Abstract
In this paper, we study the asymptotic behavior of a singularly perturbed boundary value problem under the condition that the real parts of the roots of the defining characteristic equation have opposite signs, which ensures the emergence of a new phenomenon, the phenomenon of a boundary jump. In this regard, the proposed work is devoted to the detailed development of a general algorithm for studying the asymptotic behavior of solutions to singularly perturbed general boundary value problems that exhibit the phenomenon of boundary jumps. The introduced adjoining initial functions, as well as boundary functions, and the established asymptotic estimates made it possible to: prove the existence and uniqueness of the original boundary value problem; formulate conditions for the degenerate equation, determine the values of the initial jumps and the growth of the value of derivatives with respect to a small parameter; identify a class of boundary value problems that exhibit the phenomenon of boundary jumps; prove the limiting transition of the solution of the original singularly perturbed problem to the solution of the degenerate problem with respect to a small parameter; select an appropriate asymptotic method for constructing their approximations.