Nonlinear singularly perturbed parabolic equations with boundary conditions of the first kind

I. V. Denisov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 29-36

Résumé

This paper is a review of applications of the method of angular boundary functions to nonlinear equations. We consider the first boundary-value problem for the following singularly perturbed parabolic equation in a rectangle: \begin{equation*} \epsilon^2\left(a^2\frac{\partial^2 u}{\partial x^2} -\frac{\partial u}{\partial t}\right)=F(u,x,t,\epsilon), \end{equation*} where the function $F$ is nonlinear with respect to the variable $u$. We consider the case where the function $F$ is quadratic or cubic in the variable $u$ at the corner points of the rectangle and examine the possibility of constructing a complete asymptotic expansion of the solution of the problem as $\epsilon\rightarrow 0$.

Détail

Keywords: boundary layer, asymptotic approximation, singularly perturbed equation.