Identification of a time-dependent source term in a nonlocal problem for time fractional diffusion equation
Abstract
This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data.
Keyword : inverse source problem, fractional diffusion equation, not strongly regular boundary condition, generalized Fourier method, weakly singular Volterra integral equation
How to Cite
Ismailov, M. I., & Çiçek, M. (2024). Identification of a time-dependent source term in a nonlocal problem for time fractional diffusion equation. Mathematical Modelling and Analysis, 29(2), 238–253. https://doi.org/10.3846/mma.2024.17791
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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N. Kinash and J. Janno. Inverse problems for a perturbed time fractional diffusion equation with final overdetermination. Mathematical Methods in the Applied Sciences, 41(5):1925–1943, 2018. https://doi.org/10.1002/mma.4719
M. Kirane and S.A. Malik. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time. Applied Mathematics and Computation, 218(1):163–170, 2011. https://doi.org/10.1016/j.amc.2011.05.084
M. Kirane, S.A. Malik and M.A. Al-Gwaiz. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Mathematical Methods in the Applied Sciences, 36(9):1056–1069, 2013. https://doi.org/10.1002/mma.2661
H. Lopushanska and A. Lopushansky. Inverse problems for a time fractional diffusion equation in the Schwartz-type distributions. Mathematical Methods in the Applied Sciences, 44(3):2381–2392, 2021. https://doi.org/10.1002/mma.5894
H.P. Lopushanska. A problem with an integral boundary condition for a time fractional diffusion equation and an inverse problem. Fractional Differential Calculus, 6(1):133–145, 2016. https://doi.org/10.7153/fdc-06-09
R.L. Magin. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 59(5):1586–1593, 2010. https://doi.org/10.1016/j.camwa.2009.08.039
R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1):1–77, 2000. https://doi.org/10.1016/S0370-1573(00)00070-3
M.A. Naĭmark. Linear Differential Operators. Ungar, New York, 1967.
A. Pedas and G. Vainikko. Integral equations with diagonal and boundary singularities of the kernel. Zeitschrift für Analysis und ihre Anwendungen, 25(4):487–516, 2006. https://doi.org/10.4171/ZAA/1304
K. Pileckas and R. Čiegis. Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions. Z. Angew Math. Phys., 71:192, 2020. https://doi.org/10.1007/s00033-020-01422-5
I. Podlubny. Fractional Differential Equations. Academic Press: San Diego, 1999.
M. Sadybekov, O. Gulaiym and M.I. Ismailov. Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions. Filomat, 32(3):809–814, 2018. https://doi.org/10.2298/FIL1803809S
K. Sakamoto and M. Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control and Related Fields, 1(4):509–518, 2011. https://doi.org/10.3934/mcrf.2011.1.509
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach: Yverdon, 1993.
W.R. Schneider. Completely monotone generalized Mittag-Leffler functions. Expositiones Mathematicae, 14(7):3–16, 1996.
A.L. Skubachevskii. Nonclassical boundary-value problems. Journal of Mathematical Sciences, 155(2):199–334, 2008. https://doi.org/10.1007/s10958-008-9218-9
M. Slodička. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation. Fractional Calculus and Applied Analysis, 23(6):1702–1711, 2020. https://doi.org/10.1515/fca-2020-0084
M. Stynes, E. O’Riordan and J.L. Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM Journal on Numerical Analysis, 55(2):1057–1079, 2017. https://doi.org/10.1137/16M1082329
Y. Zhang and X. Xu. Inverse source problem for a fractional diffusion equation. Inverse Problems, 27(3):23–31, 2011. https://doi.org/10.1088/0266-5611/27/3/035010
T.S. Aleroev, M. Kirane and M. Salman. Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition. Electronic Journal of Differential Equations, 2013(270):1–16, 2013.
M. Ali, S. Aziz and S.A. Malik. Inverse problem for a space-time fractional diffusion equation: application of fractional Sturm-Liouville operator. Mathematical Methods in the Applied Sciences, 41(7):2733–2747, 2018. https://doi.org/10.1002/mma.4776
M. Ali and S.A. Malik. An inverse problem for a family of time fractional diffusion equations. Inverse Problems in Science and Engineering, 25(9):1299–1322, 2017. https://doi.org/10.1080/17415977.2016.1255738
R.R. Ashurov and M.D. Shakarova. Time-dependent source identification problem for fractional Schrodinger type equations. Lobachevskii Journal of Mathematics, 43(2):303–315, 2022. https://doi.org/10.1007/s10625-005-0242-y
E. Bazhlekova and I. Bazhlekov. Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation. Journal of Computational and Applied Mathematics, 386(1):113–213, 2021. https://doi.org/10.1016/j.cam.2020.113213
B. Berkowitz, J. Klafter, R. Metzler and H. Scher. Physical pictures of transport in heterogeneous media: advection-dispersion, random-walk, and fractional derivative formulations. Water Resources Research, 38(10):1–12, 2002. https://doi.org/10.1029/2001WR001030
K. Van Bockstal. Uniqueness for inverse source problems of determining a spacedependent source in time-fractional equations with non-smooth solutions. Fractal and Fractional, 5(4):1–11, 2021. https://doi.org/10.3390/fractalfract5040169
R. Brociek and D. Słota. Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation. Mathematical Modelling of Natural Phenomena, 13(1):1–14, 2018. https://doi.org/10.1051/mmnp/2018008
J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Problems, 25(11):115002, 2009. https://doi.org/10.1088/0266-5611/25/11/115002
K. Deithelm and N.J. Ford. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2):229–248, 2002. https://doi.org/10.1006/jmaa.2000.7194
J. Dixon and S. McKee. Weakly singular discrete Gronwall inequalities. Zeitschrift für Angewandte Mathematik und Mechanik, 66(11):535–544, 1986. https://doi.org/10.1007/s10625-005-0242-y
R. Faizi and R. Atmania. An inverse source problem for a generalized time fractional diffusion equation. Eurasian Journal of Mathematical and Computer Applications, 10(1):26–39, 2022. https://doi.org/10.32523/2306-6172-2022-10-1-26-39
K.M Furati, O.S Iyiola and M. Kirane. An inverse problem for a generalized fractional diffusion. Applied Mathematics and Computation, 249:24–31, 2014. https://doi.org/10.1007/s10625-005-0242-y
K.M. Furati, O.S. Iyiola and K. Mustapha. An inverse source problem for a two-parameter anomalous diffusion with local time datum. Computers and Mathematics with Applications, 73(6):1008–1015, 2017. https://doi.org/10.1016/j.camwa.2016.06.036
A.S Hendy and K. Van Bockstal. On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions. Journal of Scientific Computing, 90(41):1–33, 2022. https://doi.org/10.1007/s10915-021-01704-8
V.A. Il’in. How to express basis conditions and conditions for the equiconvergence with trigonometric series of expansions related to non-self-adjoint differential operators. Computers & Mathematics with Applications, 34(5):641–647, 1997. https://doi.org/10.1016/S0898-1221(97)00160-0
M.I. Ismailov and M. Çiçek. Inverse source problem for a time-farctional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling, 40(7-8):4891–4899, 2016. https://doi.org/10.1016/j.apm.2015.12.020
J. Janno. Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data. Fractional Calculus and Applied Analysis, 23(6):1678–1701, 2020. https://doi.org/10.1007/s10625-005-0242-y
B. Jin and W. Rundell. An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Problems, 28(7), 2012. https://doi.org/10.1088/0266-5611/28/7/075010
N. Kinash and J. Janno. Inverse problems for a perturbed time fractional diffusion equation with final overdetermination. Mathematical Methods in the Applied Sciences, 41(5):1925–1943, 2018. https://doi.org/10.1002/mma.4719
M. Kirane and S.A. Malik. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time. Applied Mathematics and Computation, 218(1):163–170, 2011. https://doi.org/10.1016/j.amc.2011.05.084
M. Kirane, S.A. Malik and M.A. Al-Gwaiz. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Mathematical Methods in the Applied Sciences, 36(9):1056–1069, 2013. https://doi.org/10.1002/mma.2661
H. Lopushanska and A. Lopushansky. Inverse problems for a time fractional diffusion equation in the Schwartz-type distributions. Mathematical Methods in the Applied Sciences, 44(3):2381–2392, 2021. https://doi.org/10.1002/mma.5894
H.P. Lopushanska. A problem with an integral boundary condition for a time fractional diffusion equation and an inverse problem. Fractional Differential Calculus, 6(1):133–145, 2016. https://doi.org/10.7153/fdc-06-09
R.L. Magin. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 59(5):1586–1593, 2010. https://doi.org/10.1016/j.camwa.2009.08.039
R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339(1):1–77, 2000. https://doi.org/10.1016/S0370-1573(00)00070-3
M.A. Naĭmark. Linear Differential Operators. Ungar, New York, 1967.
A. Pedas and G. Vainikko. Integral equations with diagonal and boundary singularities of the kernel. Zeitschrift für Analysis und ihre Anwendungen, 25(4):487–516, 2006. https://doi.org/10.4171/ZAA/1304
K. Pileckas and R. Čiegis. Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions. Z. Angew Math. Phys., 71:192, 2020. https://doi.org/10.1007/s00033-020-01422-5
I. Podlubny. Fractional Differential Equations. Academic Press: San Diego, 1999.
M. Sadybekov, O. Gulaiym and M.I. Ismailov. Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions. Filomat, 32(3):809–814, 2018. https://doi.org/10.2298/FIL1803809S
K. Sakamoto and M. Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control and Related Fields, 1(4):509–518, 2011. https://doi.org/10.3934/mcrf.2011.1.509
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach: Yverdon, 1993.
W.R. Schneider. Completely monotone generalized Mittag-Leffler functions. Expositiones Mathematicae, 14(7):3–16, 1996.
A.L. Skubachevskii. Nonclassical boundary-value problems. Journal of Mathematical Sciences, 155(2):199–334, 2008. https://doi.org/10.1007/s10958-008-9218-9
M. Slodička. Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation. Fractional Calculus and Applied Analysis, 23(6):1702–1711, 2020. https://doi.org/10.1515/fca-2020-0084
M. Stynes, E. O’Riordan and J.L. Gracia. Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM Journal on Numerical Analysis, 55(2):1057–1079, 2017. https://doi.org/10.1137/16M1082329
Y. Zhang and X. Xu. Inverse source problem for a fractional diffusion equation. Inverse Problems, 27(3):23–31, 2011. https://doi.org/10.1088/0266-5611/27/3/035010