Conditions for solvability of a boundary value problem for ordinary differential equations with nonlocal conditions
22 6
Abstract
In this paper, Dzhumabaev's parameterization method is used to study ordinary differential equations with nonlocal boundary conditions. The parameter was introduced and the replacement was performed. The problem under consideration is divided into two parts: the first is the Cauchy problem for an ordinary differential equation, the second is a linear equation with respect to the introduced parameter. To determine the solution to the Cauchy problem, the Mikkusinsky operator method is used, based on the convolution theory. The Mikkusinsky operator method is an effective analytical tool used to solve ordinary differential and integral equations. This method is based on the theory of convolutions and allows finding solutions to equations using operator calculations. The main feature of Mikkusinsky's method is that it is considered algebraically, by transforming the differential equation using specially defined operators. Based on this method, using the values of , we determine the solution to the Cauchy problem and, by placing it on the boundary conditions, we obtain a system of linear equations associated with the introduced parameters . Theorems on the solvability of the original nonlocal boundary value problem are formulated based on the solvability of the resulting equation with respect to its parameters. The results of the study demonstrate effective methods for solving differential equations with nonlocal conditions and clarify their theoretical foundations.
References
Nakhushev, A. M. (1995). Uravneniya matematicheskoy biologii. – Moskva: Izdatel'stvo "Vysshaya Shkola".
Ionkin, N. I. (1977). Solution of nonlocal problems for one-dimensional oscillations of a medium. Differential Equations, 12(4), 294–304.
Sadybekov, M. A., Turmetov, B. Kh. (2014). Ob odnom analoge periodicheskikh kraevykh zadach dlya uravneniya Puassona v kruge. Differentsial'nye uravneniya, 50(2), 264–264.
Kozhanov, A. I., Tarasova, G. I. (2023). Prostranstvenno-nelokal'nye kraevye zadachi dlya nekotorykh klassov uravneniy sobolevskogo tipa vysokogo poryadka.
Pokrovskiy, I. L., Martynov, D. A., Zubarev, K. M. (2024). Issledovanie sobstvennykh funktsiy kraevoy zadachi s nelokal'nymi granichnymi usloviyami. Mezhdunarodnyy zhurnal gumanitarnykh i estestvennykh nauk, 4 (98), 177–185.
Zikirov, B. Z., Mamanazarov, D. S. (2022). Lokal'nye i nelokal'nye kraevye zadachi dlya nekotorykh klassov differentsial'nykh uravneniy s neznakoopredelennym napravleniem evolyutsii.
Ezaova, A. G. i dr. (2021). Nelokal'naya vnutrennyaya kraevaya zadacha dlya smeshannogo uravneniya tret'ego poryadka. Izvestiya vysshikh uchebnykh zavedeniy. Severokavkazskiy region. Estestvennye nauki, 2 (210), 4–10.
Mardanov, M. D., Sharifov, Ya. A. (2025). Sushchestvovanie i edinstvennost' resheniy sistemy Gursa–Darbu s integral'nymi granichnymi usloviyami. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya “Fiziko-matematicheskie nauki”, 29(2), 241–255.
Dzhumabajev, D. S. (1989). Priznaki odnoznachnoy razreshimosti lineynoy kraevoy zadachi dlya obyknovennogo differentsial'nogo uravneniya. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 29(1), 50–66.
Dzhumabajev, D. S., Nazarova, K. Zh. (2006). Ob odnom variante metoda parametrizatsii dlya nelineynoy dvukhtochkevoy kraevoy zadachi. Matematicheskiy zhurnal, 6(2), 60–67.
Bakirova, E., Iskakova, N., Kadirbayeva, Z. (2023). Numerical implementation for solving the boundary value problem for impulsive integro-differential equations with parameter. Journal of Mathematics, Mechanics and Computer Science, 119(3), 19–29.
Kadirbaeva, Zh. M., Bakirova, E. A. (2020). Numerical Solution of the Boundary Value Problems for the Loaded Differential and Fredholm Integro-differential Equations. International Journal of Information and Communication Technologies, 1(2), 406941.
Usmanov, K. I. (2022). Uslovie odnoznachnoy razreshimosti nelokal'nykh kraevykh zadach dlya sistem funktsional'no-differentsial'nykh uravneniy. QA Iasayı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary, 3(22), 48–58.
Assanova, A., Molybaikyzy, A. (2025). Solution to the periodic problem for the impulsive hyperbolic equation with discrete memory. Kazakh Mathematical Journal, 25(1), 16–27.
Kazhkenova, N. Zh., Orumbaeva, N. T. (2022). Ob odnoy nelineynoy kraevoy zadache dlya differentsial'nogo uravneniya v chastnykh proizvodnykh tret'ego poryadka. Itogi nauki i tekhniki. Sovremennaya matematika i ee prilozheniya. Tematicheskie obzory, 206(0), 63–67.
Mikusinskiy, Ya. (2013). Operatornoe ischislenie. – Ripol Klassik.
