A solution of the singularly perturbed Cauchy problem in the presence of a “weak” turning point at the limit operator

Abstract

The paper proposes a method for constructing an asymptotic solution of the singularly perturbed Cauchy problem in the case of violation of the stability conditions of the spectrum of the limit operator. In particular, we consider the problem with a turning point where eigenvalues "stick together" at t = 0.

On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator

“…The solution to a singularly perturbed Cauchy problem in this case is described in [6]. Here we give a solution to the Cauchy problem 3)…”

“…Moreover, it is assumed that the operator A(t) has a diagonal form for any t ∈ [0, T]. Additionally, in [5], a rational simple turning point was considered, an irrational simple turning point was considered in [6]. Significantly special singularities from a mathematical point of view are special functions that describe the irregular dependence of the solution on ε for ε → 0, and from the point of view of the hydrodynamics of the boundary layer function generated by the spectral singularity of the point λ(t).…”

“…The simplest case of a weak turning point is the point of the first order, i.e., λ 2 (t) − λ 1 (t) = ta(t), a(t) = 0. The solution to a singularly-perturbed Cauchy problem in this case is described in [7]. It is assumed that the eigenspaces corresponding to the eigenvalues λ 1 (t), λ 2 (t) are one-dimensional.…”

Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

“…A "simple" pivot point of a limit operator (matrix A(τ)) is understood when one eigenvalue vanishes at one point (i.e., matrix A(τ) is irreversible at this point). In [1], the case was considered of when one of the eigenvalues that had the form τ n a(τ), a(τ) = 0, n was natural; in [2] the features of the solution were identified and described for a rational "simple" turning point in the one-dimensional case (when the eigenvalue had the form τ m/n a(τ), a(τ) = 0).…”