najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-2-160-169najo-30Research ArticleMATHEMATICSМАТЕМАТИКАMixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditionsСмешанная задача для линейного дифференциального уравнения параболического типа с нелинейными импульсными условиямиhttps://orcid.org/0000-0002-9346-5362ЮлдашевТ. К.YuldashevT. K.

Tursun K. Yuldashev.

Karimov street, 49, TSUE, Tashkent, 100066

t.yuldashev@tsue.uzhttps://orcid.org/0000-0001-6798-3265ФайзиевА. K.FayziyevA. K.

Aziz K. Fayziyev.

Karimov street, 49, TSUE, Tashkent, 100066

fayziyev.a@inbox.ruTashkent State University of EconomicsUzbekistan202431052025152160169Copyright © Yuldashev T.K., Fayziyev A.K., 20252025Юлдашев Т.К., Файзиев А.K.Yuldashev T.K., Fayziyev A.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/30

In this paper, we consider a linear parabolic type partial differential equation in the space of generalized functions as the equation of neutron diffusion in the presence of neutron absorption by the atomic nucleus with nonlinear impulsive effects. Spectral equation is obtained from the Dirichlet boundary value conditions and this spectral problem is studied. The Fourier method of variables separation is used. Countable system of nonlinear functional integral equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of the countable system of functional integral equations is proved. The method of successive approximations is used in combination with the method of contracting mapping. Criteria of uniqueness and existence of generalized solution of the impulsive mixed problem is obtained. Solution of the mixed problem is derived in the form of the Fourier series. It is shown that the Fourier series converges uniformly.

В данной работе в пространстве обобщенных функций рассматривается линейное уравнение в частных производных параболического типа как уравнение диффузии нейтронов при наличии поглощения нейтронов атомным ядром с нелинейными импульсными эффектами. С помощью краевых условий Дирихле получена спектральная задача и исследована эта проблема. Применен метод Фурье разделения переменных. Получена счетная система нелинейных функционально-интегральных уравнений относительно коэффициентов Фурье неизвестной функции. Доказана теорема об однозначной разрешимости счетной системы функциональных интегральных уравнений. Метод последовательных приближений используется в сочетании с методом сжимающего отображения. Получены критерии единственности и существования обобщенного решения импульсной смешанной задачи. Решение смешанной задачи получено в виде ряда Фурье. Доказана равномерная сходимость рядов Фурье.

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The authors declare that there are no conflicts of interest present.