Solution of a Singularly Perturbed Mixed Problem on the Half-Line for a Parabolic Equation with a Strong Turning Point of the Limit Operator

Abstract

We study singularly perturbed problems in the presence of spectral singularities of the limit operator using S.A. Lomov’s regularization method. In particular, a regularized asymptotic solution is constructed for a singularly perturbed inhomogeneous mixed problem on the half-line for a parabolic equation with a strong turning point of the limit operator. Based on the idea of asymptotic integration of problems with unstable spectrum, it is shown how regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for this type of singularity is described in detail, this algorithm is justified, and an asymptotic solution of any order in a small parameter is constructed.