najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-3-312-320najo-303Research ArticleMATHEMATICSМАТЕМАТИКАDetermination of the coefficient function in a Whitham type nonlinear differential equation with impulse effectsОпределения коэффициентной функции в нелинейном дифференциальном уравнении типа Уизема с импульсными воздействиямиhttps://orcid.org/0000-0002-9346-5362ЮлдашевТ. К.YuldashevT. K.

Tursun K. Yuldashev

Karimov street, 49, TSUE, Tashkent, 100066

t.yuldashev@tsue.uzhttps://orcid.org/0000-0001-6798-3265ФайзиевA. K.FayziyevA. K.

Aziz K. Fayziyev

Karimov street, 49, TSUE, Tashkent, 100066

fayziyev.a@inbox.ruTashkent State University of EconomicsUzbekistan202330062025143312320Copyright © Yuldashev T.K., Fayziyev A.K., 20252025Юлдашев Т.К., Файзиев A.K.Yuldashev T.K., Fayziyev A.K.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/303

In the article, the problems of unique solvability and determination of the redefinition coefficient function in the initial inverse problem for a nonlinear Whitham type partial differential equation with impulse effects are studied. The modified method of characteristics allows partial differential equations of the first order to be represented as ordinary differential equations that describe the change of unknown function along the line of characteristics. The unique solvability of the initial inverse problem is proved by the method of successive approximations and contraction mappings. The determination of the unknown coefficient is reduced to solving the nonlinear integral equation.

Изучены вопросы однозначной разрешимости и определения коэффициентной функции переопределения в начальной обратной задаче для нелинейного дифференциального уравнения в частных производных типа Уизема. Модифицированный метод характеристик позволяет дифференциальных уравнений в частных производных первого порядка представить как обыкновенные дифференциальные уравнения, которые описывают изменение неизвестной функции вдоль линии характеристик. Доказана однозначная разрешимость начальной обратной задачи методом последовательных приближений и сжимающих отображений. Определение неизвестного коэффициента сведено к решению нелинейного интегрального уравнения.

Обратная задачауравнения типа Уиземаопределение коэффициентной функцииметод последовательных приближенийоднозначная разрешимостьinverse problemWhitham type equationsdetermination of the coefficient functionmethod of successive approximationsunique solvabilityThe research of the first author is funded by the Ministry of Innovative development of the Republic of Uzbekistan (Grant F-FA-2021-424).ReferencesBenchohra M., Henderson J., Ntouyas S.K. Impulsive differential equations and inclusions. Contemporary Mathematics and its Application. Hindawi Publishing Corporation, New York, 2006.Benchohra M., Henderson J., Ntouyas S.K. Impulsive differential equations and inclusions. Contemporary Mathematics and its Application. Hindawi Publishing Corporation, New York, 2006.Halanay A., Veksler D. Qualitative Theory of Impulsive Systems. Mir, Moscow, 1971, 309 p. (in Russian).Halanay A., Veksler D. Qualitative Theory of Impulsive Systems. Mir, Moscow, 1971, 309 p. (in Russian).Lakshmikantham V., Bainov D.D., Simeonov P.S. Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989, 434 p.Lakshmikantham V., Bainov D.D., Simeonov P.S. Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989, 434 p.Perestyk N.A., Plotnikov V.A., Samoilenko A.M., Skripnik N.V. Differential Equations with Impulse Effect: Multivalued Right-hand DSides with Discontinuities. DeGruyter Stud. 40, Mathematics. Walter de Gruter Co., Berlin, 2011.Perestyk N.A., Plotnikov V.A., Samoilenko A.M., Skripnik N.V. Differential Equations with Impulse Effect: Multivalued Right-hand DSides with Discontinuities. DeGruyter Stud. 40, Mathematics. Walter de Gruter Co., Berlin, 2011.Samoilenko A.M., Perestyk N.A. Impulsive Differential Equations. World Scientific, Singapore, 1995.Samoilenko A.M., Perestyk N.A. Impulsive Differential Equations. World Scientific, Singapore, 1995.Stamova I., Stamov, G. Impulsive Biological Models. In: Applied Impulsive Mathematical Models. CMS Books in Mathematics. Springer, Cham., 2016.Stamova I., Stamov, G. Impulsive Biological Models. In: Applied Impulsive Mathematical Models. CMS Books in Mathematics. Springer, Cham., 2016.Catlla J., Schaeffer D.G., Witelski Th.P., Monson E.E., Lin A.L. On spiking models for synaptic activity and impulsive differential equations. SIAM Review, 2008, 50(3), P. 553–569.Catlla J., Schaeffer D.G., Witelski Th.P., Monson E.E., Lin A.L. On spiking models for synaptic activity and impulsive differential equations. SIAM Review, 2008, 50(3), P. 553–569.Fedorov E.G., Popov I.Yu. Analysis of the limiting behavior of a biological neurons system with delay. Journal of Physics: Conf. Series, 2021, 2086, 012109.Fedorov E.G., Popov I.Yu. Analysis of the limiting behavior of a biological neurons system with delay. Journal of Physics: Conf. Series, 2021, 2086, 012109.Fedorov E.G., Popov I.Yu. Hopf bifurcations in a network of Fitzhigh-Nagumo biological neurons. International Journal of Nonlinear Sciences and Numerical Simulation, 2021.Fedorov E.G., Popov I.Yu. Hopf bifurcations in a network of Fitzhigh-Nagumo biological neurons. International Journal of Nonlinear Sciences and Numerical Simulation, 2021.Fedorov E.G. Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Mathematics, 2021, 12(5), P. 553–562.Fedorov E.G. Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Mathematics, 2021, 12(5), P. 553–562.Anguraj A., Arjunan M.M. Existence and uniqueness of mild and classical solutions of impulsive evolution equations. Elect. Journal of Differential Equations, 2005, 2005(111), P. 1–8.Anguraj A., Arjunan M.M. Existence and uniqueness of mild and classical solutions of impulsive evolution equations. Elect. Journal of Differential Equations, 2005, 2005(111), P. 1–8.Ashyralyev A., Sharifov Y.A. Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions. Advances in Difference Equations, 2013, 2013(173).Ashyralyev A., Sharifov Y.A. Existence and uniqueness of solutions for nonlinear impulsive differential equations with two-point and integral boundary conditions. Advances in Difference Equations, 2013, 2013(173).Bai Ch., Yang D. Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions. Boundary Value Problems (Hindawi Publishing Corporation), 2007, 2007(41589), P. 1–13.Bai Ch., Yang D. Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions. Boundary Value Problems (Hindawi Publishing Corporation), 2007, 2007(41589), P. 1–13.Bin L., Xinzhi L., Xiaoxin L. Robust global exponential stability of uncertain impulsive systems. Acta Mathematica Scientia, 2005, 25(1), P. 161– 169.Bin L., Xinzhi L., Xiaoxin L. Robust global exponential stability of uncertain impulsive systems. Acta Mathematica Scientia, 2005, 25(1), P. 161– 169.Mardanov M.J., Sharifov Ya. A., Habib M. H. Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions. Electr. Journal of Differential Equations, 2014, 2014(259), P. 1–8.Mardanov M.J., Sharifov Ya. A., Habib M. H. Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions. Electr. Journal of Differential Equations, 2014, 2014(259), P. 1–8.Yuldashev T.K. Periodic solutions for an impulsive system of nonlinear differential equations with maxima. Nanosystems: Phys. Chem. Mathematics, 2022, 13(2), P. 135–141.Yuldashev T.K. Periodic solutions for an impulsive system of nonlinear differential equations with maxima. Nanosystems: Phys. Chem. Mathematics, 2022, 13(2), P. 135–141.Yuldashev T.K. Periodic solutions for an impulsive system of integro-differential equations with maxima. Vestnik Sam. Gos. Universiteta. Seria: Fiziko.-Matem. Nauki, 2022, 26(2), P. 368–379.Yuldashev T.K. Periodic solutions for an impulsive system of integro-differential equations with maxima. Vestnik Sam. Gos. Universiteta. Seria: Fiziko.-Matem. Nauki, 2022, 26(2), P. 368–379.Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima. Nanosystems: Phys. Chem. Mathematics, 2022, 13(1), P. 36–44.Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima. Nanosystems: Phys. Chem. Mathematics, 2022, 13(1), P. 36–44.Yuldashev T.K., Fayziev A.K. Integral condition with nonlinear kernel for an impulsive system of differential equations with maxima and redefinition vector. Lobachevskii Journal of Mathematics, 2022, 43(8), P. 2332–2340.Yuldashev T.K., Fayziev A.K. Integral condition with nonlinear kernel for an impulsive system of differential equations with maxima and redefinition vector. Lobachevskii Journal of Mathematics, 2022, 43(8), P. 2332–2340.Yuldashev T.K., Ergashev T.G., Abduvahobov T.A. Nonlinear system of impulsive integro-differential equations with Hilfer fractional operator and mixed maxima. Chelyab. Physical Math. Journal, 2022, 7(3), P. 312–325.Yuldashev T.K., Ergashev T.G., Abduvahobov T.A. Nonlinear system of impulsive integro-differential equations with Hilfer fractional operator and mixed maxima. Chelyab. Physical Math. Journal, 2022, 7(3), P. 312–325.Goritskiy A.Yu., Kruzhkov S.N. Chechkin G.A. Partial Differential Equations of the First Order. Mekhmat MGU, Moscow, 1999. 95 p. (in Russian).Goritskiy A.Yu., Kruzhkov S.N. Chechkin G.A. Partial Differential Equations of the First Order. Mekhmat MGU, Moscow, 1999. 95 p. (in Russian).Imanaliev M.I., Ved Yu.A. On a partial differential equation of the first order with an integral coefficient. Differentional equations, 1989, 23(3), P. 325–335.Imanaliev M.I., Ved Yu.A. On a partial differential equation of the first order with an integral coefficient. Differentional equations, 1989, 23(3), P. 325–335.Imanaliev M.I., Alekseenko S.N. On the theory of systems of nonlinear integro-differential partial differential equations of the Whitham type. Doklady Mathematics, 1993, 46(1), P. 169–173.Imanaliev M.I., Alekseenko S.N. On the theory of systems of nonlinear integro-differential partial differential equations of the Whitham type. Doklady Mathematics, 1993, 46(1), P. 169–173.Alekseenko S.N., Dontsova M.V. Study of the solvability of a system of equations describing the distribution of electrons in the electric field of a sprite. Matematicheskiy vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2012, 14, P. 34–41 (in Russian).Alekseenko S.N., Dontsova M.V. Study of the solvability of a system of equations describing the distribution of electrons in the electric field of a sprite. Matematicheskiy vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2012, 14, P. 34–41 (in Russian).Alekseenko S.N., Dontsova M.V. Local existence of a bounded solution of a system of equations describing the distribution of electrons in a weakly ionized plasma in the electric field of a sprite. Matematicheskiy vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2013, 15, P. 52–59 (in Russian).Alekseenko S.N., Dontsova M.V. Local existence of a bounded solution of a system of equations describing the distribution of electrons in a weakly ionized plasma in the electric field of a sprite. Matematicheskiy vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2013, 15, P. 52–59 (in Russian).Alekseenko S.N., Dontsova M.V. Solvability conditions for a system of equations describing long waves in a rectangular water channel whose depth varies along the axis. Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 2016, 18(2), P. 115–124 (in Russian).Alekseenko S.N., Dontsova M.V. Solvability conditions for a system of equations describing long waves in a rectangular water channel whose depth varies along the axis. Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 2016, 18(2), P. 115–124 (in Russian).Alekseenko S.N., Dontsova M.V., Pelinovsky D.E. Global solutions to the shallow water system with a method of an additional argument. Applicable Analysis, 2017, 96(9), P. 1444–1465.Alekseenko S.N., Dontsova M.V., Pelinovsky D.E. Global solutions to the shallow water system with a method of an additional argument. Applicable Analysis, 2017, 96(9), P. 1444–1465.Dontsova M.V. Nonlocal solvability conditions for the Cauchy problem for a system of partial differential equations of the first order with continuous and bounded right-hand sides. Vestnik Voronezh Gos. Universiteta. Ser. Fizika. Matematika, 2014, 4, P. 116–130 (in Russian).Dontsova M.V. Nonlocal solvability conditions for the Cauchy problem for a system of partial differential equations of the first order with continuous and bounded right-hand sides. Vestnik Voronezh Gos. Universiteta. Ser. Fizika. Matematika, 2014, 4, P. 116–130 (in Russian).Dontsova M.V. Nonlocal solvability conditions for Cauchy problem for a system of first order partial differential equations with special right-hand sides. Ufa Mathematical Journal, 2014, 6(4), P. 68–80.Dontsova M.V. Nonlocal solvability conditions for Cauchy problem for a system of first order partial differential equations with special right-hand sides. Ufa Mathematical Journal, 2014, 6(4), P. 68–80.Imanaliev M.I., Alekseenko S.N. To the problem of the existence of a smooth bounded solution for a system of two nonlinear partial differential equations of the first order. Doklady Mathematics, 2001, 64(1), P. 10–15.Imanaliev M.I., Alekseenko S.N. To the problem of the existence of a smooth bounded solution for a system of two nonlinear partial differential equations of the first order. Doklady Mathematics, 2001, 64(1), P. 10–15.Yuldashev T.K. On the inverse problem for a quasilinear partial differential equation of the first order. Vestnik Tomsk. Gos. Universiteta. Matematika i mekhanika, 2012, 2(18), P. 56–62 (in Russian).Yuldashev T.K. On the inverse problem for a quasilinear partial differential equation of the first order. Vestnik Tomsk. Gos. Universiteta. Matematika i mekhanika, 2012, 2(18), P. 56–62 (in Russian).Yuldashev T.K. Initial problem for a quasilinear partial integro-differential equation of higher order with a degenerate kernel. Izvestia Instituta Mahematiki i Informatiki Udmurt. Gos. Universitteta, 2018, 52, P. 116–130 (in Russian).Yuldashev T.K. Initial problem for a quasilinear partial integro-differential equation of higher order with a degenerate kernel. Izvestia Instituta Mahematiki i Informatiki Udmurt. Gos. Universitteta, 2018, 52, P. 116–130 (in Russian).Yuldashev T.K., Shabadikov K.Kh. Initial-value problem for a higher-order quasilinear partial differential equation. Journal of Mathematical Sciences, 2021, 254(6), P. 811–822.Yuldashev T.K., Shabadikov K.Kh. Initial-value problem for a higher-order quasilinear partial differential equation. Journal of Mathematical Sciences, 2021, 254(6), P. 811–822.Aliev F.A., Ismailov N.A., Namazov A.A., Magarramov I.A. Asymptotic method for determining the coefficient of hydraulic resistance in different sections of the pipeline during oil production. Proceedings of IAM, 2017, 6(1), P. 3–15 (in Russian).Aliev F.A., Ismailov N.A., Namazov A.A., Magarramov I.A. Asymptotic method for determining the coefficient of hydraulic resistance in different sections of the pipeline during oil production. Proceedings of IAM, 2017, 6(1), P. 3–15 (in Russian).Gamzaev Kh.M. Numerical method for solving the coefficient inverse problem for the diffusion-convection-reaction equation. Vestnik Tomsk. Gos. Universiteta. Matematika i Mekhanika, 2017, 50, P. 67–78 (in Russian).Gamzaev Kh.M. Numerical method for solving the coefficient inverse problem for the diffusion-convection-reaction equation. Vestnik Tomsk. Gos. Universiteta. Matematika i Mekhanika, 2017, 50, P. 67–78 (in Russian).Kostin A.B. Recovery of the coefficient of ut in the heat equation from the condition of nonlocal observation in time. Computational Mathem. and Math. Physics, 2015, 55(1), P. 85–100.Kostin A.B. Recovery of the coefficient of ut in the heat equation from the condition of nonlocal observation in time. Computational Mathem. and Math. Physics, 2015, 55(1), P. 85–100.Romanov V.G. On the determination of the coefficients in the viscoelasticity equations. Sibirian Math. J., 2014, 55(3), P. 503–510.Romanov V.G. On the determination of the coefficients in the viscoelasticity equations. Sibirian Math. J., 2014, 55(3), P. 503–510.Yuldashev T.K. Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel. Lobachevskii Journal of Mathematics, 2017, 38(3), P. 547–553.Yuldashev T.K. Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel. Lobachevskii Journal of Mathematics, 2017, 38(3), P. 547–553.Yuldashev T.K. On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameter. Lobachevskii Journal of Mathematics, 2019, 40(2), P. 230–239.Yuldashev T.K. On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameter. Lobachevskii Journal of Mathematics, 2019, 40(2), P. 230–239.

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