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)ReferencesIntroduction Models that incorporate fractional derivatives offer a more precise mathematical representation compared to classical integer-order models, as fractional differentiation a nd i ntegration make it possible to account for memory and hereditary properties characteristic of many materials and processes [1]. This capability is a key advantage of fractional-order approaches, since such effects a re t ypically b eyond t he s cope of t raditional d ifferential eq uations of in teger or der. The advancement of fractional differential e quation t heory h as s ignificantly ex panded th eir us e in modeling a wide range of phenomena in science and engineering. In particular, these equations have been effectively a pplied in a reas s uch as c ontrol t heory, t he p hysics of c omplex systems, neural modeling, electrochemistry, the mechanics of porous media, electromagnetism, and various other disciplines (see [2]-[6]). The growing interest in fractional calculus is reflected i n numerous fundamental monographs and review papers [7]-[9], as well as in the references cited therein. Significant p rogress h as b een a chieved i n t he s tudy of i nitial a nd b oundary v alue problems for fractional differential e quations w ith R iemann–Liouville o perators of o rder (0 < α ≤ 1) [10, 11]. In papers [12]-[15], certain classes of Cauchy problems for functional differential equations with fractional derivatives of both Riemann–Liouville and Caputo types of the same order were examined. Loaded differential e quations o ccupy a d istinctive p osition i n t he t heory of fractional differential e quations. O ne of t he m ost g eneral d efinitions of su ch eq uations wa s in troduced by Nakhushev [16]. The rapid development of research in optimal control of agro-economic systems, as well as in problems related to groundwater level regulation and soil moisture management, has led to an increased interest in boundary value problems for this class of equations [17]-[19]. Important results in the theory of boundary value problems for loaded equations of parabolic, parabolic–hyperbolic, and elliptic–hyperbolic types were obtained in [20, 21]. This paper is devoted to the study of boundary value problems for loaded differential e quations w ith a fractional derivative. The main objective is to establish conditions for the existence and uniqueness of solutions, as well as to develop numerical algorithms for their computation. Motivated by the above considerations, on the interval [0, T] we consider the following linear boundary value problem for a loaded differential e quation with a Caputo derivative of o rder (0, 1). Where and are continuous on Let is the space of absolutely continuous functions with the norm A function is called a solution of satisfies together with the condition (2). Mаtеriаl аnd mеthоds The main purpose of this work is to apply the parameterization method for solving problem (1), (2). Based on this method, an algorithm for finding its solution will be constructed, and sufficient conditions for its solvability will be established. Let We take a number ℓ ∈ N and construct a partition the interval where and so on Let us introduce the space consisting of vector-valued functions where each function is absolutely continuous on the interval and possesses a finite left-hand limit lim This space is endowed with the norm Then we denote the restriction of the function In this case the problem (1), (2) is reduced to a multipoint boundary value problem Here an equality (5) is conditions for matching the solution at the interior points and Let a solution to problem (3)-(5) is the system of functions where with a Caputo derivative that satisfy the system (3) and conditions (4), (5). Lemma 1. The problems (1),(2) and (3)-(5) are equivalent. Дәлелдеме. Let be a solution of problem (1),(2), and be its restriction to the intervals Then, from the absolute continuity of on it follows that is absolutely continuous on and possesses finite left-hand limits By virtue of (1), on each interval the equality holds That is the restriction of the function x∗(t) to the intervals satisfies equation (3). Condit ion (4) follows from condition (2). Conversely, let be a solution of(3)–(5), where Define From (4) and (5), it follows that x˜(t) is absolutely continuous on [0,T] and satisfies the boundary condition (2) Since the system of functions x˜[t] is a solution of equation (3), the function x˜(t) has a Caputo derivative and satisfies 1) for all t on (0,T) except at the loading points; that is, there exists a function Since it follows from (5) that CDαx(t) exists for all and satisfies equation (1). and substituting we transform the boundary value problem (3)-(5) into the equivalent multi-point boundary value problem with parameters A solution to the problem (6)-(9) is a pair with components that satisfy (6)-(9). If is a solution to problem (1), (2), then the pair with components where satisfies to problem (1), (2). Conversely, if the pair is a solution to problem (6)-(9), then the function defined by the equalities is a solution to origin problem (1), (2). On interval the initial value problem (6), (7) to the equivalent system of integral equations By taking the limit as in the right-hand side of (10) and substituting the resulting expressions into condition (9), as well as multiplying equation (8) by we obtain a system of equations with respect to the unknown parameters where for all The matrix corresponding to the left part of the system (11), (12) will be denoted by Let us write this system in the following form with the right sides of the view f the matrix is invertible, then to find the pair we use the method of successive approximations, which is carried out according to the following algorithm. Step 0. (a) Provided that, for the chosen he matrix is invertible, the initial approximation with respect to the parameter is obtained from the equation, that is, (b) Using the components of the vector we solve the initial value problem (6), (7) with on each interval As a result, we obtain the functions Step 1. (a) Substituting the obtained functions into the right-hand side of (13), we determine the next approximation λ(1) from the equation (b) On each interval, the initial value problem (6), (7) are then solved with yielding the functions And so on. Continuing the process, at the k-th step we obtain a system of pairs We introduce the following auxiliary definitions and theorem Definition 2. (Mittag–Leffler Function, [22]). Let n> 0. The function En defined by whevener the series converges is called the Mittag–Leffler function of order n. This function has been introduced by Mittag-Leffler. We immediately notice that is just the well known exponential function. Definition 3. Let. The function defined by whenever the series converges is called the two-parameter Mittag-Leffler function with parameters n1 and n2. Theorem 4. (Theorem 2.2, [23]) Let a(t) and g(t) be nondecreasing functions for all and let satisfies the inequality then where Eβ denotes the Mittag–Leffler function. Let us introduce the notations Theorem 5. Let the matrix be invertible and suppose that the following inequalities hold: Then the sequence of pairs converges to Дәлелдеме. Under the assumptions of the theorem, we define at the zeroth step of the algorithm. Then, the following estimate holds: Let be defined as Applying Gr¨onwall’s inequality, we obtain Taking into account (14), we obtain After determining from an equation we estimate the difference aving obtained as the solution of the Cauchy problem (10) for let us estimate the following difference Therefore, by the fractional Gr¨onwall inequality, we have that is, Proceeding with the iterative process, we obtain a sequence of pairs The following estimates hold for the differences and Since by the assumptions of the theorem 2, it follows from inequalities (15) and (16) that the sequence converges to , and the sequence of functions system converges to the function system in the norm of the space Numerical experiment We consider on [0, 1] the boundary value problem for a loaded fractional differential equation where and The exact solution to the problem at hand is the function We partition the given interval as follows and introduce parameters Consider the following substitution Then the test problem (17), (18) reduces to a problem with parameters where For fixed parameters, the solution to the initial problem (19), (20) is given by From (23)-(26) we determine lim Based on (21), (22), we construct the system of linear equations for unknown parameters Here where The results of implementing the proposed algorithm are as follows: Step 0. And Step 1. And And so on. Then All computations were performed in Spyder (Python 3.13). Conclusion In this work, an analysis of a two-point boundary value problem for a loaded differential equation with a fractional derivative is carried out. Within the framework of this study, the development of the parametrization method is presented, which makes it possible to reduce the original problem to an equivalent multipoint boundary value problem by introducing additional parameters. Based on the obtained formulation, the construction of approximate solutions is performed using an iterative procedure. The proposed algorithm provides successive refinement of the solution and allows one to establish sufficient conditions for the existence of a solution as well as for the convergence of the constructed iterative process. The obtained results demonstrate the effectiveness of the developed parametrization method for studying boundary value problems of the considered class. To ensure the continuity of the present research in this direction, it is planned in future work to prove that the obtained sufficient conditions for the solvability of the problem (1), (2) are also necessary.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.