Convergence Analysis of Nonlinear Hyperbolic Problem with Periodic Boundary Conditions

Year 2024, Volume: 1 Issue: 2, 84 - 91

Akbala Yernazar , İrem Bağlan

Abstract

In this paper, an inverse problem for the determination of time-dependent coefficients for a one-dimensiоnаl nonlinеаr hypеrbоlic equation with periоdic bоundаrу conditions is considered. The periodic boundary conditions involve eigenfunctions in the form of sine, cosine and each eigenvalue corresponds to two eigenfunctions, which complicates the solution process. The generalised Fourier method, effective for solving boundary value problems in domains with such non-local boundary conditions, is used. The existence, uniqueness and convergence of the solution are proved using the Picard method of successive approximations.

Keywords

Inverse problem, Hyperbolic equation, Periodic boundary condition, Fourier method

Periyodik Sınır Koşullu Lineer Olmayan Hiperbolik Problemin Yakınsaklık Analizi

Abstract

Bu çalışmada, periyodik sınır koşullarına sahip bir boyutlu lineer olmayan hiperbolik denklemi için zamana bağlı katsayıların belirlenmesine yönelik bir ters problem ele alınmıştır. Periyodik sınır koşullar sinüs, kosinüs biçimindeki özfonksiyonları içerir ve her özdeğer iki özfonksiyona karşılık geldiği için çözüm sürecini karmaşıklaştırmaktadır. Bu tür lokal olmayan sınır koşullarına sahip alanlarda sınır değer problemlerini çözmekte etkili olan genelleştirilmiş Fourier yöntemi kullanılmıştır. Picard ardışık yaklaşımlar yöntemi ile çözümün varlığı, yakınsaklığı ve tekliği kanıtlanmıştır.

Keywords

Ters problem, Hiperbolik denklem, Periyodik sınır koşulu, Fourier yöntemi

References

• Baglan, I. (2015). Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition. Inverse Problems in Science and Engineering, 23(5), 884-900. doi: 10.1080/17415977.2014.947479
• Baglan, I., Kanca, F. (2021). Fourier method for higher dimensional inverse quasi‐linear parabolic problem. Numerical Methods for Partial Differential Equations, 37(3), 2222-2234. doi: 10.1002/num.22682
• Cannon, J. R. (1963). The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics, 21(2), 155-160.
• Denisov, A. M. (2019). Existence of a solution of the inverse coefficient problem for a quasilinear hyperbolic equation. Computational Mathematics and Mathematical Physics, 59, 550-558. doi: 10.1134/S096554251904002X
• Denisov, A. M., Shirkova, E. Y. (2013). Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument. Differential Equations, 49, 1053-1061. doi: 10.1134/S0012266113090012
• Huang, X., Imanuvilov, O. and Yamamoto, M.(2020). Stability for inverse source problems by Carleman estimates. Inverse Problems, 36(12), doi: 10.1088/1361-6420/aba892
• Ionkin, N.I. (1977). Solution of a boundary value problem in heat conduction with a nonclassical boundary condition, Differential Equations, 13, 204-211.
• Ismailov, M.I., Tekin, I. (2016). Inverse coefficient problems for a first order hyperbolic system. Applied Numerical Mathematics, 106, 98-115. doi: 10.1016/j.apnum.2016.02.008
• Jiang, D., Liu, Y., Yamamoto, Y. (2017). Inverse source problem for the hyperbolic equation with a time-dependent principal part. Journal of Differential Equations, 262( 1), 653-681. doi: 10.1016/j.jde.2016.09.036
• Kamynin, L. I. (1964). A boundary-value problem in the theory of heat conduction with non-classical boundary conditions. Zh. Vychisl. Mat. Mat. Fiz, 4(6), 1006-1024. doi: 10.1016/0041-5553(64)90080-1
• Kanca, F., Baglan, I. (2017). Solution of two-dimensional non-linear Burgers’ equations with nonlocal boundary condition. Malaya Journal of Matematik, 5(04), 675-679. doi: 10.26637/MJM0504/0010
• Kanca, F., Baglan, I. (2018). Inverse problem for Euler-Bernoulli equation with periodic boundary condition. Filomat, 32(16). doi: 10.2298/FIL1816691K
• Loc Hoang, N. (2019). An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method. Inverse Problems,35(3), doi: 10.1088/1361-6420/aafe8f
• Protsakh, N. (2024). Inverse problem for semilinear wave equation with strong damping. Frontiers in Applied Mathematics and Statistics, 10, 1-12, doi: 10.3389/fams.2024.1467441
• Romanov, V.G. , Bugueva T.V. (2024) . An inverse problem for a nonlinear hyperbolic equation. Eurasian Journal of Mathematical and Computer Applications. 12(2) , 134–154.
• Sieradzan, A. K. (2015). Introduction of periodic boundary conditions into unres force field. Journal of Computational Chemistry, 36(12), 940-946. doi:10.1002/jcc.23864
• Tekin, I. (2019). Determination of a time-dependent coefficient in a wave equation with unusual boundary condition. Filomat, 33(9), 2653-2665. doi: 10.2298/FIL1909653T
• Yıldız, M. (2014). Hiperbolik Türden Bir Denklem için Bir Katsayı Ters Problemi. Karaelmas Fen ve Mühendislik Dergisi, 4(2): 59-63.