Solution of Singularly Perturbed Cauchy Problem for Ordinary Differential Equation of Second Order with Constant Coefficients by Fourier Method

In this paper, using the Fourier method in the Krein space, a singularly perturbed Cauchy problem for a second-order differential equation with constant coefficients is solved, and an asymptotic expansion of this solution is found using the theory of linear operators and functional analysis. The peculiarity of the method consists in the fact that the operator of the corresponding Cauchy problem does not have a spectrum, but, nevertheless, even in this case it is possible to expand its solution in a Fourier series in the Krein space and obtain an asymptotic expansion with an estimate of the remainder term. The estimate of the remainder term is obtained in the form of a convolution operator through the right-hand side of the equation and through the coefficients of the equation itself.