ASYMPTOTIC ESTIMATES OF THE SOLUTION OF THE BVP FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

In this paper, we study a boundary value problem for a higher-order linear singularly perturbed integro-differential equation, focusing on the case where all roots of the associated “additional characteristic equation” have negative real parts. The main objective is to derive an asymptotic estimate for the solution of this boundary value problem as the perturbation parameter tends to zero. To achieve this, we construct an explicit analytical representation of the solution and rigorously prove a theorem describing its asymptotic behavior. Additionally, a modified degenerate boundary value problem is formulated, which serves as the limiting case of the original perturbed problem. It is shown that the solution of the singularly perturbed problem converges uniformly to the solution of this degenerate problem. The study also addresses the determination of the magnitude of the initial jump in the integral terms, which plays a crucial role in the convergence analysis. The results provide a clear understanding of the solution structure and its dependence
on the perturbation parameter.