On solvability of the initial-boundary value problems for a nonlocal hyperbolic equation with periodic boundary conditions
DOI:
https://doi.org/10.31489/2025m3/46-56Keywords:
antiperiodic condition, Dirichlet problem, eigenfunctions, eigenvalues, Fourier series, hyperbolic equation, initial-boundary value problem, involution, Neumann problem, nonlocal operator, periodic conditionAbstract
In this paper, the solvability of initial-boundary value problems for a nonlocal analogue of a hyperbolic equation in a cylindrical domain is studied. The elliptic part of the considered equation involves a nonlocal Laplace operator, which is introduced using involution-type mappings. Two types of boundary conditions are considered. These conditions are specified as a relationship between the values of the unknown function at points in one half of the lateral part of the cylinder and the values at points in the other part of the cylinder boundary. The boundary conditions specified in this form generalize known periodic and antiperiodic boundary conditions for circular domains. The unknown function is represented in the form u(x) = v(x)+w(x), where v(x) is the even part of the function and w(x) is the odd part of the function with respect to the mapping. Using the properties of these functions, we obtain auxiliary initial-boundary value problems with classical hyperbolic equations. In this case, the boundary conditions of these problems are specified in the form of the Dirichlet and Neumann conditions. Further, using the known assertions for the auxiliary problems, theorems on the existence and uniqueness of the solution to the main problems are proved. The solutions to the problems are constructed as a series in systems of eigenfunctions of the Dirichlet and Neumann problems for the classical Laplace operator.
References
Dibrov, B.F., Livshits, M.A., & Volkenstein, M.V. (1977). Mathematical model of immune processes. Journal of theoretical biology, 65(4), 609–631. https://doi.org/10.1016/0022-5193(77)90012-1
Dushek, O. (2008). Mathematical modeling in cellular immunology: T cell activation and parameter estimation. PhD Thesis. University of British Columbia.
Nisbet, R.M. (1997). Delay-differential equations for structured populations. Structured-population models in marine, terrestrial, and freshwater systems. Boston: Springer. https://doi.org/10.1007/978-1-4615-5973-3_4
Srividhya, J., & Gopinathan, M.S. (2006). A simple time delay model for eukaryotic cell cycle. Journal of Theoretical Biology, 241(3), 617–627. https://doi.org/10.1016/j.jtbi.2005.12.020
Akhmanov, S.A., Vorontsov, M.A., Ivanov, V.Yu., Larichev, A.V., & Zheleznykh, N.I. (1992). Controlling transverse-wave interactions in nonlinear optics: generation and interaction of spatiotemporal structures. Journal of the Optical Society of America B, 9(1), 78–90. https://doi.org/10.1364/JOSAB.9.000078
Razgulin, A.V. (2008). Nelineinye modeli opticheskoi sinergetiki [Nonlinear models of optical synergetics]. Moscow: MAX-Press [in Russian].
Al-Salti, N., Kerbal, S., & Kirane, M. (2019). Initial-boundary value problems for a timefractional differential equation with involution perturbation. Mathematical Modelling of Natural Phenomena, 14(3), 312. https://doi.org/10.1051/mmnp/2019014
Al-Salti, N., Kirane, M., & Torebek, B.T. (2019). On a class of inverse problems for a heat equation with involution perturbation. Hacettepe Journal of Mathematics and Statistics, 48(3), 669–681. https://doi.org/10.48550/arXiv.1612.01419
Ashyralyev, A., & Sarsenbi, A. (2017). Well-Posedness of a Parabolic Equation with Involution. Numerical Functional Analysis and Optimization, 38(10), 1295–1304. https://doi.org/10.1080/01630563.2017.1316997
Ashyralyev, A., & Sarsenbi, A. (2015). Well-posedness of an elliptic equation with involution. Electronic Journal of Differential Equations, 2015(284), 1–8.
Baranetskij, Y.O., Demkiv, I.I., & Kaleniuk, P.I. (2024). Inverse problems of determination of the time-dependent coefficient of a parabolic equation with involution and antiperiodicity conditions. Journal of Mathematical Sciences, 282(5), 699–717. https://doi.org/10.1007/s10958-024-07210-0
Borikhanov, M.B., & Mambetov, S.A. (2024). Initial-boundary value problems to the timespace nonlocal diffusion equation. Fractional Differential Calculus, 14, 39–59. http://dx.doi.org/10.7153/fdc-2024-14-06
Kozhanov, A.I., & Bzheumikhova, O.I. (2022). Elliptic and parabolic equations with involution and degeneration at higher derivatives. Mathematics, 10(18), 3325. https://doi.org/10.3390/math10183325
Mussirepova, E., Sarsenbi, A.A., & Sarsenbi, A.M. (2022). Solvability of mixed problems for the wave equation with reflection of the argument. Mathematical Methods in the Applied Sciences, 45(17), 11262–11271. https://doi.org/10.1002/mma.8448
Mussirepova, E., Sarsenbi, A.A., & Sarsenbi, A.M. (2022). The inverse problem for the heat equation with reflection of the argument and with a complex coefficient. Boundary Value Problem, 2022, 99. https://doi.org/10.1186/s13661-022-01675-1
Yarka, U., Fedushko, S., & Vesely´, P. (2020). The Dirichlet Problem for the Perturbed Elliptic Equation. Mathematics, 8(12), 2108. https://doi.org/10.3390/math8122108
Ashyralyev, A., & Agirseven, D. (2019). Bounded solutions of semilinear time delay hyperbolic differential and difference equation. Mathematics, 7(12), 1163. https://doi.org/10.3390/math7121163
Kirane, M., & Al-Salti, N. (2016). Inverse problems for a nonlocal wave equation with an involution perturbation. Journal of Nonlinear Sciences and Applications, 9(3), 1243–1251. http://dx.doi.org/10.22436/jnsa.009.03.49
Tapdigoglu, R., & Torebek, B.T. (2018). Inverse source problems for a wave equation with involution. Bulletin of the Karaganda University. Mathematics Series, 3(91), 75–82. https://doi.org/10.31489/2018m3/75-82
Sadybekov, M.A, & Turmetov, B.Kh. (2014). On an analog of periodic boundary value problems for the Poisson equation in the disk. Differential Equations, 50(2), 268–273. https://doi.org/10.1134/S0012266114020153
Sadybekov, M.A, & Turmetov, B.Kh. (2012). On analogues of periodic boundary value problems for the Laplace operator in ball. Eurasian Mathematical Journal, 50, 143–146.
Turmetov, B., Koshanova, M., & Muratbekova, M. (2023). On periodic problems for the nonlocal Poisson equation in the circle. International Journal of Applied Mathematics, 36(5), 735–746. http://dx.doi.org/10.12732/ijam.v36i5.11
Tokmagambetova, T.D., & Orumbayeva, N.T. (2023). On one solution of a periodic boundary value problem for a hyperbolic equations. Bulletin of the Karaganda University. Mathematics Series, 1(109), 141–155. http://dx.doi.org/10.31489/2023M1/141-155
Il’in, V.A. (1957). K voprosu ob obosnovanii metoda Fure dlia uravneniia kolebanii [On the foundations of the Fourier method for the wave equation]. Uspekhi matematicheskikh nauk — Russian Mathematical Surveys, 4(76), 289–296 [in Russian].
Turmetov, B.Kh., & Karachik, V.V. (2021). On eigenfunctions and eigenvalues of a nonlocal Laplace operator with multiple involution. Symmetry, 13(10), 1781. https://doi.org/10.3390/sym13101781
