kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-12-23kaz29-3171Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESДВУХФАЗНЫЕ ЗАДАЧИ ТЕПЛОПРОВОДНОСТИ С ГРАНИЧНЫМИ УСЛОВИЯМИ ТИПА ШТУРМА И С ДРОБНОЙ ПРОИЗВОДНОЙ ПО ВРЕМЕНИTWO-PHASE HEAT CONDUCTION PROBLEMS WITH STURM-TYPE BOUNDARY CONDITIONS AND WITH A FRACTIONAL TIME DERIVATIVEhttps://orcid.org/0000-0003-1752-7848КойлышовУ.KoilyshovU.

К.ф.-м.н., ассоциированный профессор

Алматы

Cand.Phys.-Math.Sc., Associate Professor

Almaty

koylyshov@math.kzhttps://orcid.org/0009-0007-4580-0983ЖаппароваС.ZhapparovaS.

Докторант

Алматы

PhD student

Almaty

zhapparova.saule@gmail.comhttps://orcid.org/0009-0006-5677-5464НиетбаевА.NietbayevA.

К.ф.-м.н., ассоциированный профессор

Тараз

Cand.Phys.-Math.Sc., Associate Professor

Taraz

a.niet57@mail.ruИнститут математики и математического моделирования; Казахский национальный университет им. аль-ФарабиКазахстанInstitute of Mathematics and Mathematical Modeling; Al-Farabi Kazakh National UniversityKazakhstanТаразский региональный университет им. М.Х. ДулатиКазахстанDulaty UniversityKazakhstan2026240920262331223Copyright © Койлышов У., Жаппарова С., Ниетбаев А., 20262026Койлышов У., Жаппарова С., Ниетбаев А.Koilyshov U., Zhapparova S., Nietbayev A.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/3171

В работе исследуется начально-краевая задача типа Штурма для уравнения теплопроводности с ку­сочно-постоянным коэффициентом и производной Капуто порядка < α < 1 по времени. Рассматрива­емая область представляет собой интервал :", , содержащий строго внутреннюю точку разрыва x = x0. С физической точки зрения эта задача моделирует процесс распространения темпепатупиого поля в тонком стержне длины . Стержень является составным и состоит из двух участков (0, x0)  и (x0, l),  обладающих различными теплофизическими характеристиками. В точке контакта двух сред x = x0 задаются условия идеального контакта, подразумевающие непрерывность температуры и непрерывность теплового потока при переходе из одной среды в другую. Основной целью работы является обоснование решения поставлен­ной начально-краевой задачи методом разделения переменных (методом Фурье). Применение этого метода приводит к необходимости исследования соответствующей спектральной задачи для обыкновенного диффе­ренциального оператора с разрывным коэффициентом при старшей производной.

The paper investigates an initial-boundary value problem of the Sturm type for the heat equation with a piecewise-constant coefficient and a Caputo derivative of order 0 < a < 1 with respect to time. The domain under consideration is an interval (0, l) containing a strictly internal point of discontinuity x = x0. From a physical point of view, this problem models the process of temperature field propagation in a thin rod of length l. The rod is composite and consists of two sections (0, x0) and (x0, l), possessing different thermophysical characteristics. At the contact point of the two media x = x0 , perfect contact conditions are prescribed, implying continuity of temperature and continuity of heat flux when passing from one medium to another. The main goal of the work is to substantiate the solution of the posed initial-boundary value problem by the method of separation of variables (Fourier method). Application of this method leads to the necessity of investigating the corresponding spectral problem for an ordinary differential operator with a discontinuous coefficient at the highest derivative.

уравнение теплопроводностиразрывные коэффициентысобственные функциисобственные значениядробные производныеметод разделения переменныхheat equationfractional derivativesdiscontinuous coefficientseigenvalueseigenfunctionsmethod of separation of variablesThis research was carried out under the project “Financing of scientific organizations carrying out fundamental scientific research for 2026-2028” (Project No.BR31714735)ReferencesGorenflo, R., Mainardi, F. Fractional calculus. In: Carpinteri, A., Mainardi, F. (eds). Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences, vol. 378. Springer, Vienna (1997). https://doi.org/10.1007/978-3-7091-2664-6_5.Gorenflo, R., Mainardi, F. Fractional calculus. In: Carpinteri, A., Mainardi, F. (eds). Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences, vol. 378. Springer, Vienna (1997). https://doi.org/10.1007/978-3-7091-2664-6_5.Wyss, W. The fractional diffusion equation. Journal of Mathematical Physics, 27, 2782–2785 (1986). https://doi.org./10.1063/1.527251.Wyss, W. The fractional diffusion equation. Journal of Mathematical Physics, 27, 2782–2785 (1986). https://doi.org./10.1063/1.527251.Mainardi, F. The fundamental solution for the fractional diffusion-wave equation. Applied Mathematics Letters, 9(6), 23-28 (1996). https://www.sciencedirect.com/science/article/pii/0893965996000894Mainardi, F. The fundamental solution for the fractional diffusion-wave equation. Applied Mathematics Letters, 9(6), 23-28 (1996). https://www.sciencedirect.com/science/article/pii/0893965996000894Luchko, Y. Maximum principle for the generalized time-fractional diffusion equation. Journal of Mathematical Analysis and Applications, 351(1), 218–223 (2009).Luchko, Y. Maximum principle for the generalized time-fractional diffusion equation. Journal of Mathematical Analysis and Applications, 351(1), 218–223 (2009).Agrawal, O.P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynamics, 29(1), 145–155 (2002).Agrawal, O.P. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dynamics, 29(1), 145–155 (2002).Псху, А.В. Уравнения в частных производных дробного порядка. М.: Наука (2005).Псху, А.В. Уравнения в частных производных дробного порядка. М.: Наука (2005).Samarskiy, A.A. Parabolic equations with discontinuous coefficients. Doklady Akademii Nauk SSSR, 121(2), 225–228 (1958).Samarskiy, A.A. Parabolic equations with discontinuous coefficients. Doklady Akademii Nauk SSSR, 121(2), 225–228 (1958).Hald, O.H. Discontinuous inverse eigenvalue problems. Communications on Pure and Applied Mathematics, 37, 539–577 (1984). https://doi.org/10.1002/cpa.3160370502.Hald, O.H. Discontinuous inverse eigenvalue problems. Communications on Pure and Applied Mathematics, 37, 539–577 (1984). https://doi.org/10.1002/cpa.3160370502.Il’in, V.A., Kritskov, L.V. Properties of spectral expansions corresponding to non-self-adjoint differential operators. In: Functional Analysis. Itogi Nauki i Tekhniki. Series: Sovremennaya Matematika. Prilozheniya. Tematicheskie Obzory, Vol. 96, VINITI, Moscow, 5–105 (2006).Il’in, V.A., Kritskov, L.V. Properties of spectral expansions corresponding to non-self-adjoint differential operators. In: Functional Analysis. Itogi Nauki i Tekhniki. Series: Sovremennaya Matematika. Prilozheniya. Tematicheskie Obzory, Vol. 96, VINITI, Moscow, 5–105 (2006).English transl.: Journal of Mathematical Sciences (New York), 116(5), 3489–3550 (2003).English transl.: Journal of Mathematical Sciences (New York), 116(5), 3489–3550 (2003).Kapustin, N.Y., Moiseev, E.I. The basis property in Lp of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition. Differential Equations, 36, 1498–1501 (2000). https://doi.org/10.1007/BF02757389.11.Kapustin, N.Y., Moiseev, E.I. The basis property in Lp of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition. Differential Equations, 36, 1498–1501 (2000). https://doi.org/10.1007/BF02757389.11.Koilyshov, U., Sadybekov, M., Beisenbayeva, K. Solution to initial-boundary value problem for the heat conductivity equation with a discontinuous coefficient and general conjugation conditions. International Journal of Mathematics and Mathematical Sciences, 2025, Article ID 2756189, 9 pp. (2025). https://doi.org/10.1155/ijmm/2756189.Koilyshov, U., Sadybekov, M., Beisenbayeva, K. Solution to initial-boundary value problem for the heat conductivity equation with a discontinuous coefficient and general conjugation conditions. International Journal of Mathematics and Mathematical Sciences, 2025, Article ID 2756189, 9 pp. (2025). https://doi.org/10.1155/ijmm/2756189.Koilyshov, U.K., Sadybekov, M.A., Beisenbayeva, K.A. Solution of nonlocal boundary value problems for the heat equation with discontinuous coefficients in the case of two discontinuity points. Bulletin of the Karaganda University. Mathematics Series, 117(1), 81–91 (2025). https://doi.org/10.31489/2025M1/81-91.Koilyshov, U.K., Sadybekov, M.A., Beisenbayeva, K.A. Solution of nonlocal boundary value problems for the heat equation with discontinuous coefficients in the case of two discontinuity points. Bulletin of the Karaganda University. Mathematics Series, 117(1), 81–91 (2025). https://doi.org/10.31489/2025M1/81-91.Koilyshov, U.K., Sadybekov, M.A., Beisenbayeva, K.A. Solution of initial-boundary value problem for heat equation with a discontinuous coefficient and general conjugation condition. Filomat, 39(23), 7997– 8005 (2025).Koilyshov, U.K., Sadybekov, M.A., Beisenbayeva, K.A. Solution of initial-boundary value problem for heat equation with a discontinuous coefficient and general conjugation condition. Filomat, 39(23), 7997– 8005 (2025).Koilyshov, U., Sadybekov, M. Two-phase heat conduction problems with Sturm-type boundary conditions. Boundary Value Problems, 2024, Article ID 149 (2024). https://doi.org/10.1186/s13661-024- 01964-x.Koilyshov, U., Sadybekov, M. Two-phase heat conduction problems with Sturm-type boundary conditions. Boundary Value Problems, 2024, Article ID 149 (2024). https://doi.org/10.1186/s13661-024- 01964-x.Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204. Elsevier, New York (2006). https://doi.org/10.1016/S0304-0208(06)80001-0.Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204. Elsevier, New York (2006). https://doi.org/10.1016/S0304-0208(06)80001-0.

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