maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2016-3-370-376mais-352Research ArticleОригинальные статьиArticlesДвухточечная краевая задача Капуто: существование, единствен- ность и регулярность решенияA Caputo Two-Point Boundary Value Problem: Existence, Uniqueness and Regularity of a SolutionСтайнсМ.StynesM.m.stynes@csrc.ac.cnПекинский Исследовательский Центр Вычислительных Наук, район Хайдянь, Пекин 100193, КитайРоссияBeijing Computational Science Research Center, Haidian District, Beijing 100193, ChinaRussian Federation201620062016233370376Copyright © Стайнс М., 20162016Стайнс М.Stynes M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/352

Рассматривается двухточечная краевая задача на промежутке [0,1], в которой старшая производная является дробной производной Капуто порядка 2−δ при 0 < δ < 1. Получено необходимое и достаточное условие существования и единственности решения u. Производная uꌐ этого решения оказывается абсолютно непрерывной на [0,1]. Показано, что предположение о большей регулярности ꐀ что u принадлежит C2[0,1] ꋀ накладывает довольно тонкое ограничение на данные задачи.

A two-point boundary value problem on the interval [0, 1] is considered, where the highest-order derivative is a Caputo fractional derivative of order 2 − δ with 0 < δ < 1. A necessary and sufficient condition for existence and uniqueness of a solution u is derived. For this solution the derivative uŐ is absolutely continuous on [0, 1]. It is shown that if one assumes more regularity — that u lies in C2[0, 1] — then this places a subtle restriction on the data of the problem.

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