An Inverse Initial Problem for a Fourth Order Time-Fractional Space Degenerate Partial Differential Equation
Keywords:
degenerate equation, an inverse initial problem, spectral method, Green’s function, integral equation with symmetric kernelAbstract
In this paper, we investigate inverse initial problem for a fourth-order differential equation that becomes degenerate along the boundary of a rectangular domain. By applying the method of separation of variables, we reduce the problem to a spectral problem for an ordinary differential equation. We then construct the Green’s function for this spectral problem, which allows us to transform it into a Fredholm integral equation of the second kind with a symmetric kernel. Using the theory of integral equations with symmetric kernels, we establish the existence and some properties of the eigenvalues and eigenfunctions of the spectral problem. The solution to the original problem is expressed as a Fourier series expansion in terms of these eigenfunctions, and we prove the uniform convergence of this series.
References
1. DuChateau, P., Zachmann, D.: Applied Partial Differential Equations. Harper & Row, New York (1989)
2. Ivanov, V.K.: On linear problems which are not well-posed. Dokl. Akad. Nauk SSSR 145, 270–272 (1962)
3. Lavrentiev, M.M., Romanov, V.G., Vasiliev, V.G.: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1970)
4. Tikhonov, A.N.: On the stability of inverse problems. Dokl. Akad. Nauk SSSR 39, 195–198 (1943)
5. Huntul, M., Tamsir, M.: Identifying an unknown potential term in the fourth-order Boussinesq–Love equation from mass measurement. Eng. Comput. (2021).
6. Huntul, M.J., Tamisr, M., Ahmadini, A.: An inverse problem of determining the time-dependent potential in the higher-order Boussinesq–Love equation from boundary data. Eng. Comput. (2021).
7. Huntul, M.J., Tamsir, M., Dhiman, N.: An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition. Numer. Methods Partial Differ. Equ. (2021).
8. Megraliev, Y.T., Alizade, F.K.: Inverse boundary value problem for a Boussinesq type equation of fourth order with nonlocal time integral conditions of the second kind. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauk. 26, 503–514 (2016)
9. Yang, H.: An inverse problem for the sixth-order linear Boussinesq-type equation. UPB Sci. Bull., Ser. A 82, 27–36 (2020)
10. Yuldashev, T.K.: Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel. J. Math. Sci. 250, 847–858 (2020)
11. Abbasova, K.E., Mehraliyev, Y.T., Azizbayov, E.I.: Inverse boundary-value problem for linearized equation of motion of a homogeneous elastic beam. Int. J. Appl. Comput. Math. 33, 157–170 (2020)
12. Z.Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Probl. Imaging,11(5): 875–900,2017.
13. Z.M. Odibat, N.T. Shawagfeh. Generalized Taylor’s formula. Appl. Math. Comput., 186: 286–293,2007.
14. Dzhrbashyan M.M., Nersesyan A.B. Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. AN ArmSSR. Mat. 1968. Vol. 3. No. 1. pp 3-29.
15. Mikhlin S. G. Linear Integral Equations. Moscow: Fizmatgiz, 1959.
16. Mamanazarov A., Sobirjonova M. An initial boundary-value problem for a fourth order space degenerate partial differential equation. Scientific bulletin: physical and mathematical research. 2024. Vol. 6, No. 2, pp. 118-126.
17. H.Pollard. The completely monotonic character of the Mittag-Leffler function Eα(−x), Bull. Amer. Math. Soc., 54: 1115–1116,1948.
