Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps

Nurgabyl D.N. Uaissov A.B.
May 2021 John Wiley and Sons Inc

Numerical Methods for Partial Differential Equations
2021 #37 Issue 3 2375 - 2392 pp.

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.

asymptotic estimates , boundary functions , limit transition , phenomenon of boundary jumps

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Department of Mathematics, Zhetysu State University named after I. Zhansugurov, Taldykorgan, Kazakhstan
Department of Differential Equations and Control Theory, Al-Farabi Kazakh National University, Almaty, Kazakhstan

Department of Mathematics
Department of Differential Equations and Control Theory

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