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

Abstract

The article deals with the singularly perturbed boundary value problem for third order linear integro-differential equation with a small parameter in the highest derivatives, provided that the roots of additional distinctive equation have opposite signs. The work is focused on the evaluation and asymptotic behavior of solutions of singularly - perturbed boundary value problem in the initial points jumps. In this paper for a singularly - perturbed homogeneous differential equation are constructed a fundamental system of solutions, initial and boundary functions and their asymptotic estimates are derived. With initial and boundary functions are obtained explicit analytical formula solutions. The theorem about asymptotic estimate of a solution of boundary value problem is proved. The theorem implies that the solution of boundary value problem of both sides of given segment has the initial jumps with different orders. For instanse, in the left point t = 0 the solution of third order linear integro-differential equation has a first order initial jump and in the right side t = 1 has null order initial jump.