Boundary layer phenomenon in a first-order algebraic-differential equation

V. I. Uskov

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 144, pp. 436-446

Résumé

The Cauchy problem for the first-order algebraic differential equation is considered \begin{equation*} A\frac{du}{dt}=(B+\varepsilon C+\varepsilon^2 D)u(t,\varepsilon), \end{equation*} \begin{equation*} u(t_0,\varepsilon)=u^0(\varepsilon)\in E_1, \end{equation*} where $A,B,C,D$ are closed linear operators acting from a Banach space $E_1$ to a Banach space $E_2$ with domains everywhere dense in $E_1,$ $u^0$ is a holomorphic function at the point $\varepsilon=0,$ $\varepsilon$ is a small parameter, $t\in[t_0;t_{max}].$ Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator $A$ is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton's diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained — these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space $\mathbb{R}^4.$

Détail

Keywords: first-order algebraic-differential equation, small parameter, Fredholm operator, boundary layer phenomenon, regularity conditions for degeneracy
Mots-clés : bifurcation equation