Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
An Inverse Conductivity Problem in Multifrequency Electric Impedance Tomography Read chapter Read first chapter

2020 | OriginalPaper | Chapter

5. Inverse Moving Source Problem for Fractional Diffusion(-Wave) Equations: Determination of Orbits

Authors : Guanghui Hu, Yikan Liu, Masahiro Yamamoto

Published in: Inverse Problems and Related Topics

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter On the Inverse Source Problem with Boundary Data at Many Wave Numbers
next chapter Inverse Problems for a Compressible Fluid System
Literature
1.
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975)MATH
2.
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)
3.
A.E. Badia, T. Ha-Duong, Determination of point wave sources by boundary measurements. Inverse Probl. 17, 1127–1139 (2001)MathSciNetCrossRef
4.
A.L. Bukhgeim, M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Sov. Math. Dokl. 24, 244–247 (1981)MATH
5.
J. Cheng, V. Isakov, S. Lu, Increasing stability in the inverse source problem with many frequencies. J. Differ. Equ. 260, 4786–4804 (2016)MathSciNetCrossRef
6.
M. Choulli, M. Yamamoto, Some stability estimates in determining sources and coefficients. J. Inverse Ill Posed Probl. 14, 355–373 (2006)MathSciNetCrossRef
7.
S.D. Eidelman, A.N. Kochubei, Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199, 211–255 (2004)MathSciNetCrossRef
8.
K. Fujishiro, Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations. Math. Control Relat. Fields 6, 251–269 (2016)MathSciNetCrossRef
9.
D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, Berlin, 1981)CrossRef
10.
G. Hu, Y. Kian, P. Li, Y. Zhao, Inverse moving source problems in electrodynamics. Inverse Probl. 35, 075001 (2019)MathSciNetCrossRef
11.
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
12.
V. Isakov, Inverse Source Problems (AMS, Providence, RI, 1989)MATH
13.
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)MathSciNetCrossRef
14.
15.
V. Komornik, M. Yamamoto, Upper and lower estimates in determining point sources in a wave equation. Inverse Probl. 18, 319–329 (2002)MathSciNetCrossRef
16.
V. Komornik, M. Yamamoto, Estimation of point sources and applications to inverse problems. Inverse Probl. 21, 2051–2070 (2005)MathSciNetCrossRef
17.
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)MathSciNetCrossRef
18.
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)MathSciNetMATH
19.
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)MathSciNet
20.
T. Nara, Algebraic reconstruction of the general-order poles of a meromorphic function. Inverse Probl. 28, 025008 (2012)MathSciNetCrossRef
21.
T. Ohe, Real-time reconstruction of moving point/dipole wave sources from boundary measurements. Inverse Probl. Sci. Eng. (accepted)
22.
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)MathSciNetCrossRef
23.
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)MATH
24.
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)MathSciNetCrossRef
25.
S.R. Umarov, E.M. Saidamatov, A generalization of Duhamel’s principle for differential equations of fractional order. Dokl. Math. 75, 94–96 (2007)MathSciNetCrossRef
26.
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)MathSciNetCrossRef
27.
M. Yamamoto, Stability reconstruction formula and regularization for an inverse source hyperbolic problem by control method. Inverse Probl. 11, 481–496 (1995)MathSciNetCrossRef
28.
M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pure Appl. 78, 65–98 (1999)MathSciNetCrossRef
29.
Metadata
Title
Inverse Moving Source Problem for Fractional Diffusion(-Wave) Equations: Determination of Orbits
Authors
Guanghui Hu
Yikan Liu
Masahiro Yamamoto
Copyright Year
2020
Publisher
Springer Singapore
Book
Inverse Problems and Related Topics

Print ISBN: 978-981-15-1591-0

Electronic ISBN: 978-981-15-1592-7

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-981-15-1592-7_5

Premium Partner

    Image Credits