Projector Approach to the Butuzov–Nefedov Algorithm for Finding Asymptotic Solutions for a Class of Discrete Problems with a Small Step

Abstract

V.F. Butuzov and N.N. Nefedov proposed an algorithm for constructing asymptotics with boundary functions of two types for solving a discrete initial value problem with a small step \({{\varepsilon }^{2}}\) and a nonlinear term of order \(\varepsilon \) in the critical case, i.e., when the degenerate equation with \(\varepsilon = 0\) is not solvable uniquely for the unknown variable. In this paper, an asymptotic solution of the same problem is constructed by applying a new approach based on orthogonal projectors onto \(\ker (B(t) - I)\) and \(\ker (B(t) - I){\kern 1pt} '\) , where \(B(t)\) is the matrix premultiplying the unknown variable in the linear part of the equation, \(I\) is the identity matrix of suitable size, and the prime denotes transposition. This approach considerably simplifies the understanding of the asymptotics-constructing algorithm and makes it possible to represent the problems of finding asymptotic terms of any order in explicit form, which is convenient for researchers applying asymptotic methods for real-world problems.