Solution to the periodic problem for the impulsive hyperbolic equation with discrete memory
DOI:
https://doi.org/10.70474/kmj-25-1-02Keywords:
hyperbolic equation, impulse effects, periodic condition, discrete memory, partition of domain, problem with parameters, solvability conditionsAbstract
In this article, we consider the periodic problem for the impulsive hyperbolic equation with discrete memory. Impulsive hyperbolic equations with discrete memory arise as a mathematical model for describing physical processes in the neural networks, discontinuous dynamical systems, hybrid systems, and etc. Questions of the existence and construction of solutions to periodic problems for impulsive hyperbolic equations with discrete memory remain important issues in the theory of discontinuous partial differential equations. To find the solvability conditions of this problem we apply Dzhumabaev’s parametrization method. The coefficient conditions for the existence and uniqueness of the periodic problem for the impulsive hyperbolic equation with discrete memory are established. We offer an algorithm for determining the approximate solution to this problem and show its convergence to the exact solution of the periodic problem for the impulsive hyperbolic equation with discrete memory.
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References
Wiener J. Generalized Solutions of Functional Differential Equations, World Scientific, 1993 DOI: https://doi.org/10.1142/9789814343183
Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations, World Scientific, 1995 DOI: https://doi.org/10.1142/9789812798664
Akhmet M.U. Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonl. Anal., 66:3 (2007), 367-383 10.1016/j.na.2005.11.032 DOI: https://doi.org/10.1016/j.na.2005.11.032
Akhmet M.U. Almost periodic solutions of differential equations with piecewise constant argument of generalized type, J. Math. Anal. Appl., 336:4 (2007), 646-663, 10.1016/j.jmaa.2007.03.010 DOI: https://doi.org/10.1016/j.jmaa.2007.03.010
Akhmet M.U. Principles of discontinuous dynamical systems, Springer, 2010 DOI: https://doi.org/10.1007/978-1-4419-6581-3_8
Akhmet M.U. Nonlinear hybrid continuous/discrete-time models, Atlantis Press, 2011 DOI: https://doi.org/10.2991/978-94-91216-03-9
Akhmet M.U., Yilmaz E. Neural Networks with Discontinuous/Impact Activations, Springer, 2013
Nieto J.J., Rodriguez-Lopez R. Green's function for second order periodic BVPs with piecewise constant argument, J. Math. Anal. Appl., 304:1 (2005), 33-57, 10.1016/j.jmaa.2004.09.023 DOI: https://doi.org/10.1016/j.jmaa.2004.09.023
Assanova A.T. Hyperbolic equation with piecewise-constant argument of generalized type and solving boundary value problems for it, Lobachevskii J. Math., 42:15 (2021), 3584-3593, 10.1134/S1995080222030040 DOI: https://doi.org/10.1134/S1995080222030040
Assanova A.T., Uteshova R.E. Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type, Chaos Solitons & Fractals, 165-2:12 (2022), 112816, 10.1016/j.chaos.2022.112816 DOI: https://doi.org/10.1016/j.chaos.2022.112816
Abildayeva A., Assanova A.T., Imanchiyev A. A multi-point problem for a system of differential equations with piecewise-constant argument of generalized type as a neural network model, Eurasian Math. J., 13:2 (2022), 8-17, 10.32523/2077-9879-2022-13-2-08-17 DOI: https://doi.org/10.32523/2077-9879-2022-13-2-08-17
Assanova A.T., Kadirbayeva Z.M. Periodic problem for an impulsive system of the loaded hyperbolic equations, Electr. J. Differ. Equ., 2018, 72
Imanchiyev A.E., Assanova A.T., Molybaikyzy A. Properties of a nonlocal problem for hyperbolic equations with impulse discrete memory, Lobachevskii J. Math., 44:10 (2023), 4299-4309, 10.1134/S1995080223100177 DOI: https://doi.org/10.1134/S1995080223100177
Dzhumabayev D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, U.S.S.R. Comput. Math. Math. Phys., 29:1 (1989), 34-46 DOI: https://doi.org/10.1016/0041-5553(89)90038-4
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