Investigation of inverse problem for unknown coefficient in a time fractional diffusion problem with periodic boundary conditions

Year 2025, Volume: 74 Issue: 3, 403 - 410, 23.09.2025

Süleyman Çetinkaya , İrem Bağlan , Ali Demir

https://doi.org/10.31801/cfsuasmas.1639047

Abstract

This paper is about the identification of time dependent unknown function in a time fractional diffusion problem with periodic boundary conditions. Nonlocal over-determined condition in integral form is taken into account in the determination of unknown function as well as the solution of the problem. The existence, uniqueness as well as continuous dependence on data for the inverse problem is presented with certain regularity and consistency conditions. Generalized Fourier method is utilized in this study. The algorithm of the method and an example are also presented to show the effectiveness and accuracy of the proposed method.

Keywords

Inverse problem , Fourier method , time fractional diffusion problem , nonlocal condition , periodic boundary conditions

References

• Baglan, I., Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition, Inverse Probl. Sci. Eng., 23(5) (2015), 884–900.
• Cannon, J. R., The solution of the heat conduction subject to the specification of energy, Q. Appl. Math., 21 (2) (1963), 155–160.
• Choi, Y. S., Chan, K. Y. , A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal., 18 (4) (1992), 317–331.
• Dehghan, M., Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition, Math. Comput. Simul., 49 (4-5) (1999), 331–349.
• Durdiev, D. K., Inverse coefficient problem for the time-fractional diffusion equation, Eurasian J. Math. Comput. Appl., 9 (1) (2021), 44–54.
• Durdiev, D. K., Rahmonov, A. A., Bozorov, Z. R., A two-dimensional diffusion coefficient determination problem for the time-fractional equation, Math. Meth. Appl. Sci., 44 (13) (2021), 10753–10761.
• Durdiev, D., Rahmanov, A., A multidimensional diffusion coefficient determination problem for the time-fractional equation, Turk. J. Math., 46 (6) (2022), 2250–2263.
• Durdiev, U. D., Problem of determining the reaction coefficient in a fractional diffusion equation, Differ. Equ., 57 (2021), 1195–1204.
• Hill, G. W., On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math., 8 (1886), 1–36.
• Ionkin, N. I., The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differentsial’nye Uravneniya, 13 (2) (1977), 294–304.
• Kanca, F., The inverse coefficient problem of the heat equation with periodic boundary and integral overdetermination conditions, J. Inequal. Appl., 108 (2013).
• Kanca, F., Baglan, I., An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions, Bound. Value Probl., 213 (2013).
• Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
• Miller, L., Yamamoto, M., Coefficient inverse problem for a fractional diffusion equation, Inverse Probl., 29 (7) (2013).
• Mohebbi, A., A numerical algorithm for determination of a control parameter in two-dimensional parabolic inverse problems, Acta Math. Appl. Sin., 31 (2015), 213–224.
• Podlubny, I., Fractional Differential Equations, Academic Press, 1999.
• Ramm, G., Mathematical and Analytical Techniques with Application to Engineering, Springer, 2005.
• Sharma, P. R., Methi, G., Solution of two-dimensional parabolic equation subject to non-local boundary conditions using homotopy perturbation method, J. Appl. Comput. Sci. Math., 6 (2012), 64–68.
• Subhonova, Z. A., Rahmonov, A. A., Problem of determining the time dependent coefficient in the fractional diffusion-wave equation, Lobachevskii J. Math., 42 (2021), 3747–3760.
• Wang, S., Lin, Y., A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation, Inverse Prob., 5 (1989).
• Xiong, X., Guo, H., Liu, X., An inverse problem for a fractional diffusion equation, J. Comput. Appl. Math., 236 (17) (2012), 4474–4484.
• Xiong, X., Zhou, Q., Hon, Y. C., An inverse problem for fractional diffusion equation in 2-dimensional case: Stability analysis and regularization, J. Math. Anal. Appl., 393 (1) (2012), 185–199.