kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-80-92kaz29-3177Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО КРАЕВОЙ ЗАДАЧЕ ДЛЯ НАГРУЖЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ С ДРОБНОЙ ПРОИЗВОДНОЙON A BOUNDARY VALUE PROBLEM FOR A LOADED FRACTIONAL DIFFERENTIAL EQUATIONhttps://orcid.org/0000-0002-0680-4099ИскаковаН. Б.IskakovaN. B.

Кандидат физико-математических наук, ассоциированный профессор

Алматы

Almaty

narkesh77@gmail.comhttps://orcid.org/0000-0002-3820-5373БакироваЭ. А.BakirovaE. A.

Кандидат физико-математических наук, профессор

Алматы

Candidate of Physical and Mathematical Sciences, Professor

Almaty

bakirova.elmira74@gmail.comhttps://orcid.org/0000-0002-4311-5807УсмановК. И.UsmanovK. I.

Кандидат физико-математических наук, ассоциированный профессор

Туркестан

Candidate of Physical and Mathematical Sciences, Associate Professor

Turkistan

kairat.usmanov@ayu.edu.kzhttps://orcid.org/0000-0003-1714-6850ОрумбаеваН. Т.OrumbayevaN. T.

Кандидат физико-математических наук, ассоциированный профессор

Алматы, Караганда

Candidate of Physical and Mathematical Sciences, Associate Professor

Almaty, Karaganda

Orumbayevanurgul@gmail.comИнститут математики и математического моделированияКазахстанInstitute of Mathematics and Mathematical ModelingKazakhstanИнститут математики и математического моделирования; Казахский национальны женский педагогический университетКазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women’s Teacher Training UniversityKazakhstanКарагандинский национальный исследовательский университет имени академикаКазахстанAhmet Yassawi University, Turkistan, KazakhstanKazakhstanИнститут математики и математического моделирования; Карагандинский национальный исследовательский университет имени академика Е.А. БукетоваКазахстанInstitute of Mathematics and Mathematical Modeling; Karaganda Buketov National Research UniversityKazakhstan2026250920262338092Copyright © Искакова Н.Б., Бакирова Э.А., Усманов К.И., Орумбаева Н.Т., 20262026Искакова Н.Б., Бакирова Э.А., Усманов К.И., Орумбаева Н.Т.Iskakova N.B., Bakirova E.A., Usmanov K.I., Orumbayeva N.T.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/3177

В работе исследуется двухточечная краевая задача для нагруженного дифференциального уравнения с дробной производной Капуто. Для исследования поставленной задачи применяется метод параметризации Джумабаева. В рамках данного подхода исходный промежуток разбивается на конечное число частей, вводятся дополнительные параметры и новые функции, что позволяет свести исходную краевую задачу к эквивалентной многоточечной краевой задаче с неизвестными параметрами. Устанавливается эквивалентность исходной и преобразованной задач. На основе полученных интегральных представлений строится система алгебраических уравнений относительно введённых параметров. Устанавливаются достаточные условия существования и единственности решения, связанные с обратимостью соответствующей матрицы. Для нахождения неизвестных параметров и функций разработан алгоритм, основанный на методе последовательных приближений. Приведён численный пример, демонстрирующий применение предложенного алгоритма и подтверждающий его эффективность. Полученные результаты показывают эффективность метода параметризации при исследовании краевых задач для нагруженных дифференциальных уравнений дробного порядка и построении их приближённых решений.

The paper investigates a two-point boundary value problem for a loaded differential e quation with a Caputo fractional derivative. The Dzhumabaev parametrization method is applied to study the problem. Within this approach, the initial interval is divided into a finite n umber of s ubintervals, a nd additional parameters and new functions are introduced, which makes it possible to reduce the original boundary value problem to an equivalent multipoint boundary value problem with unknown parameters. The equivalence of the original and transformed problems is established. Based on the obtained integral representations, a system of algebraic equations with respect to the introduced parameters is constructed. Sufficient conditions for the existence and uniqueness of a solution are established and related to the invertibility of the corresponding matrix. An algorithm based on the method of successive approximations is developed to determine the unknown parameters and functions. A numerical example demonstrating the application of the proposed algorithm and confirming its e ffectiveness is presented. The ob tained results de monstrate the efficiency of the pa rametrization me thod for in vestigating bo undary va lue pr oblems for lo aded fractionalorder differential e quations and c onstructing t heir a pproximate solutions.

задача для нагруженного дифференциального уравненияпроизводная Капутометод параметризацииприближенное решениеproblem for loaded differential e quationC aputo d erivativep arametrization’s methodapproximate solutionThis research is partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP26194447)ReferencesPodlubny I., Fractional Differential Equations. Academic Press, New York, 1999.Podlubny I., Fractional Differential Equations. Academic Press, New York, 1999.Diethelm K., Freed A.D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.). Scientific Computing in Chemical Engineering II Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 217–224 (1999).Diethelm K., Freed A.D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in: F. Keil, W. Mackens, H. Voss, J. Werther (Eds.). Scientific Computing in Chemical Engineering II Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, 217–224 (1999).Glockle W.G., Nonnenmacher T.F., A fractional calculus approach of self-similar protein dynamics. Biophys. J., 68, 46–53 (1995).Glockle W.G., Nonnenmacher T.F., A fractional calculus approach of self-similar protein dynamics. Biophys. J., 68, 46–53 (1995).Kirchner J.W., Feng X., Neal V., Fractal streamchemistry and its implications for contaminant transport in catchments. Nature, 403, 524–526 (2000). doi.org/10.1038/35000537Kirchner J.W., Feng X., Neal V., Fractal streamchemistry and its implications for contaminant transport in catchments. Nature, 403, 524–526 (2000). doi.org/10.1038/35000537Lundstrom B.N., Higgs M.H., Spain W.J., Fairhall A.L., Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci, 11, 1335–1342 (2008). doi.org/10.1038/nn.2212Lundstrom B.N., Higgs M.H., Spain W.J., Fairhall A.L., Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci, 11, 1335–1342 (2008). doi.org/10.1038/nn.2212Mainardi F., Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.). Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 291–348 (1997).Mainardi F., Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.). Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 291–348 (1997).Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 204 (2006).Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies. Elsevier Science B.V., Amsterdam, 204 (2006).Miller K.S., Ross B., An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993).Miller K.S., Ross B., An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993).Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Longhorne, PA (1993).Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Longhorne, PA (1993).Lakshmikantham V., Vatsala A.S., Basic theory of fractional differential equations. Nonlinear Anal, 69(8), 2677–2682 (2008). doi.org/10.1016/j.na.2007.08.042Lakshmikantham V., Vatsala A.S., Basic theory of fractional differential equations. Nonlinear Anal, 69(8), 2677–2682 (2008). doi.org/10.1016/j.na.2007.08.042Lakshmikantham V., Vatsala A.S., Theory of fractional differential inequalities and applications. Commun. Appl. Anal., 11(3–4), 395–402 (2007).Lakshmikantham V., Vatsala A.S., Theory of fractional differential inequalities and applications. Commun. Appl. Anal., 11(3–4), 395–402 (2007).Belarbi A., Benchohra M., Ouahab A., Uniqueness results for fractional functional differential equations with in finite delay in Frechet spaces. Appl. Anal., 85(12), 1459–1470 (2006). doi.org/10.1080/00036810601066350Belarbi A., Benchohra M., Ouahab A., Uniqueness results for fractional functional differential equations with in finite delay in Frechet spaces. Appl. Anal., 85(12), 1459–1470 (2006). doi.org/10.1080/00036810601066350Benchohra M., Henderson J., Ntouyas S.K., Ouahab A., Existence results for fractional order functional differential equations with in finite delay. J. Math. Anal.Appl., 338(2), 1340–1350 (2008). doi.org/10.1016/j.jmaa.2007.06.021Benchohra M., Henderson J., Ntouyas S.K., Ouahab A., Existence results for fractional order functional differential equations with in finite delay. J. Math. Anal.Appl., 338(2), 1340–1350 (2008). doi.org/10.1016/j.jmaa.2007.06.021Usmanov K.I., Nazarova K.Zh., Yerkisheva Zh.S., Solvability of a boundary value problem for fractional-order integro-differential equations with involution. Herald of the Kazakh-British Technical University, 23(1), 231–239 (2026). doi.org/10.55452/1998-6688-2026-23-1-231-239Usmanov K.I., Nazarova K.Zh., Yerkisheva Zh.S., Solvability of a boundary value problem for fractional-order integro-differential equations with involution. Herald of the Kazakh-British Technical University, 23(1), 231–239 (2026). doi.org/10.55452/1998-6688-2026-23-1-231-239Torebek B.T., Turmetov B.K., On solvability of a boundary value problem for the Poisson equation with the boundary operator of a fractional order. Bound Value Probl 2013, 93 (2013). doi:10.1186/1687-2770-2013-93Torebek B.T., Turmetov B.K., On solvability of a boundary value problem for the Poisson equation with the boundary operator of a fractional order. Bound Value Probl 2013, 93 (2013). doi:10.1186/1687-2770-2013-93Nakhushev A. M. The loaded equations and their applications. M. Nauka, 2012, p. 232.Nakhushev A. M. The loaded equations and their applications. M. Nauka, 2012, p. 232.Iskakova N.B., Bakirova E.A., Khanzharova B.S., Kadirbayeva, Zh.M., An Algorithm for Solving Boundary Value Problems for Delay Differential Equations with Loadings. Lobachevskii Journal of Mathematics, 45(10), 5032–5042 (2024). doi.org/10.1134/S1995080224606064.Iskakova N.B., Bakirova E.A., Khanzharova B.S., Kadirbayeva, Zh.M., An Algorithm for Solving Boundary Value Problems for Delay Differential Equations with Loadings. Lobachevskii Journal of Mathematics, 45(10), 5032–5042 (2024). doi.org/10.1134/S1995080224606064.Bakirova E.A., Iskakova N.B., Kadirbayeva Zh.M., Numerical implementation of solving a boundary value problem including both delay and parameter. Mathematica Slovaca, 75(3), 601–610 (2025). doi.org/10.26577/JMMCS2023v119i3a2Bakirova E.A., Iskakova N.B., Kadirbayeva Zh.M., Numerical implementation of solving a boundary value problem including both delay and parameter. Mathematica Slovaca, 75(3), 601–610 (2025). doi.org/10.26577/JMMCS2023v119i3a2Dzhumabaev D.S., Bakirova E.A., Mynbayeva S.T., A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation. Mathematical Method in Applied Science, 43(4), 1788–1802 (2020). doi.org/10.1002/mma.6003.Dzhumabaev D.S., Bakirova E.A., Mynbayeva S.T., A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation. Mathematical Method in Applied Science, 43(4), 1788–1802 (2020). doi.org/10.1002/mma.6003.Abdullaev O.Kh., About a method of research of the non-local problem for the loaded mixed type equation in double-connected domain. Bulletin KRASEC. Phys. Math. Sci, 9(2), 11–16 (2014). doi.org/10.18454/2313-0156-2014-9-2-3-12Abdullaev O.Kh., About a method of research of the non-local problem for the loaded mixed type equation in double-connected domain. Bulletin KRASEC. Phys. Math. Sci, 9(2), 11–16 (2014). doi.org/10.18454/2313-0156-2014-9-2-3-12Islomov B., Baltaeva U., Boudary-value problems for a third-order loaded parabolic-hyperbolic type equation with variable coefficients. Electron. J. Differential Equ., 2015(221), 1–10 (2015). URL: http://ejde.math.txstate.eduIslomov B., Baltaeva U., Boudary-value problems for a third-order loaded parabolic-hyperbolic type equation with variable coefficients. Electron. J. Differential Equ., 2015(221), 1–10 (2015). URL: http://ejde.math.txstate.eduDiethelm K., Ford N. J., The analysis of fractional differential equations. Lecture notes in mathematics, Springer Heidelberg Dordrecht London New York, (2010).Diethelm K., Ford N. J., The analysis of fractional differential equations. Lecture notes in mathematics, Springer Heidelberg Dordrecht London New York, (2010).Ye H., Gao J., Ding Y., A generalized and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328(2), 1075-1081. (2007). 10.1016/j.jmaa.2006.05.061Ye H., Gao J., Ding Y., A generalized and its application to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328(2), 1075-1081. (2007). 10.1016/j.jmaa.2006.05.061

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