Asymptotic estimates of the solution of the boundary value problem for singularly perturbed integro-differential equations

Abstract

The mathematical models of many processes in physics, astrophysics, chemistry, biology, mechanics and technology are differential and integro-differential equations containing small parameters at the highest derivatives. Such equations currently are called singularly perturbed equations. The paper considers a two-point boundary value problem for a third-order linear integro-differential equation with a small parameter at the two highest derivatives, when the roots of the «additional characteristic equation» are negative and the boundary conditions contain terms with small perturbations. The aim of the study is to obtain asymptotic estimates of the solution and to obtain the asymptotic behavior of the solution in a neighborhood of points where additional conditions are given that are lost during degeneracy. The boundary functions of the boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using boundary functions and Cauchy function an analytical formula for the solutions of the boundary value problem is obtained. A theorem on the asymptotic estimate of the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution and the growth order of its derivatives with respect to a small parameter are established. It is shown that the solution of the boundary value problem at the left point of given segment has the phenomenon of an initial first-order jump. Distinctive features in the asymptotic properties of the solutions of this boundary-value problem are shown in comparison with similar works in the field of singularly perturbed differential and integro-differential equations with initial jumps. The obtained results make it possible to construct a uniform asymptotic expansion of the solutions of boundary value problems for singularly perturbed integro-differential equations with any degree of accuracy with respect to a small parameter.

Keywords

• singular perturbation
• small parameter
• initial jump
• asymptotic estimates