nucВестник НЯЦ РКNNC RK Bulletin1729-75161729-7885Национальный ядерный центр Республики Казахстан10.52676/1729-7885-2025-3-156-163nuc-871Research ArticleСтатьиМОДЕЛИРОВАНИЕ И РЕШЕНИЕ УРАВНЕНИЙ ВОЛНОВОГО ПРОЦЕССА С ИСПОЛЬЗОВАНИЕМ РЕКУРРЕНТНО-ОПЕРАТОРНОГО МЕТОДАMODELING AND SOLVING EQUATIONS OF THE WAVE PROCESS USING THE RECURRENT OPERATOR METHODhttps://orcid.org/0000-0002-2636-9717ПирниязоваП. М.PirniyazovaP. M.

Пирниязова Периуза Мамбетниязовна.

Алматы

Peiuza M. Pirniyazova - Department Computer Engineering.

Almaty

pirniyazova1974@gmail.comАлматинский технологический университетКазахстанAlmaty Technological UniversityKazakhstan20252210202503156163Copyright © Пирниязова П.М., 20252025Пирниязова П.М.Pirniyazova P.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://journals.nnc.kz/jour/article/view/871

Эта обзорная статья, в которой дается решение одномерной и трехмерной задачи о волновом процессе. Рассматриваемое уравнение является гиперболическим уравнением, которое может быть получено только для одномерного случая и описывает процесс речного стока, переноса вещества, диффузионного массопереноса и течения через пористую среду. Решение уравнения ищется в виде интегро-дифференциальных операторов, представленных в виде специального ряда с постоянными коэффициентами, определяемыми из рекуррентного соотношения. При таком подходе общие решения выражаются в терминах произвольных функций и не связаны с решением другого уравнения. Результирующий вид общих решений позволяет применять метод начальных функций для решения краевых задач, поскольку произвольные аналитические функции, входящие в общие решения, могут быть выражены в терминах начальных функций, заданных условием задачи. Из этих формул общих решений легко найти все частные решения в различных классах аналитических функций. Далее, рассматривается решение частного случая уравнения Ламе, описывающего волновой процесс. Решение частного случая уравнения Ламе также ищется в виде ряда с использованием рекуррентно-операторного метода. Алгоритм решения трехмерной задачи о волновом процессе аналогичен алгоритму решения одномерной задачи. Такой подход очень эффективен для построения решений уравнений теории теплопроводности и теории упругости. Полученные результаты проиллюстрированы графически и представлены в таблицах.

This article is a review, which provides a solution to the one-dimensional and three-dimensional problem of the wave process. The equation in question is a hyperbolic equation that can be derived only for the one-dimensional case and describes the process of river flow, matter transfer, diffusive mass transfer, and flow through a porous medium. The solution of the equation is sought in the form of integro-differential operators, represented as a special series with constant coefficients determined from the recurrence relation. With this approach, general solutions are expressed in terms of arbitrary functions and are not related to solving another equation. The resulting form of general solutions makes it possible to apply the method of initial functions to solve boundary value problems, since arbitrary analytical functions included in general solutions can be expressed in terms of initial functions specified by the problem condition. From these formulas of general solutions, it is easy to find all particular solutions in various classes of analytical functions. Next, the solution of a special case of the Lame equation describing the wave process is considered. The solution of a special case of the Lame equation is also sought in the form of a series, using the recurrent operator method. The algorithm for solving the three-dimensional problem of the wave process is similar to that for the one-dimensional problem. This approach is very effective for constructing solutions to the equations of the theory of thermal conductivity and the theory of elasticity. The results obtained are illustrated graphically and shown in the tables.

волновой процессчастный случай уравнения Ламерекуррентный операторный методwave processspecial case of the Lame equationrecurrent operator methodReferencesP.M. Pirniyazova. Mathematical modelling of ecologically unfavourable pollution hotspots. Monography. – Almaty, 2020. – 101 p.P.M. Pirniyazova. Mathematical modelling of ecologically unfavourable pollution hotspots. Monography. – Almaty, 2020. – 101 p.Pirniyazova, P. M. Modelling of a three-dimensional problem of distribution of harmful impurity in the river a recurrently-operational method / P. M. Pirniyazova // Евразийский союз ученых. – 2020. – No. 6–1 (75). – P. 8–12.Pirniyazova, P. M. Modelling of a three-dimensional problem of distribution of harmful impurity in the river a recurrently-operational method / P. M. Pirniyazova // Евразийский союз ученых. – 2020. – No. 6–1 (75). – P. 8–12.V. N. Frolov. Specialnye klassi funciy v anizotropnoy teoriy uprugosti. Monography. – Tashkent, 1981. – 223 p.V. N. Frolov. Specialnye klassi funciy v anizotropnoy teoriy uprugosti. Monography. – Tashkent, 1981. – 223 p.Spivakov Y.L. Special classes of solutions of linear differential equations and their application in anisotropic and heterogeneous elastic theory. Monography. – Tashkent: Fan, 1987. – 296 p.Spivakov Y.L. Special classes of solutions of linear differential equations and their application in anisotropic and heterogeneous elastic theory. Monography. – Tashkent: Fan, 1987. – 296 p.B.А. Bondarenko. Operator algorithms in differential equations. «FANӘ Publishing House.1984г.B.А. Bondarenko. Operator algorithms in differential equations. «FANӘ Publishing House.1984г.B.А. Bondarenko, P. M. Pirniyazova. Normalized systems of functions and their application to the solution of problems of diffusion equations. https://zbmath.org/pdf/05873058.pdfB.А. Bondarenko, P. M. Pirniyazova. Normalized systems of functions and their application to the solution of problems of diffusion equations. https://zbmath.org/pdf/05873058.pdfXu, D. An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections / D. Xu, J. Du, Z. Liu // Journal of Low Frequency Noise, Vibration and Active Control. – 2019. – Vol. 38, No. 2. – P. 403–414.Xu, D. An accurate and efficient series solution for the longitudinal vibration of elastically restrained rods with arbitrarily variable cross sections / D. Xu, J. Du, Z. Liu // Journal of Low Frequency Noise, Vibration and Active Control. – 2019. – Vol. 38, No. 2. – P. 403–414.P.M. Pirniyazova, D. Koshanova. Mathematical Modeling to Estimate the Diffusion of Traffic-Related Air Pollutants in Almaty City. 2023 IEEE Smart Information Systems and Technologies (SIST) 4-6 May, 2023, Astana, Kazakhstan.P.M. Pirniyazova, D. Koshanova. Mathematical Modeling to Estimate the Diffusion of Traffic-Related Air Pollutants in Almaty City. 2023 IEEE Smart Information Systems and Technologies (SIST) 4-6 May, 2023, Astana, Kazakhstan.E.V Rojkova. Longitudinal vibrations of the rod in a generalized elastic medium // Mechanics. Research and innovation. – Gomel, 2022. – Issue 15. ISSN 2519-8742.E.V Rojkova. Longitudinal vibrations of the rod in a generalized elastic medium // Mechanics. Research and innovation. – Gomel, 2022. – Issue 15. ISSN 2519-8742.Kachiashvili K.I., Gordeziani D.G., Melikjanyan D. I. Modern modeling and computer technologies for research and quality control of river water. – Tbilisi: GTU, 2007 – 251 p.Kachiashvili K.I., Gordeziani D.G., Melikjanyan D. I. Modern modeling and computer technologies for research and quality control of river water. – Tbilisi: GTU, 2007 – 251 p.Rozhkova, E. V. Recurrent operator method in problems of oscillations of rod systems / E. V. Rozhkova // News of the Russian Academy of Sciences. Solid state mechanics. – 2009. – No. 6. – P. 124–138.Rozhkova, E. V. Recurrent operator method in problems of oscillations of rod systems / E. V. Rozhkova // News of the Russian Academy of Sciences. Solid state mechanics. – 2009. – No. 6. – P. 124–138.Rozhkova, E. V. Solution of the generalized equation of oscillation of rods by the recurrent operator method / E. V. Rozhkova // News of the Russian Academy of Sciences. Solid state mechanics. – 2018. – No. 5. – P. 90–105.Rozhkova, E. V. Solution of the generalized equation of oscillation of rods by the recurrent operator method / E. V. Rozhkova // News of the Russian Academy of Sciences. Solid state mechanics. – 2018. – No. 5. – P. 90–105.Map of earthquakes in Almaty, Kakhastan City of Almaty (www.gov.kz ).Map of earthquakes in Almaty, Kakhastan City of Almaty (www.gov.kz ).

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