A boundary jumps phenomenon in the integral boundary value problem for singularly perturbed differential equations

Abstract

The article is devoted to the study of the asymptotic behavior of solving an integral boundary value problem for a third-order linear differential equation with a small parameter for two higher derivatives, provided that the roots of the "additional characteristic equation" have opposite signs. In the work are constructed the fundamental system of solutions, boundary functions for singularly perturbed homogeneous differential equation and are provided their asymptotic representations. An analytical formula of solution for a given singularly perturbed integral boundary value problem is obtained. Theorem about asymptotic estimates of solution is proved. For a singularly perturbed integral boundary value problem, the growth of the solution and its derivatives at the boundary points of this segment is obtained when the small parameter tends to zero. It is established that the solution of a singularly perturbed integral boundary value problem has initial jumps at both ends of this segment. In this case, we say that there is a phenomenon of boundary jumps, which is a feature of the considered singularly perturbed integral boundary value problem. Moreover, the orders of initial jumps were different. Namely, at the point t = 0 , there is a phenomenon of the initial jump of the first order, and at the point t = 1, the order of the initial jump was equal to zero. The results obtained allow us to construct uniform asymptotic expansions of solutions of nonlinear singularly perturbed integral boundary value problems.