Nonlocal problem for а system of partial differential equations of higher order with pulsed actions
Abstract
We consider a nonlocal problem for а system of partial differential equations of higher order with pulsed actions.
By introducing new unknown functions, the analyzed problem is reduced to an equivalent problem formed by a nonlocal problem for impulsive system of hyperbolic equations of the second order and integral relations.
We propose an algorithm for finding the solutions of the equivalent problem based on the solution of a nonlocal problem for a system of hyperbolic equations of the second order with pulsed action for fixed values of the introduced additional functions, which are then determined from the integral relations in terms of the obtained solution.
Sufficient conditions for the existence of a unique solution to the nonlocal problem for an impulsive system of hyperbolic equations of the second order are obtained by method of introduction functional parameters.
The algorithms for finding its solutions are constructed.
Conditions for the unique solvability of a nonlocal problem for the system of partial differential equations of higher order with pulsed actions are established in terms of the coefficients of the system and boundary matrices.
References
Ptashnik, B. I. Некорректные граничные задачи для дифференциальных уравнений с частными производными. (Russian) [[Ill-posed boundary value problems for partial differential equations]] "Naukova Dumka'', Kiev, 1984. 264 pp. MR0772024
Ptashnik B. J., Ilkiv V. S., Kmit I. Ya., Polishuk V. M. Нелокальнi крайовi задачi для рiвнянь iз частинними похiдними(Russian) [[Nelokalni krajovi zadachi dlya rivnyan iz chastinnimi pohidnimi]] – Kiyiv: Nauk. dumka, 2002. – 415 c.
Kiguradze, T.; Lakshmikantham, V. On the Dirichlet problem in a characteristic rectangle for higher order linear hyperbolic equations. Nonlinear Anal. 50 (2002), no. 8, Ser. A: Theory Methods, 1153--1178. MR1914234, DOI = 10.1016/S0362-546X(01)00806-9
Kiguradze, T. I.; Kusano, T. On the well-posedness of initial-boundary value problems for higher-order linear hyperbolic equations with two independent variables. (Russian) ; translated from Differ. Uravn. 39 (2003), no. 4, 516--526, 575 Differ. Equ. 39 (2003), no. 4, 553--563 MR2132998, DOI = 10.1023/A:1026071112728
Kiguradze, T. I.; Kusano, T. On ill-posed initial-boundary value problems for higher-order linear hyperbolic equations with two independent variables. (Russian) ; translated from Differ. Uravn. 39 (2003), no. 10, 1379--1394, 1438--1439 Differ. Equ. 39 (2003), no. 10, 1454--1470 MR1955035, DOI =10.1023/B:DIEQ.0000017918.27580.bd
Kiguradze, Ivan; Kiguradze, Tariel. On solvability of boundary value problems for higher order nonlinear hyperbolic equations. Nonlinear Anal. 69 (2008), no. 7, 1914--1933. MR2435115, DOI = 10.1016/j.na.2007.07.033
Kiguradze, Tariel. The Vallée-Poussin problem for higher order nonlinear hyperbolic equations. Comput. Math. Appl. 59 (2010), no. 2, 994--1002. MR2575587, DOI = 10.1016/j.camwa.2009.09.009
Kiguradze, I. T.; Kiguradze, T. I. Analog of the first Fredholm theorem for higher-order nonlinear differential equations. Translation of Differ. Uravn. 53 (2017), no. 8, 1024–1032. Differ. Equ. 53 (2017), no. 8, 1024--1032. MR3706487, DOI = 10.1134/S0012266117080043
Nakhushev A. M. Задачи со смещением для дифференциальных уравнений в частных производных (Russian) [[ Zadachi so smeshheniem dlya differenczial`ny`kh uravnenij v chastny`kh proizvodny`kh ]]– M.: Nauka, 2006. – 200 с.
Samoilenko A. M., Perestyuk N. A. Дифференциальные уравнения с импульсными воздействиями (Russian) [[Differenczial`ny`e uravneniya s impul`sny`mi vozdejstviyami]] – Kiev: Vishha shk., 1987. – 287 s.
Rogovchenko, Yuri V. Impulsive evolution systems: main results and new trends. Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 1, 57--88. MR1435816
Akhmetov, M. U.; Perestyuk, N. A. Stability of periodic solutions of differential equations with impulse action on surfaces. (Russian) ; translated from Ukrain. Mat. Zh. 41 (1989), no. 12, 1596--1601, 1725 Ukrainian Math. J. 41 (1989), no. 12, 1368--1372 (1990) MR1042953, DOI = 10.1007/BF01056101
Baĭnov, D. D.; Simeonov, P. S. Systems with impulse effect. Stability, theory and applications. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989. 255 pp. ISBN: 0-470-21437-6 MR1010418
Lakshmikantham, V.; Baĭnov, D. D.; Simeonov, P. S. Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. xii+273 pp. ISBN: 9971-50-970-9 MR1082551, DOI = 10.1142/0906
Bainov, Drumi D.; Kamont, Zdzislaw; Minchev, Emil. Monotone iterative methods for impulsive hyperbolic differential-functional equations. J. Comput. Appl. Math. 70 (1996), no. 2, 329--347. MR1399877, DOI = 10.1016/0377-0427(95)00209-X
Perestyuk, N. A.; Tkach, A. B. Periodic solutions of a weakly nonlinear system of partial differential equations with impulse action. (Russian) ; translated from Ukraïn. Mat. Zh. 49 (1997), no. 4, 601--605 Ukrainian Math. J. 49 (1997), no. 4, 665--671 (1998) MR1473717, DOI = 10.1007/BF02487331
Bainov, D.; Minchev, E.; Myshkis, A. Periodic boundary value problems for impulsive hyperbolic systems. Commun. Appl. Anal. 1 (1997), no. 1, 1--14. MR1453936
Liu, X.; Zhang, Shenghai. A cell population model described by impulsive PDEs—existence and numerical approximation. Comput. Math. Appl. 36 (1998), no. 8, 1--11. MR1653819, DOI = 10.1016/S0898-1221(98)00178-3
Tkach, A. B. Numerical-analytic method of finding periodic solutions for systems of partial differential equations with pulse influence. Nonlinear Oscil. 4 (2001), no. 2, 278--288. MR1890303
Tkach, A. B. A numerical-analytical method for investigating periodic solutions of integro-partial differential equations with impulse action. (Russian) ; translated from Nelīnīĭnī Koliv. 8 (2005), no. 1, 123--131 Nonlinear Oscil. (N. Y.) 8 (2005), no. 1, 122--130 MR2190971, DOI = 10.1007/s11072-005-0041-7
Benchohra, M.; Henderson, J.; Ntouyas, S. Impulsive differential equations and inclusions. Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. xiv+366 pp. ISBN: 977-5945-50-X MR2322133, DOI = 10.1155/9789775945501
Akhmet, Marat. Principles of discontinuous dynamical systems. Springer, New York, 2010. xii+176 pp. ISBN: 978-1-4419-6580-6 MR2681108 DOI 10.1007/978-1-4419-6581-3
Akhmet, Marat; Fen, Mehmet Onur. Replication of chaos in neural networks, economics and physics. Nonlinear Physical Science. Higher Education Press, Beijing; Springer, Heidelberg, 2016. xv+457 pp. ISBN: 978-7-04-043102-5; 978-3-662-47499-0; 978-3-662-47500-3 MR3381294, DOI = 10.1007/978-3-662-47500-3
Asanova A. T. О нелокальной краевой задаче для систем гиперболических уравнений с импульсными воздействиями (Russian) [[O nelokal`noj kraevoj zadache dlya sistem giperbolicheskikh uravnenij s impul`sny`mi vozdejstviyami]] // Ukr. mat. zhurn.– 2013. – 65, № 3. – S. 315 – 328.
Asanova A. T. О корректной разрешимости нелокальной краевой задачи для систем гиперболических уравнений с импульсными воздействиями (Russian) [[O korrektnoj razreshimosti nelokal`noj kraevoj zadachi dlya sistem giperbolicheskikh uravnenij s impul`sny`mi vozdejstviyami]]// Ukr. mat. zhurn. – 2015. – 67, № 3. – S. 291 – 303.
Asanova, A. T. Criteria for the unique solvability of a nonlocal boundary value problem for systems of hyperbolic equations with mixed derivatives. (Russian) ; translated from Izv. Vyssh. Uchebn. Zaved. Mat. 2016, , no. 5, 3--21 Russian Math. (Iz. VUZ) 60 (2016), no. 5, 1--17 MR3588348 , DOI = 10.3103/s1066369x16050017
Akhmedov K. T., Akhiev S. S. Необходимое условие оптимальности для некоторых задач теории оптимального управления (Russian) [[Neobkhodimoe uslovie optimal`nosti dlya nekotory`kh zadach teorii optimal`nogo upravleniya]] // Dokl. AN AzSSR. – 1972. – 28, № 5. – С. 12 – 15.
Novozhenov M. M., Sumin V. I., Sumin M. I. Методы оптимального управления системами математической физики (Russian) [[Metody` optimal`nogo upravleniya sistemami matematicheskoj fiziki]] – Gor`kij: Izd-vo Gor`k. un-ta, 1986.
Denisov A. M., Lukshin A. V. Математические модели однокомпонентной динамики сорбции(Russian) [[Matematicheskie modeli odnokomponentnoj dinamiki sorbczii]] – M.: Izd-vo Mosk. un-ta, 1989.
Bokmelʹder, E. P.; Dykhta, V. A.; Moskalenko, A. I.; Ovsyannikova, N. A. Условия èкстремума и конструктивные методы решения в задачах оптимизации гиперболических систем. (Russian) [[Extremum conditions and constructive solution methods in problems of the optimization of hyperbolic systems]] VO ``Nauka'', Novosibirsk, 1993. 197 pp. ISBN: 5-02-030317-8 MR1268884
Vasil`ev O. V. Лекции по методам оптимизации (Russian) [[Lekczii po metodam optimizaczii]] – Irkutsk: Izd-vo Irkut. un-ta, 1994.
Vasil`ev F. P. Методы оптимизации (Russian) [[Metody` optimizaczii]] – M.: Izd-vo Faktorial Press, 2002. – 824 s.
Asanova A. T., Dzhumabaev D. S. Однозначная разрешимость краевой задачи с данными на характеристиках для систем гиперболических уравнений (Russian) [[Odnoznachnaya razreshimost` kraevoj zadachi s danny`mi na kharakteristikakh dlya sistem giperbolicheskikh uravnenij ]]// Zhurn. vy`chislit. matematiki i mat. fiziki. – 2002. – 42, № 11. – C. 1673 – 1685.
Asanova A. T., Dzhumabaev D. S. Однозначная разрешимость нелокальной краевой задачи для систем гиперболических уравнений (Russian) [[Odnoznachnaya razreshimost` nelokal`noj kraevoj zadachi dlya sistem giperbolicheskikh uravnenij]] // Differencz. uravneniya. – 2003. – 39, № 10. – S. 1343 – 1354.
Asanova A. T., Dzhumabaev D. S. Корректная разрешимость нелокальной краевой задачи для систем гиперболических уравнений (Russian) [[Korrektnaya razreshimost` nelokal`noj kraevoj zadachi dlya sistem giperbolicheskikh uravnenij ]]// Differencz. uravneniya. – 2005. – 41, № 3. – C. 337 – 346.
Asanova A. T., Dzhumabaev D. S. Признаки корректной разрешимости линейной нелокальной краевой задачи для систем гиперболических уравнений (Russian) [[Priznaki korrektnoj razreshimosti linejnoj nelokal`noj kraevoj zadachi dlya sistem giperbolicheskikh uravnenij]] // Dop. NAN Ukrayini. – 2010. – # 4. – S. 7 – 11.
Asanova, Anar T.; Dzhumabaev, Dulat S. Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations. J. Math. Anal. Appl. 402 (2013), no. 1, 167--178. MR3023246, DOI = 10.1016/j.jmaa.2013.01.012
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