Solvability of a coefficient recovery problem for a time-fractional diffusion equation with periodic boundary and overdetermination conditions
- Authors: Durdiev D.K.1,2, Jumaev J.J.1,2
-
Affiliations:
- Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan
- Bukhara State University
- Issue: Vol 29, No 1 (2025)
- Pages: 21-36
- Section: Differential Equations and Mathematical Physics
- URL: https://journals.rcsi.science/1991-8615/article/view/311027
- DOI: https://doi.org/10.14498/vsgtu2083
- EDN: https://elibrary.ru/YZQBWZ
- ID: 311027
Cite item
Full Text
Abstract
This article investigates the inverse problem for time-fractional diffusion equations with periodic boundary conditions and integral overdetermination conditions on a rectangular domain. First, the definition of a classical solution to the problem is introduced. Using the Fourier method, the direct problem is reduced to an equivalent integral equation. The existence and uniqueness of the solution to the direct problem are established by employing estimates for the Mittag–Leffler function and generalized singular Gronwall inequalities.
In the second part of the work, the inverse problem is examined. This problem is reformulated as an equivalent integral equation, which is then solved using the contraction mapping principle. Local existence and global uniqueness of the solution are rigorously proven. Furthermore, a stability estimate for the solution is derived.
The study contributes to the theory of inverse problems for fractional differential equations by providing a framework for analyzing problems with periodic boundary conditions and integral overdetermination. The methods developed in this work can be applied to a wide range of problems in mathematical physics and engineering, where time-fractional diffusion models are increasingly used to describe complex phenomena.
Full Text
##article.viewOnOriginalSite##About the authors
Durdimurod K. Durdiev
Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University
Email: d.durdiev@mathinst.uz
ORCID iD: 0000-0002-6054-2827
https://www.mathnet.ru/person29112
Dr. Phys. & Math. Sci., Professor; Head of Branch1; Professor, Dept. of Differential Equations2
Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11Jonibek J. Jumaev
Bukhara Branch of the Institute of Mathematics named after V. I. Romanovskiy at the Academy of Sciences of the Republic of Uzbekistan; Bukhara State University
Author for correspondence.
Email: jonibekjj@mail.ru
ORCID iD: 0000-0001-8496-1092
https://www.mathnet.ru/person159031
Cand. Phys. & Math. Sci., Associate Professor; Senior Researcher1; Associate Professor, Dept. of Differential Equations2
Uzbekistan, 705018, Bukhara, Muhammad Igbol st., 11; 705018, Bukhara, Muhammad Igbol st., 11References
- Agarwal R., Sharma U. P., Agarwal R. P. Bicomplex Mittag–Leffler function and associated properties, J. Nonlinear Sci. Appl., 2022, vol. 15, no. 1, pp. 48–60, arXiv: 2103.10324 [math.CV]. DOI: https://doi.org/10.22436/jnsa.015.01.04.
- Haddouchi F., Guendouz C., Benaicha S. Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem, Matemat. Vesn., 2021, vol. 73, no. 1, pp. 25–36, arXiv: 1908.08598 [math.CA].
- Sharma S. Molecular Dynamics Simulation of Nanocomposites Using BIOVIA Materials Studio, Lammps and Gromacs. Amsterdam, Netherlands, Elsevier, 2019, xvi+349 pp. DOI: https://doi.org/10.1016/C2017-0-04396-3.
- Baglan I. Determination of a coefficient in a quasilinear parabolic equation with periodic boundary condition, Inverse Probl. Sci. Eng., 2015, vol. 23, no. 5, pp. 884-900. DOI: https://doi.org/10.1080/17415977.2014.947479.
- Kanca F. Inverse coefficient problem of the parabolic equation with periodic boundary and integral overdetermination conditions, Abstr. Appl. Anal., 2013, no. 5, 659804. DOI: https://doi.org/10.1155/2013/659804.
- Kanca F., Baglan I. An inverse coefficient problem for a quasilinear parabolic equation with nonlocal boundary conditions., Bound. Value Probl., 2013, vol. 2013, 213. DOI: https://doi.org/10.1186/1687-2770-2013-213.
- Kanca F. The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions, J. Inequal. Appl., 2013, vol. 2013, 108. DOI: https://doi.org/10.1186/1029-242X-2013-108.
- Ivanchov N. I. Inverse problems for the heat-conduction equation with nonlocal boundary condition, Ukr. Math. J., 1993, vol. 45, no. 8, pp. 1186–1192. DOI: https://doi.org/10.1007/BF01070965.
- Ivanchov M. I., Pabyrivska N. Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions, Ukr. Math. J., 2001, vol. 53, no. 5, pp. 674–684. DOI: https://doi.org/10.1023/A:1012570031242.
- Liao W., Dehghan M., Mohebbi A. Direct numerical method for an inverse problem of a parabolic partial differential equation, J. Comput. Appl. Math., 2009, vol. 232, no. 2, pp. 351–360. DOI: https://doi.org/10.1016/j.cam.2009.06.017.
- Oussaeif T. E., Abdelfatah B. An inverse coefficient problem for a parabolic equation under nonlocal boundary and integral overdetermination conditions, Int. J. Part. Dif. Equ. Appl., 2014, vol. 2, no. 3, pp. 38–43. DOI: https://doi.org/10.12691/ijpdea-2-3-1.
- Cannon J. R., Lin Y., Wang S. Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc., Ser. B, 1991, vol. 33, no. 2, pp. 149–163. DOI: https://doi.org/10.1017/S0334270000006962.
- Hazanee A., Lesnic D., Ismailov M. I., Kerimov N. B. Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions, Appl. Math. Comput., 2019, vol. 346, pp. 800–815. DOI: https://doi.org/10.1016/j.amc.2018.10.059.
- Huzyk N. Nonlocal inverse problem for a parabolic equation with degeneration, Ukr. Math. J., 2013, vol. 65, no. 6, pp. 847–863. DOI: https://doi.org/10.1007/s11253-013-0822-6.
- Durdiev D. K., Jumaev J. J. Inverse coefficient problem for a time-fractional diffusion equation in the bounded domain, Lobachevskii J. Math., 2023, vol. 44, no. 2, pp. 548–557. DOI: https://doi.org/10.1134/S1995080223020130.
- Durdiev D. K., Rahmonov A. A., Bozorov Z. R. A two-dimensional diffusion coefficient determination problem for the time-fractional equation, Math. Methods Appl. Sci., 2021, vol. 44, no. 13, pp. 10753–10761. DOI: https://doi.org/10.1002/mma.7442.
- Ionkin N. I. Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition, Differ. Equ., 1977, vol. 13, no. 2, pp. 204–211.
- Subhonova Z. A., Rahmonov A. A. Problem of determining the time dependent coefficient in the fractional diffusion-wave equation, Lobachevskii J. Math., 2021, vol. 42, no. 15, pp. 3747–3760. DOI: https://doi.org/10.1134/S1995080222030209.
- Sultanov M. A., Durdiev D. K., Rahmonov A. A. Construction of an explicit solution of a time-fractional multidimensional differential equation, Mathematics, 2021, vol. 9, no. 17, 2052. DOI: https://doi.org/10.3390/math9172052.
- Colombo F. An inverse problem for a parabolic integrodifferential model in the theory of combustion, Phys. D, 2007, vol. 236, no. 2, pp. 81–89. DOI: https://doi.org/10.1016/j.physd.2007.07.012.
- Durdiev D. K., Nuriddinov J. Z. Determination of a multidimensional kernel in some parabolic integro-differential equation, J. Sib. Fed. Univ., Math. Phys., 2021, vol. 14, no. 1, pp. 117–127. EDN: RMPPXU. DOI: https://doi.org/10.17516/1997-1397-2021-14-1-117-127.
- Durdiev D. K., Zhumaev Zh. Zh. Memory kernel reconstruction problems in the integrodifferential equation of rigid heat conductor, Math. Methods Appl. Sci., 2022, vol. 45, no. 14, pp. 8374–8388. DOI: https://doi.org/10.1002/mma.7133.
- Durdiev D. K., Zhumaev Zh. Zh. One-dimensional inverse problems of finding the kernel of the integro-differential heat equation in a bounded domain, Ukr. Math. J., 2022, vol. 73, no. 11, pp. 1723–1740. DOI: https://doi.org/10.1007/s11253-022-02026-0.
- Durdiev D. K., Zhumaev Zh. Zh. Problem of determining a multidimensional thermal memory in a heat conductivity equation, Methods Funct. Anal. Topol., 2019, vol. 25, no. 3, pp. 219–226.
- Durdiev D. K., Zhumaev Zh. Zh. Problem of determining the thermal memory of a conducting medium, Differ. Equ., 2020, vol. 56, no. 6, pp. 785–796. DOI: https://doi.org/10.1134/S0012266120060117.
- Dyatlov G. V. Determination for the memory kernel from boundary measurements on a finite time interval, J. Inverse Ill-Posed Probl., 2003, vol. 11, no. 1, pp. 59–66. DOI: https://doi. org/10.1515/156939403322004937.
- Janno J., Wolfersdorf L. Inverse problems for identification of memory kernels in heat flow, Ill-Posed Probl., 1996, vol. 4, no. 1, pp. 39–66. DOI: https://doi.org/10.1515/jiip.1996.4.1.39.
- Lorenzi A., Paparoni E. Direct and inverse problems in the theory of materials with memory, Rend. Semin. Mat. Univ. Padova, 1992, vol. 87, pp. 105–138.
- Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Amsterdam, Elsevier, 2006, xx+523 pp. DOI: https://doi.org/10.1016/s0304-0208(06)x8001-5.
- Kang B., Koo N. A note on generalized singular Gronwall inequalities, J. Chungcheong Math. Soc., 2018, vol. 31, no. 1, pp. 161–167. DOI: https://doi.org/10.14403/jcms.2018.31.1.161.
- Lakshmikantham V., Leela S., Vasundhara Devi J. Theory of Fractional Dynamic Systems. Cambridge, Cambridge Scientific Publ., 2009, vi+170 pp.
- Budak B. M., Samarskii A. A., Tikhonov A. N. Sbornik zadach po matematicheskoi fizike [A collection of Problems on Mathematical Physics]. Moscow, Nauka, 1979, 685 pp. (In Russian)
- Kolmogorov A. N., Fomin S. V. Elementy teorii funktsii i funktsional’nogo analiza [Elements of Function Theory and Functional Analysis]. Moscow, Nauka, 1972, 496 pp. (In Russian)
Supplementary files
