Abstract
This paper is concerned with the inverse problem on determining the orbit of a moving source in fractional diffusion(-wave) equations either in a connected bounded domain of \({\mathbb R}^d\) or in the whole space \({\mathbb R}^d\). Based on a newly established fractional Duhamel’s principle, we derive a Lipschitz stability estimate in the case of a localized moving source by the observation data at d interior points. The uniqueness for the general non-localized moving source is verified with additional data of more interior observations.
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R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975)
Y.E. Anikonov, J. Cheng, M. Yamamoto, A uniqueness result in an inverse hyperbolic problem with analyticity. Eur. J. Appl. Math. 15, 533–543 (2004)
A.E. Badia, T. Ha-Duong, Determination of point wave sources by boundary measurements. Inverse Probl. 17, 1127–1139 (2001)
A.L. Bukhgeim, M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Sov. Math. Dokl. 24, 244–247 (1981)
J. Cheng, V. Isakov, S. Lu, Increasing stability in the inverse source problem with many frequencies. J. Differ. Equ. 260, 4786–4804 (2016)
M. Choulli, M. Yamamoto, Some stability estimates in determining sources and coefficients. J. Inverse Ill Posed Probl. 14, 355–373 (2006)
S.D. Eidelman, A.N. Kochubei, Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199, 211–255 (2004)
K. Fujishiro, Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations. Math. Control Relat. Fields 6, 251–269 (2016)
D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, Berlin, 1981)
G. Hu, Y. Kian, P. Li, Y. Zhao, Inverse moving source problems in electrodynamics. Inverse Probl. 35, 075001 (2019)
V. Isakov, Stability in the continuation for the Helmholtz equation with variable coefficient, in Control Methods in PDE Dynamical Systems, Contemporary Mathematics, vol. 426 (AMS, Providence, RI, 2007), pp. 255–269
V. Isakov, Inverse Source Problems (AMS, Providence, RI, 1989)
D. Jiang, Y. Liu, M. Yamamoto, Inverse source problem for the hyperbolic equation with a time-dependent principal part. J. Differ. Equ. 262, 653–681 (2017)
M.V. Klibanov, Inverse problems and Carleman estimates. Inverse Probl. 8, 575–596 (1992)
V. Komornik, M. Yamamoto, Upper and lower estimates in determining point sources in a wave equation. Inverse Probl. 18, 319–329 (2002)
V. Komornik, M. Yamamoto, Estimation of point sources and applications to inverse problems. Inverse Probl. 21, 2051–2070 (2005)
Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Comput. Math. Appl. 73, 96–108 (2017)
Y. Liu, W. Rundell, M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fract. Calc. Appl. Anal. 19, 888–906 (2016)
Y. Liu, Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation. J. Phys. A 50, 305–203 (2017)
T. Nara, Algebraic reconstruction of the general-order poles of a meromorphic function. Inverse Probl. 28, 025008 (2012)
T. Ohe, Real-time reconstruction of moving point/dipole wave sources from boundary measurements. Inverse Probl. Sci. Eng. (accepted)
T. Ohe, H. Inui, K. Ohnaka, Real-time reconstruction of time-varying point sources in a three-dimensional scalar wave equation. Inverse Probl. 27, 115011 (2011)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)
S.R. Umarov, E.M. Saidamatov, A generalization of Duhamel’s principle for differential equations of fractional order. Dokl. Math. 75, 94–96 (2007)
T. Wei, X.L. Li, Y.S. Li, An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Probl. 32, 085003 (2016)
M. Yamamoto, Stability reconstruction formula and regularization for an inverse source hyperbolic problem by control method. Inverse Probl. 11, 481–496 (1995)
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pure Appl. 78, 65–98 (1999)
Y. Zhang, X. Xu, Inverse source problem for a fractional diffusion equation. Inverse Probl. 27, 035010 (2011)
Acknowledgements
This work is partly supported by the A3 Foresight Program “Modeling and Computation of Applied Inverse Problems”, Japan Society for the Promotion of Science (JSPS) and National Natural Science Foundation of China (NSFC). G. Hu is supported by the NSFC grant (No. 11671028) and NSAF grant (No. U1930402). Y. Liu and M. Yamamoto are supported by JSPS KAKENHI Grant Number JP15H05740. M. Yamamoto is partly supported by NSFC (Nos. 11771270, 91730303) and RUDN University Program 5-100.
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Hu, G., Liu, Y., Yamamoto, M. (2020). Inverse Moving Source Problem for Fractional Diffusion(-Wave) Equations: Determination of Orbits. In: Cheng, J., Lu, S., Yamamoto, M. (eds) Inverse Problems and Related Topics. ICIP2 2018. Springer Proceedings in Mathematics & Statistics, vol 310. Springer, Singapore. https://doi.org/10.1007/978-981-15-1592-7_5
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DOI: https://doi.org/10.1007/978-981-15-1592-7_5
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