential equations, which are widely used as mathematical models in many important applications
We developed the asymptotic method of differential inequalities for those classes of singularly perturbed problems where the principle of comparison of elliptic boundary value problems, parabolic initialboundary value problems, and time-periodic parabolic boundary value problems, as well as problems for integro-differential equations, works
Summary
One of the topical problems in the singular perturbation theory is the study of nonlinear singularly perturbed partial differential equations, the solutions of which have boundary and interior layers. A number of interesting results concerning theory and their application for the numerical solution of singularly perturbed problems with transition layers were obtained in [13,14,15,16,17,18] Note that this actively developing area is based on the results of an asymptotic analysis of the works presented in this review. The results presented in this paper form the basis for further development of methods for asymptotic analysis of new classes of nonlinear singularly perturbed problems for partial differential equations, which are widely used as mathematical models in many important applications. This paper highlights the line of studies pursued by the scientific school to which the author belongs and the method to the development of which he is directly related
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