Метод регуляризации решений бисингулярно возмущенной задачи в пространстве обобщенных функций

Abstract

When singularly perturbed problems are investigated, in the case of a change in stability, all work was performed in the space of analytical functions. Naturally, questions will arise whether it is possible to obtain an estimate of solutions to a singularly perturbed problem without moving to the complex plane. In the work, the first results obtained are the solutions of the singularly motivated task, not moving into the complex plane. For this purpose, a method of regularization in the space of generalized functions has been developed and corresponding estimates have been obtained. If we choose the starting point in a stable interval, then up to the transition point, the asymptotic proximity of solutions to the perturbed and undisturbed problem is in the order of a small parameter ε. The problem will appear when the point belongs to an unstable interval. Therefore, prior to this, the works moved to the complex plane. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. In such problems, there is a concept of the delay time of solutions to the perturbed and undisturbed problem. Level lines will appear in complex planes. At special points, these lines have critical level lines. Therefore, it is impossible to choose the starting point so as to get the maximum delay time. But the asymptotic proximity of solutions of perturbed and undisturbed problems is possible with limited time delays. If we study the solution in the space of generalized functions, then we can choose the starting point with the maximum time delay. And also, without passing to the complex plane, it is possible to establish the asymptotic proximity of solutions to the perturbed and undisturbed problem. For this purpose, a method of regularization of solutions of a singularly perturbed problem has been developed for the first time.