The Initialization Problem for a Singularly Perturbed Integral Equation with the Diagonally Degenerated Kernel

Abstract

The article considers a singularly perturbed integral equation with a diagonally degenerated first-order kernel containing not only the unknown function under the integral sign, but also its derivative. Just as in the case when there is no derivative under the integral sign, the problem is formulated as follows: construct a regularized (in the Lomov sense) asymptotic solution of this problem and investigate the limiting transition in it as the small parameter tends to zero over the entire interval of time, including the boundary-layer zone (the initialization problem). Unlike the previously considered case, in which the positivity of the diagonal kernel and the independence of the integral operator from the derivative of the unknown function led to constructing a regularized asymptotics with rapidly oscillating functions, the problem with a derivative admits solutions the asymptotic behavior of which may contain both rapidly oscillating and rapidly decreasing components. It should be noted that a similar problem was previously studied only for an integro-differential equation. Not only the inhomogeneity and the kernel of the integral operator, but also the initial vector serve in this case as initial data for the initialization class. As a result, the study of the initialization problem becomes significantly easier. However, this is not the case for a singularly perturbed integral equation; therefore, the initialization class will be "poorer" than it is for an integro-differential equation. The principal term of the asymptotics in its regularized (in the Lomov sense) form is used as the object for studying the limiting transition. It is exactly this form that is closest to the exact solution of the problem under consideration; therefore, the study of the principal term of the asymptotics in its regularized form makes it possible to eliminate the components in the exact solution that impede the solution in its tending to the limiting one. All manipulations are given for a scalar equation; nonetheless, they will also hold in a vector case.