Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation (Q2078986)

In this paper, the authors extend and generalize the basic ideas in the case of problems with discontinuous time variables on the basis of contrast structure theory. They use the Butuzov method to construct a unified asymptotic expansion to describe the behavior of the solution in three different zones in the neighborhoods of boundary layers and internal layers. It is showed, that under certain assumptions, the scalar singularly perturbed boundary value problem \[ \mu^2\frac{d^2y}{dx^2}=f(y,x,\mu),\ 0 < x < 1,\ 0 < \mu \ll 1, \] \[ \frac{dy}{dx}(0,\mu) = 0, \ \frac{dy}{dx}(1,\mu) = 0, \] has a contrast structure solution with an internal layer in the neighborhood of the transition point \((x^*, d(x^*))\) on the discontinuous curve. The derived results have a potential to provide a useful tool for the applications and numerical methods of piecewise-smooth differential equations in many areas.