ОБ АСИМПТОТИКЕ РЕШЕНИЯ ОДНОЙ СИНГУЛЯРНО ВОЗМУЩЕННОЙ КРАЕВОЙ ЗАДАЧИ С НАЧАЛЬНЫМ СКАЧКОМ
Journal Title: Science Review - Year 2017, Vol 1, Issue 4
Abstract
This work focuses on building the asymptotic decomposition to solve the singularly disturbed boundary value problem with an initial leap of the first order for the linear differential equation of the third order with constant coefficients. The variable, which the solution of the differential equation depends upon, is considered on the range from zero to one. The conditions, under which the asymptotic decomposition will be built by the small parameter to solve the studied boundary value problem, have been formulated. The undisturbed initial problem derived from the initial one by zero value of the small parameter has been indicated. The solution of the set singularly disturbed boundary value problem has been studied by way of reducing it to Cauchy problem solution with an initial leap of the first order. By means of boundary curves the asymptotic decomposition to solve this initial problem as the sum of two power serious by the small parameter the first of which consists of regular functions and the second one consists of boundary functions. The regular functions are determined from linear differential equations of the second order and they meet the obtained initial conditions. The boundary functions are determined from linear differential equations of the third order with their initial conditions. It has been proved that regular functions are limited and boundary functions assume exponential estimates. It has been found out the dominant term of the regular asymptotics coincides with solution of the undisturbed initial problem. The value of the initial leap of the first order to solve the boundary value problem has been obtained. Suppose we substitute the asymptotic decomposition to solve auxiliary Cauchy problem for the boundary conditions for the initial singularly disturbed differential equation with variable value at the end of the studied range then we can obtain some system of algebraic equations. Doing this system we obtain parameters values when the solution for the auxiliary initial problem will also be the only solution for the studied singularly disturbed boundary value problem.
Authors and Affiliations
А. Т. Есимова, М. А. Абдиманапова
Keywords
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