Смешанные интегро-дифференциальные уравнения с операторами Капуто дробного порядка и спектральными параметрами
Ключевые слова:
интегро-дифференциальное уравнение, уравнение смешанного типа, малый параметр, спектральные параметры, дробные операторы Капуто, однозначная разрешимость
Аннотация
Рассматриваются вопросы однозначной разрешимости краевой задачи для интегро-дифференциального уравнения смешанного типа с двумя операторами Капуто дробного порядка и спектральными параметрами. Интегро-дифференциальное уравнение смешанного типа является интегро-дифференциальным уравнением с частными производными дробного порядка как в положительной, так и в отрицательной частях рассматриваемой многомерной прямоугольной области. Порядок дробного оператора Капуто меньше в положительной части области, чем порядок соответствующего оператора в отрицательной части области. Используя метод рядов Фурье, получены две системы счетных систем обыкновенных дробных интегро-дифференциальных уравнений с вырожденными ядрами. Далее используется метод вырожденных ядер. Для определения произвольных постоянных интегрирования получена система алгебраических уравнений. Из этой системы были вычислены регулярные и нерегулярные значения спектральных параметров. Решение рассматриваемой задачи получено в виде рядов Фурье. Доказана однозначная разрешимость задачи для регулярных значений спектральных параметров. При доказательстве сходимости рядов Фурье используются свойства функции Миттаг-Леффлера, неравенство Коши-Шварца и неравенство Бесселя. Также изучена непрерывная зависимость решения задачи от малого параметра при регулярных значениях спектральных параметров. Результаты сформулированы в виде теоремы.
Литература
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https://doi.org/10.1134/S0012266118120108
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https://doi.org/10.1134/S0965542519020167
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http://mi.mathnet.ru/eng/uzeru453
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https://doi.org/10.20537/2226-3594-2019-54-09
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https://doi.org/10.1134/S1995080220010151
2. Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics // Fractals and fractional calculus in continuum mechanics. Wien: Springer, 1997.
https://doi.org/10.1007/978-3-7091-2664-6_7
3. Area I., Batarfi H., Losada J., Nieto J.J., Shammakh W., Torres A. On a fractional order Ebola epidemic model // Advances in Difference Equations. 2015. Vol. 2015. Issue 1. Article 278.
https://doi.org/10.1186/s13662-015-0613-5
4. Hussain A., Baleanu D., Adeel M. Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model // Advances in Difference Equations. 2020. Vol. 2020. Issue 1. Article 384.
https://doi.org/10.1186/s13662-020-02845-0
5. Ullah S., Khan M.A., Farooq M., Hammouch Z., Baleanu D. A fractional model for the dynamics of tuberculosis infection using Caputo-Fabrizio derivative // Discrete and Continuous Dynamical Systems - S. 2020. Vol. 13. No. 3. P. 975-993.
https://doi.org/10.3934/dcdss.2020057
6. Tenreiro Machado J.A. Handbook of fractional calculus with applications in 8 volumes. Berlin-Boston: Walter de Gruyter GmbH, 2019.
7. Kumar D., Baleanu D. Editorial: fractional calculus and its applications in physics // Frontiers in Physics. 2019. Vol. 7.
https://doi.org/10.3389/fphy.2019.00081
8. Sun H., Chang A., Zhang Y., Chen W. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications // Fractional Calculus and Applied Analysis. 2019. Vol. 22. Issue 1. P. 27-59.
https://doi.org/10.1515/fca-2019-0003
9. Saxena R.K., Garra R., Orsingher E. Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives // Integral Transforms and Special Functions. 2016. Vol. 27. Issue 1. P. 30-42.
https://doi.org/10.1080/10652469.2015.1092142
10. Patnaik S., Hollkamp J.P., Semperlotti F. Applications of variable-order fractional operators: a review // Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2020. Vol. 476. No. 2234. Article 20190498.
https://doi.org/10.1098/rspa.2019.0498
11. Sandev T., Tomovski Z. Fractional equations and models. Cham: Springer, 2019.
https://doi.org/10.1007/978-3-030-29614-8
12. Gel'fand I.M. Some questions of analysis and differential equations // Uspekhi Matematicheskikh Nauk. 1959. Vol. 14. Issue 3 (87). P. 3-19 (in Russian).
http://mi.mathnet.ru/eng/umn7309
13. Uflyand Ya.S. On oscillation propagation in compound electric lines // Inzhenerno-Phizicheskiy Zhurnal. 1964. Vol. 7. No. 1. P. 89-92 (in Russian).
14. Terlyga O., Bellout H., Bloom F. A hyperbolic-parabolic system arising in pulse combustion: existence of solutions for the linearized problem // Electronic Journal of Differential Equations. 2013. Vol. 2013. No. 46. P. 1-42.
https://ejde.math.txstate.edu/Volumes/2013/46/terlyga.pdf
15. Abdullaev O.Kh., Sadarangani K.S. Non-local problems with integral gluing condition for loaded mixed type equations involving the Caputo fractional derivative // Electronic Journal of Differential Equations. 2016. Vol. 2016. No. 164. P. 1-10.
https://ejde.math.txstate.edu/Volumes/2016/164/abdullaev.pdf
16. Agarwal P., Berdyshev A.S., Karimov E.T. Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative // Results in Mathematics. 2017. Vol. 71. Issue 3-4. P. 1235-1257.
https://doi.org/10.1007/s00025-016-0620-1
17. Zarubin A.N. Boundary value problem for a differential-difference mixed-compound equation with fractional derivative and with functional delay and advance // Differential Equations. 2019. Vol. 55. Issue 2. P. 220-230.
https://doi.org/10.1134/S0012266119020071
18. Karimov E., Al-Salti N., Kerbal S. An inverse source non-local problem for a mixed type equation with a Caputo fractional differential operator // East Asian Journal on Applied Mathematics. 2017. Vol. 7. Issue 2. P. 417-438.
https://doi.org/10.4208/eajam.051216.280217a
19. Karimov E.T., Kerbal S., Al-Salti N. Inverse source problem for multi-term fractional mixed type equation // Advanes in real and complex analysis with applications. Singapore: Birkhäuser, 2017. P. 289-301.
https://doi.org/10.1007/978-981-10-4337-6_13
20. Repin O.A. Nonlocal problem with Saigo operators for mixed type equation of the third order // Russian Mathematics. 2019. Vol. 63. Issue 1. P. 55-60.
https://doi.org/10.3103/S1066369X19010067
21. Repin O.A. On a problem for a mixed-type equation with fractional derivative // Russian Mathematics. 2018. Vol. 62. Issue 8. P. 38-42.
https://doi.org/10.3103/S1066369X18080066
22. Salakhitdinov M.S., Karimov ET. Uniqueness of an inverse source non-local problem for fractional order mixed type equations // Eurasian Mathematical Journal. 2016. Vol. 7. No. 1. P. 74-83.
http://mi.mathnet.ru/rus/emj/v7/i1/p74
23. Yuldashev T.K., Kadirkulov B.J. Boundary value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator // Axioms. 2020. Vol. 9. Issue 2. Article 68.
https://doi.org/10.3390/axioms9020068
24. Yuldashev T.K., Kadirkulov B.J. Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator // Ural Mathematical Journal. 2020. Vol. 6. No. 1. P. 153-167.
https://doi.org/10.15826/umj.2020.1.013
25. Yuldashev T.K. Nonlocal boundary value problem for a nonlinear Fredholm integro-differential equation with degenerate kernel // Differential Equations. 2018. Vol. 54. Issue 12. P. 1646-1653.
https://doi.org/10.1134/S0012266118120108
26. Yuldashev T.K. On the solvability of a boundary value problem for the ordinary Fredholm integrodifferential equation with a degenerate kernel // Computational Mathematics and Mathematical Physics. 2019. Vol. 59. Issue 2. P. 241-252.
https://doi.org/10.1134/S0965542519020167
27. Yuldashev T.K. On an integro-differential equation of pseudoparabolic-pseudohyperbolic type with degenerate kernels // Proceedings of the Yerevan State University. Physical and Mathematical Sciences. 2018. Vol. 52. Issue 1. P. 19-26.
http://mi.mathnet.ru/eng/uzeru453
28. Yuldashev T.K. Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations // Axioms. 2020. Vol. 9. Issue 2. Article 45.
https://doi.org/10.3390/axioms9020045
29. Yuldashev T.K. Spectral singularities of solutions to a boundary-value problem for the Fredholm integro-differential equation of the second order with reflection of argument // Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. 2019. Vol. 54. P. 122-134 (in Russian).
https://doi.org/10.20537/2226-3594-2019-54-09
30. Yuldashev T.K. On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument // Lobachevskii Journal of Mathematics. 2020. Vol. 41. Issue 1. P. 111-123.
https://doi.org/10.1134/S1995080220010151
Поступила в редакцию
2020-08-09
Опубликована 2021-05-20
Опубликована 2021-05-20