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Kazakhstan Shymkent
Year
2023
Volume
33
Issue
3
Pages
452-466
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Section Mathematics
Title Inverse problems for the beam vibration equation with involution
Author(-s) Imanbetova A.B.a, Sarsenbi A.A.a, Seilbekov B.ab
Affiliations M. Auezov South Kazakhstan State Universitya, South Kazakhstan State Pedagogical Universityb
Abstract This article considers inverse problems for a fourth-order hyperbolic equation with involution. The existence and uniqueness of a solution of the studied inverse problems is established by the method of separation of variables. To apply the method of separation of variables, we prove the Riesz basis property of the eigenfunctions for a fourth-order differential operator with involution in the space ${{L}_{2}}(-1,1)$. For proving theorems on the existence and uniqueness of a solution, we widely use the Bessel inequality for the coefficients of expansions into a Fourier series in the space ${{L}_{2}}(-1,1)$. A significant dependence of the existence of a solution on the equation coefficient $\alpha$ is shown. In each of the cases $\alpha <-1$, $\alpha >1$, $-1<\alpha<1$ representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are written out.
Keywords differential equations with involution, inverse problem, eigenvalue, eigenfunction, Fourier method
UDC 517.927.21, 517.927.25
MSC 34L34, 35D35, 35Q70
DOI 10.35634/vm230305
Received 17 May 2023
Language English
Citation Imanbetova A.B., Sarsenbi A.A., Seilbekov B. Inverse problems for the beam vibration equation with involution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 3, pp. 452-466.
References
  1. Whitham G.B. Linear and nonlinear waves, New York: John Wiley and Sons, 1999. https://doi.org/10.1002/9781118032954
  2. Sabitov K.B. Inverse problems of determining the right-hand side and the initial conditions for the beam vibration equation, Differential Equations, 2020, vol. 56, issue 6, pp. 761-774. https://doi.org/10.1134/S0012266120060099
  3. Imanbetova A., Sarsenbi A., Seilbekov B. Inverse problem for a fourth-order hyperbolic equation with a complex-valued coefficient, Mathematics, 2023, vol. 11, issue 15, 3432. https://doi.org/10.3390/math11153432
  4. Durdiev D.K., Jumayev J.J., Atoev D.D. Kernel determination problem in an integro-differential equation of parabolic type with nonlocal condition, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 90-102. https://doi.org/10.35634/vm230106
  5. Durdiev D.K., Nuriddinov Zh.Z. On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 572-584. https://doi.org/10.35634/vm200403
  6. Wiener J. Generalized solutions of functional differential equations, Hackensack: World Scientific, 1993.
  7. Przeworska–Rolewicz D. Equations with transformed argument: an algebraic approach, Amsterdam: Elsevier, 1973.
  8. Cabada A., Tojo F.A.F. Differential equations with involutions, Atlantis Press, 2015. https://doi.org/10.2991/978-94-6239-121-5
  9. Kirane M., Sadybekov M., Sarsenbi AA. On an inverse problem of reconstructing a subdiffusion process from nonlocal data, Mathematical Methods in the Applied Sciences, 2019, vol. 42, issue 6, pp. 2043-2052. https://doi.org/10.1002/mma.5498
  10. Kirane M., Al-Salti N. Inverse problems for a nonlocal wave equation with an involution perturbation, Journal of Nonlinear Sciences and Applications, 2016, vol. 9, issue 3, pp. 1243-1251. https://doi.org/10.22436/jnsa.009.03.49
  11. Ruzhansky M., Tokmagambetov N., Torebek B.T. Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations, Journal of Inverse and Ill-posed Problems, 2019, vol. 27, issue 6, pp. 891-911. https://doi.org/10.1515/jiip-2019-0031
  12. Ahmad B., Alsaedi A., Kirane M., Tapdigoglu R.G. An inverse problem for space and time fractional evolution equations with an involution perturbation, Quaestiones Mathematicae, 2017, vol. 40, issue 2, pp. 151-160. https://doi.org/10.2989/16073606.2017.1283370
  13. Mussirepova E., Sarsenbi A.A., Sarsenbi A.M. Solvability of mixed problems for the wave equation with reflection of the argument, Mathematical Methods in the Applied Sciences, 2022, vol. 45, issue 17, pp. 11262-11271. https://doi.org/10.1002/mma.8448
  14. Mussirepova E., Sarsenbi A.A., Sarsenbi A.M. The inverse problem for the heat equation with reflection of the argument and with a complex coefficient, Boundary Value Problems, 2022, vol. 2022, issue 1, article number: 99. https://doi.org/10.1186/s13661-022-01675-1
  15. Al-Salti N., Kerbal S., Kirane M. Initial-boundary value problems for a time-fractional differential equation with involution perturbation, Mathematical Modelling of Natural Phenomena, 2019, vol. 14, no. 3, pp. 312-327. https://doi.org/10.1051/mmnp/2019014
  16. Ashyralyev A., Erdogan A.S., Sarsenbi A. A note on the parabolic identification problem with involution and Dirichlet condition, Bulletin of the Karaganda University-Mathematics, 2020, vol. 99, issue 3, pp. 130-139.
  17. Karachik V.V., Sarsenbi A.M., Turmetov B.K. On the solvability of the main boundary value problems for a nonlocal Poisson equation, Turkish Journal of Mathematics, 2019, vol. 43, issue 3, pp. 1604-1625. https://doi.org/10.3906/mat-1901-71
  18. Ilyas A., Malik S.A., Saif S. Inverse problems for a multi-term time fractional evolution equation with an involution, Inverse Problems in Science and Engineering, 2021, vol. 29, issue 13, pp. 3377-3405. https://doi.org/10.1080/17415977.2021.2000606
  19. Yarka U., Fedushko S., Veselý P. The Dirichlet problem for the perturbed elliptic equation, Mathematics, 2020, vol. 8, issue 12, 2108. https://doi.org/10.3390/math8122108
  20. Sarsenbi A.A. A solvability conditions of mixed problems for equations of parabolic type with involution, Bulletin of the Karaganda University-Mathematics, 2018, vol. 92, issue 4, pp. 87-93. https://doi.org/10.31489/2018m4/87-93
  21. Turmetov B.Kh., Karachik V.V. On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 4, pp. 651-667. https://doi.org/10.35634/vm210409
  22. Kozhanov A.I., Bzheumikhova O.I. Elliptic and parabolic equations with involution and degeneration at higher derivatives, Mathematics, 2022, vol. 10, issue 18, 3325. https://doi.org/10.3390/math10183325
  23. Burlutskaya M.Sh., Khromov A.P. Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution, Computational Mathematics and Mathematical Physics, 2011, vol. 51, issue 12, pp. 2102-2114. https://doi.org/10.1134/S0965542511120086
  24. Ashyralyev A., Ashyralyyev A., Abdalmohammed B. On the hyperbolic type differential equation with time involution, Bulletin of the Karaganda University-Mathematics, 2023, vol. 109, issue 1, pp. 38-47. https://doi.org/10.31489/2023m1/38-47
  25. Kirane M., Sarsenbi A.A. Solvability of mixed problems for a fourth-order equation with involution and fractional derivative, Fractal and Fractional, 2023, vol. 7, issue 2, 131. https://doi.org/10.3390/fractalfract7020131
  26. Sarsenbi A.A., Turmetov B.Kh. Basis property of a system of eigenfunctions of a second-order differential operator with involution, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 183-196. https://doi.org/10.20537/vm190204
  27. Moiseev E.I. On the basis property of systems of sines and cosines, Soviet Mathematics. Doklady, 1984, vol. 29, pp. 296-300. https://zbmath.org/0587.42006
  28. Sarsenbi A.A., Sarsenbi A.M. On eigenfunctions of the boundary value problems for second order differential equations with involution, Symmetry, 2021, vol. 13, issue 10, 1972. https://doi.org/10.3390/sym13101972
  29. Imanbaev N.S. On nonlocal perturbation of the problem on eigenvalues of differentiation operator on a segment, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 186-193. https://doi.org/10.35634/vm210202
  30. Imanbaev N.S., Kanguzhin B.E., Kalimbetov B.T. On zeros of the characteristic determinant of the spectral problem for a third-order differential operator on a segment with nonlocal boundary conditions, Advances in Difference Equations, 2013, vol. 2013, issue 1, article number: 110. https://doi.org/10.1186/1687-1847-2013-110
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