ipolytechiPolytech JournaliPolytech Journal2782-40042782-6341Irkutsk National Research Technical University10.21285/1814-3520-2021-5-623-632ipolytech-529Research ArticleМЕТАЛЛУРГИЯ И МАТЕРИАЛОВЕДЕНИЕMETALLURGY AND MATERIALS SCIENCEЧисленная валидация метода разложения Адомиана для решения нелинейного уравнения мелкой водыDynamical control on the Adomian decomposition method for solving shallow water wave equationНойягдамЛ.NoeiaghdamL.

Нойягдам Лале, аспирант, факультет гражданского строительства и охраны окружающей среды

1591634311, г. Тегеран, пр. Хафеза, 350, пл. Валиасра

Laleh Noeiaghdam, Postgraduate Student, Department of Civil and Environmental Engineering

350, Hafez Ave, Valiasr Square, Tehran, 1591634311

l_noeiaghdam@yahoo.comhttps://orcid.org/0000-0002-2307-0891НойягдамС.NoeiaghdamS.

Нойягдам Самад, старший преподаватель, Лаборатория промышленной математики, Байкальский институт БРИКС, Иркутский национальный исследовательский технический университет; старший научный сотрудник кафедры прикладной математики и программирования, Южно-Уральский государственный университет

664074, г. Иркутск, ул. Лермонтова, 83454080, г. Челябинск, пр. Ленина, 76

Samad Noeiaghdam, Senior Lecturer, Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University; Senior Researcher of the Department of Applied Mathematics and Programming, South Ural State University

83 Lermontov St., Irkutsk 66407476, Lenin pr., Chelyabinsk 454080

snoei@istu.eduhttps://orcid.org/0000-0002-3131-1325СидоровД. Н.SidorovD. N.

Сидоров Денис Николаевич, доктор физико-математических наук, профессор РАН, заведующий Лабораторией промышленной математики, Байкальский институт БРИКС, Иркутский национальный исследовательский технический университет; главный научный сотрудник Отдела прикладной математики, Институт систем энергетики им. Л. А. Мелентьева СО РАН

664074, г. Иркутск, ул. Лермонтова, 83664033, г. Иркутск, ул. Лермонтова, 130

Denis N. Sidorov, Dr. Sci. (Physics & Mathematics), Professor of RAS, Head of the Laboratory of Industrial Mathematics, Baikal School of BRICS, Irkutsk National Research Technical University; Chief Researcher of the Department of Applied Mathematics, L. A. Melentiev Institute of Energy Systems of Siberian Branch of the Russian Academy of Sciences,

83 Lermontov St., Irkutsk 664074130, Lermontov St., Irkutsk 664033

dsidorov@isem.irk.ruТехнологический университет АмиркабираИранAmirkabir University of TechnologyIslamic Republic of IranИркутский национальный исследовательский технический университет; Южно-Уральский государственный университетРоссияIrkutsk National Research Technical University; South Ural State UniversityRussian FederationИркутский национальный исследовательский технический университет; Институт систем энергетики им. Л. А. Мелентьева СО РАНРоссияIrkutsk National Research Technical University; L. A. Melentiev Institute of Energy Systems of Siberian Branch of the Russian Academy of SciencesRussian Federation202109112021255623632Copyright © Нойягдам Л., Нойягдам С., Сидоров Д.Н., 20212021Нойягдам Л., Нойягдам С., Сидоров Д.Н.Noeiaghdam L., Noeiaghdam S., Sidorov D.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://ipolytech.elpub.ru/jour/article/view/529

Целью данной работы является применение новой методики для проведения эффективного контроля точности и анализа погрешности применения метода разложения Адомиана для решения нелинейного волнового уравнения мелкой воды, возникающего при выполнении ряда важных задач в различных областях машиностроения и теории материалов. Рассмотрены три важных случая: набегание жидкости на полуплоские берега, на берега с умеренным уклоном и на берега с крутым уклоном. В исследованиях применялся метод разложения Адомиана, являющийся полуаналитическим эффективным методом, обладающим большей гибкостью, чем прямое разложение в ряд Тейлора. В данном исследовании мы применяем метод CESTAC (Contrôle et Estimation STochastique des Arrondis de Calculs ), основанный на стохастической арифметике. Также вместо применения стандартных математических пакетов в работе эффективно использована библиотека контроля точности и отладки для численных приложений CADNA (Control of Accuracy and Debugging for Numerical Applications ). Программная реализация подхода с использованием библиотеки CADNA выполнена на C++ под операционную с истему LINUX. Вместо использования традиционной абсолютной ошибки, которая основана на точном решении и небольшом пороговом значении, используем новое правило останова, которое основано на двух последовательных приближениях. Основная теорема метода CESTAC показывает, что количество общих значащих цифр двух последовательных приближений практически равно количеству общих значащих цифр точного и приближенного решений. Применение представленной в работе методики позволило получить оптимальные численные результаты: найти погрешность и оптимальный шаг метода разложения Адомиана, чего не позволяли делать классические подходы. В этом заключается основная новизна работы. Приведены результаты тестирования разработанной численной модели для решения уравнения мелкой воды. Таким образом, при проведении численных расчетов с использованием предлагаемого метода разложения Адомиана продемонстрирована высокая точность и эффективность разработанного подхода для решения нелинейного волнового уравнения мелкой воды.

The aim of this study is to apply a novel technique to control the accuracy and error of the Adomian decomposition method (ADM) for solving nonlinear shallow water wave equation. The ADM is among semi-analytical and powerful methods for solving many mathematical and engineering problems. We apply the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method which is based on stochastic arithmetic (SA). Also instead of applying mathematical packages we use the Control of Accuracy and Debugging for Numerical Applications (CADNA) library. In this library we will write all codes using C++ programming codes. Applying the method we can find the optimal numerical results, error and step of the ADM and they are the main novelties of this research. The numerical results show the accuracy and efficiency of the novel scheme.

волновая задача на мелководьеметод разложения Адомианаметод CESTACбиблиотека CADNAshallow water wave problemAdomian decomposition methodCESTAC methodCADNA libraryReferencesAbdelsalam U. M. Traveling wave solutions for shallow water equations. Journal of Ocean Engineering and Science. 2017;2(1):28-33. https://doi.org/10.1016/j.joes.2017.02.002.Abdelsalam U. M. Traveling wave solutions for shallow water equations. Journal of Ocean Engineering and Science. 2017;2(1):28-33. https://doi.org/10.1016/j.joes.2017.02.002.Adeyemo O. D., Khalique C. M. Stability analysis, symmetry solutions and conserved currents of a two-dimensional extended shallow water wave equation of fluid mechanics. Partial Differential Equations in Applied Mathematics. 2021;4:100134. https://doi.org/10.1016/j.padiff.2021.100134.Adeyemo O. D., Khalique C. M. Stability analysis, symmetry solutions and conserved currents of a two-dimensional extended shallow water wave equation of fluid mechanics. Partial Differential Equations in Applied Mathematics. 2021;4:100134. https://doi.org/10.1016/j.padiff.2021.100134.Aydin B., Kânoğlu U. New analytical solution for nonlinear shallow water-wave equations. Pure and Applied Geophysics . 2017;174:3209-3218. https://doi.org/10.1007/s00024-017-1508-z.Aydin B., Kânoğlu U. New analytical solution for nonlinear shallow water-wave equations. Pure and Applied Geophysics . 2017;174:3209-3218. https://doi.org/10.1007/s00024-017-1508-z.Boháček J., Kharicha A., Ludwig A., Wu Menghuai. Shallow water model for horizontal centrifugal casting. In: Materials Science and Engineering: IOP Conference S eries. 2012;33. https://doi.org/10.1088/1757-899X/33/1/012032.Boháček J., Kharicha A., Ludwig A., Wu Menghuai. Shallow water model for horizontal centrifugal casting. In: Materials Science and Engineering: IOP Conference S eries. 2012;33. https://doi.org/10.1088/1757-899X/33/1/012032.Bresch D., Desjardins B., Lin Chi-Kun. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Communications in Partial Differential Equations . 2003:28(3-4):843-868. https://doi.org/10.1081/PDE-120020499.Bresch D., Desjardins B., Lin Chi-Kun. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Communications in Partial Differential Equations . 2003:28(3-4):843-868. https://doi.org/10.1081/PDE-120020499.Scott N. S., Jézéquel F., Denis C., Chesneaux J. M. Numerical ‘health check’for scientific codes: the CADNA approach. Computer Physics Communications. 2007;176(8):507-521. https://doi.org/10.1016/j.cpc.2007.01.005.Scott N. S., Jézéquel F., Denis C., Chesneaux J. M. Numerical ‘health check’for scientific codes: the CADNA approach. Computer Physics Communications. 2007;176(8):507-521. https://doi.org/10.1016/j.cpc.2007.01.005.Jézéquel F., Chesneaux J. M. CADNA: a library for estimating round-off error propagation. Computer Physics Communications. 2008;178(12):933-955. https://doi.org/10.1016/j.cpc.2008.02.003.Jézéquel F., Chesneaux J. M. CADNA: a library for estimating round-off error propagation. Computer Physics Communications. 2008;178(12):933-955. https://doi.org/10.1016/j.cpc.2008.02.003.Chesneaux J.-M. The equality relations in scientific computing. Numerical Algorithms. 1994;7:129-143. https://doi.org/10.1007/BF02140678.Chesneaux J.-M. The equality relations in scientific computing. Numerical Algorithms. 1994;7:129-143. https://doi.org/10.1007/BF02140678.Chesneaux J.-M., Jézéquel F. Dynamical control of computations using the Trapezoidal and Simpson's rules. Journal of Universal Computer Scienc е. 1998;4(1):2-10. https://doi.org/10.3217/jucs-004-01-0002.Chesneaux J.-M., Jézéquel F. Dynamical control of computations using the Trapezoidal and Simpson's rules. Journal of Universal Computer Scienc е. 1998;4(1):2-10. https://doi.org/10.3217/jucs-004-01-0002.Fariborzi Araghi M. A., Fallahzadeh A. Dynamical Control of accuracy using the stochastic arithmetic to estimate the solution of the ordinary differential equations via Adomian decomposition method. Asian Journal of Mathematics and Computer Research. 2016;8(2):128-135.Fariborzi Araghi M. A., Fallahzadeh A. Dynamical Control of accuracy using the stochastic arithmetic to estimate the solution of the ordinary differential equations via Adomian decomposition method. Asian Journal of Mathematics and Computer Research. 2016;8(2):128-135.Fallahzadeh A., Fariborzi Araghi M. A., Discrete Adomian decomposition method for fuzzy convection-diffusion equation. British Journal of Mathematics & Computer Science. 2015;9(3):279-287. https://doi.org/10.9734/BJMCS/2015/14727.Fallahzadeh A., Fariborzi Araghi M. A., Discrete Adomian decomposition method for fuzzy convection-diffusion equation. British Journal of Mathematics & Computer Science. 2015;9(3):279-287. https://doi.org/10.9734/BJMCS/2015/14727.Imani A. A., Ganji D. D., Rokni H. B., Latifizadeh H., Hesameddini E., Hadi Rafiee M. Approximate traveling wave solution for shallow water wave equation. Applied Mathematical Modelling. 2012;36(4):1550-1557. https://doi.org/10.1016/j.apm.2011.09.030.Imani A. A., Ganji D. D., Rokni H. B., Latifizadeh H., Hesameddini E., Hadi Rafiee M. Approximate traveling wave solution for shallow water wave equation. Applied Mathematical Modelling. 2012;36(4):1550-1557. https://doi.org/10.1016/j.apm.2011.09.030.Noble P., Vila J.-P. Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations // Journal of Fluid Mechanics. 2013;735:29-60. https://doi.org/10.1017/jfm.2013.454.Noble P., Vila J.-P. Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations // Journal of Fluid Mechanics. 2013;735:29-60. https://doi.org/10.1017/jfm.2013.454.Noeiaghdam S., Fariborzi Araghi M. A. A novel algorithm to evaluate definite integrals by the Gauss-Legendre integration rule based on the stochastic arithmetic: application in the model of osmosis system // Mathematical Modelling of Engineering Problems. 2020;7(4):577-586. https://doi.org/10.18280/mmep.070410.Noeiaghdam S., Fariborzi Araghi M. A. A novel algorithm to evaluate definite integrals by the Gauss-Legendre integration rule based on the stochastic arithmetic: application in the model of osmosis system // Mathematical Modelling of Engineering Problems. 2020;7(4):577-586. https://doi.org/10.18280/mmep.070410.Noeiaghdam L., Noeiaghdam S., Sidorov D. Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation // Journal of Physics: Conference Series. 2021;1847:012010. https://doi.org/10.1088/1742-6596/1847/1/012010.Noeiaghdam L., Noeiaghdam S., Sidorov D. Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation // Journal of Physics: Conference Series. 2021;1847:012010. https://doi.org/10.1088/1742-6596/1847/1/012010.Noeiaghdam S., Micula S., Nieto J. J. A novel technique to control the accuracy of a nonlinear fractional order model of COVID-19: application of the CESTAC method and the CADNA library. Mathematics. 2021;9(12):1321. https://doi.org/10.3390/math9121321.Noeiaghdam S., Micula S., Nieto J. J. A novel technique to control the accuracy of a nonlinear fractional order model of COVID-19: application of the CESTAC method and the CADNA library. Mathematics. 2021;9(12):1321. https://doi.org/10.3390/math9121321.Noeiaghdam S., Sidorov D. Integral equations: theories, approximations, and applications. Symmetry. 2021;13(8):1402. https://doi.org/10.3390/sym13081402.Noeiaghdam S., Sidorov D. Integral equations: theories, approximations, and applications. Symmetry. 2021;13(8):1402. https://doi.org/10.3390/sym13081402.Noeiaghdam S., Sidorov D., Wazwaz A.-M., Sidorov N., Sizikov V. The numerical validation of the Adomian decomposition method for solving Volterra integral equation with discontinuous kernel using the CESTAC method. Mathematics. 2021;9(3):260. https://doi.org/10.3390/math9030260.Noeiaghdam S., Sidorov D., Wazwaz A.-M., Sidorov N., Sizikov V. The numerical validation of the Adomian decomposition method for solving Volterra integral equation with discontinuous kernel using the CESTAC method. Mathematics. 2021;9(3):260. https://doi.org/10.3390/math9030260.Noeiaghdam S., Sidorov D., Zamyshlyaeva A., Tynda A., Dreglea A. A valid dynamical control on the reverse osmosis system using the CESTAC method. Mathematics. 2021;9(1). https://dx.doi.org/10.3390/math9010048.Noeiaghdam S., Sidorov D., Zamyshlyaeva A., Tynda A., Dreglea A. A valid dynamical control on the reverse osmosis system using the CESTAC method. Mathematics. 2021;9(1). https://dx.doi.org/10.3390/math9010048.Novin R., Fariborzi Araghi M. A., Mahmoudi Ya. A novel fast modification of the Adomian decomposition method to solve integral equations of the first kind with hypersingular kernels. Journal of Computational and Applied Mathematics. 2018;343:619-634. https://doi.org/10.1016/j.cam.2018.04.055.Novin R., Fariborzi Araghi M. A., Mahmoudi Ya. A novel fast modification of the Adomian decomposition method to solve integral equations of the first kind with hypersingular kernels. Journal of Computational and Applied Mathematics. 2018;343:619-634. https://doi.org/10.1016/j.cam.2018.04.055.Vignes J., La Porte M. Error analysis in computing. In: Information Processing. Amsterdam; 1974, р. 610-614.Vignes J., La Porte M. Error analysis in computing. In: Information Processing. Amsterdam; 1974, р. 610-614.Yang Nan, Xu Wenlong, Zhang Kai, Zheng Bailin. Exact solutions to the space-time fractional shallow water wave equation via the complete discrimination system for polynomial method. Results in Physics. 2021;20(002):103728. https://doi.org/10.1016/j.rinp.2020.103728.Yang Nan, Xu Wenlong, Zhang Kai, Zheng Bailin. Exact solutions to the space-time fractional shallow water wave equation via the complete discrimination system for polynomial method. Results in Physics. 2021;20(002):103728. https://doi.org/10.1016/j.rinp.2020.103728.Zarei E., Noeiaghdam S. Advantages of the discrete stochastic arithmetic to validate the results of the Taylor expansion method to solve the generalized Abel’s integral equation. Symmetry. 2021;13(8):1370. https://doi.org/10.3390/sym13081370.Zarei E., Noeiaghdam S. Advantages of the discrete stochastic arithmetic to validate the results of the Taylor expansion method to solve the generalized Abel’s integral equation. Symmetry. 2021;13(8):1370. https://doi.org/10.3390/sym13081370.Urata N. Wave mode coupling and instability in the internal wave in aluminum reduction cells. In: Bearne G., Dupuis M., Tarcy G. (eds.). Essential Readings in Light Metals. Cham: Springer; 2016, р. 373–378. https://doi.org/10.1007/978-3-319-48156-2_53.Urata N. Wave mode coupling and instability in the internal wave in aluminum reduction cells. In: Bearne G., Dupuis M., Tarcy G. (eds.). Essential Readings in Light Metals. Cham: Springer; 2016, р. 373–378. https://doi.org/10.1007/978-3-319-48156-2_53.Kadkhodabeigi M., Saboohi Y. A new wave equation for MHD instabilities in aluminum reduction cells. Light Metals. 2007;345-349.Kadkhodabeigi M., Saboohi Y. A new wave equation for MHD instabilities in aluminum reduction cells. Light Metals. 2007;345-349.Радионов Е. Ю., Немчинова Н. В., Третьяков Я. А. Моделирование магнитогидродинамических процессов в электролизерах при получении первичного алюминия // Вестник Иркутского государственного технического университета. 2015. № 7. С. 112–120.Radionov E. Yu., Nemchinova N. V., Tretiakov Ya. A. Magnetohydrodynamic processes modeling in electrolyzers at primary aluminum production. Vestnik Irkutskogo gosudarstvennogo tehnicheskogo universiteta = Proceedings of Irkutsk State Technical University. 2015;7:112-120. (In Russ.).

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