It is known that the study of boundary value and mixed problems for integrable linear equations encounters significant difficulties of a fundamental nature. Exceptions are problems with boundary conditions of a special type, which are often called integrable or linearizable. The purpose of this article is to study the asymptotic behaviors of solutions of singularly perturbed general boundary value problems with boundary jumps for higher-order equations. Using the Schlesinger-Birghof theorem, we constructed a fundamental system of solutions of a homogeneous perturbed equation of conditionally stable type in the critical case. Initial boundary functions are constructed based on the fundamental system of solutions. An analytical representation is found, the existence and uniqueness of a solution to this boundary value problem are proved. Asymptotic estimates of the solution and its derivatives are derived from the analytical representation of the solution of the given boundary value problem. The limit passage of solution of the perturbed problem to the solution of the unperturbed problem is proved. The conditions of the existence of jumps are found. The values of boundary jumps are determined. As a result, a class of boundary value problems is highlighted that has possessing of phenomenon of boundary jumps.
The article is devoted to research the Cauchy problem for singularly perturbed higher-order linear integro-differential equation with a small parameters at the highest derivatives, provided that the roots of additional characteristic equation have negative signs. An explicit analytical formula of the solution of singularly perturbed Cauchy problem is obtained. The theorem about asymptotic estimate of a solution of the initial value problem is proved. The nonstandard degenerate initial value problem is constructed. We find the solution of the nonstandard degenerate initial value problem. An estimate difference of the solution of a singularly perturbed and nonstandard degenerate initial value problems is obtained. The asymptotic convergence of solution of a singularly perturbed initial value problem to the solution of the nonstandard degenerate initial value problem is established.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.