najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-1-5-15najo-25Research ArticleMATHEMATICSМАТЕМАТИКАExistence and uniqueness theorem for a weak solution of fractional parabolic problem by the Rothe methodТеорема существования и единственности слабого решения дробной параболической задачи методом Ротеhttps://orcid.org/0000-0001-5672-4348БекакраЮ.BekakraY.

Department of Mathematics and Informatics.

Laghrour University, Khenchela, 04000

youcef.bekakra@univ-khenchela.dzhttps://orcid.org/0000-0002-1216-8033БузианиА.BouzianiA.

Department of Mathematics and Informatics, ICOSI Laboratory University; Department of Mathematics, L’arbi Ben M’hidi University

Khenchela, 04000, Algeria; Oum El Bouagui, 04000

aefbouziani1963@gmail.comICOSI Laboratory, Abbes Laghrour UniversityAlgeriaICOSI Laboratory, Abbes Laghrour University; L’arbi Ben M’hidi UniversityAlgeria202431052025151515Copyright © Bekakra Y., Bouziani A., 20252025Бекакра Ю., Бузиани А.Bekakra Y., Bouziani A.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/25

This paper aims to study the existence and uniqueness of a weak solution for the boundary value problem of a time fractional equation involving the Caputo fractional derivative with an integral operator. By utilizing the discretization method, we first derive some a priori estimates for the approximate solutions at the points (x, tj). We then evaluate the accuracy of the proposed method to demonstrate that the implemented sequence of α-Rothe functions converges in a certain sense, and its limit is the solution (in a weak sense) of our problem. It must be pointed out that the constructed L1 scheme is designed to approximate the Caputo fractional derivative mentioned in the problem.

Целью данной работы является исследование существования и единственности слабого решения краевой задачи дробного уравнения по времени, включающего дробную производную Капуто с интегральным оператором. Используя метод дискретизации, мы сначала получаем некоторые априорные оценки приближенных решений в точках (x, tj). Затем мы оценим точность предложенного метода, чтобы продемонстрировать, что реализованная последовательность α-функций Роте сходится в определенном смысле, а ее предел является решением (в слабом смысле) нашей задачи. Следует отметить, что построенная схема L1 предназначена для аппроксимации упомянутой в задаче дробной производной Капуто.

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