A Method of the Study of the Cauchy Problem for a Singularly Perturbed Linear Inhomogeneous Differential Equation

TL;DR

The paper introduces a convergent sequence approach for solving the Cauchy problem of singularly perturbed linear inhomogeneous differential equations, providing a basis for asymptotic analysis and solution approximation.

Contribution

It develops a novel sequence method that converges to the solution and serves as an asymptotic sequence for singular perturbation problems of arbitrary order.

Findings

Sequence converges to the exact solution

Deviation from the solution is proportional to the perturbation parameter's power

Method justifies asymptotic solutions obtained by boundary function methods

Abstract

We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the deviation (in the norm of the space of continuous functions) of its $n$th element from the solution of the problem is proportional to the $(n+1)$th power of the parameter of perturbation. This sequence can be used for justification of asymptotics obtained by the method of boundary functions.