December  2011, 1(4): 509-518. doi: 10.3934/mcrf.2011.1.509

Inverse source problem with a final overdetermination for a fractional diffusion equation

1. 

Mathematical Science & Technology Research Lab, Advanced Technology Research Laboratories, Technical Development Bureau, Nippon Steel Corporation, 20-1 Shintomi, Futtsu, Chiba 293-8511, Japan

2. 

Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914

Received  December 2010 Revised  May 2011 Published  November 2011

For a time fractional diffusion equation with source term, we discuss an inverse problem of determining a spatially varying function of the source by final overdetermining data. We prove that this inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants.
Citation: Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
References:
[1]

Academic Press, New York, 1975.  Google Scholar

[2]

Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.  Google Scholar

[3]

Inverse Problems, 12 (1996), 195-205. doi: 10.1088/0266-5611/12/3/002.  Google Scholar

[4]

J. Inverse Ill-Posed Problems, 7 (1999), 241-254. doi: 10.1515/jiip.1999.7.3.241.  Google Scholar

[5]

Physica A, 191 (1992), 449-453. doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[6]

Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 223-276.  Google Scholar

[7]

Lec. Notes in Math. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[8]

in "Inverse Problems in Mathematical Physics," 49-54, Lecture Notes in Phys. 422, Springer-Verlag, Berlin, 1993.  Google Scholar

[9]

Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.  Google Scholar

[10]

Springer-Verlag, Berlin, 1982. Google Scholar

[11]

Springer-Verlag, Berlin, 1976.  Google Scholar

[12]

Elsevier, Amsterdam, 2006.  Google Scholar

[13]

Waves and Stability in Continuous Media, (Edited by S. Rionero and T. Ruggeri), World Scientific, Singapore, 1994, 246-251.  Google Scholar

[14]

Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[15]

Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 291-348.  Google Scholar

[16]

Phys. Stat. Sol. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

Academic Press, New York, 1974.  Google Scholar

[18]

Academic Press, San Diego, 1999.  Google Scholar

[19]

Marcel Dekker, New York, 2000.  Google Scholar

[20]

Birkhäuser, Basel, 1993.  Google Scholar

[21]

J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[22]

Ph.D. Thesis, Graduate School of Mathematical Sciences, The University of Tokyo, 2010. Google Scholar

[23]

J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

show all references

References:
[1]

Academic Press, New York, 1975.  Google Scholar

[2]

Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.  Google Scholar

[3]

Inverse Problems, 12 (1996), 195-205. doi: 10.1088/0266-5611/12/3/002.  Google Scholar

[4]

J. Inverse Ill-Posed Problems, 7 (1999), 241-254. doi: 10.1515/jiip.1999.7.3.241.  Google Scholar

[5]

Physica A, 191 (1992), 449-453. doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[6]

Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 223-276.  Google Scholar

[7]

Lec. Notes in Math. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[8]

in "Inverse Problems in Mathematical Physics," 49-54, Lecture Notes in Phys. 422, Springer-Verlag, Berlin, 1993.  Google Scholar

[9]

Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.  Google Scholar

[10]

Springer-Verlag, Berlin, 1982. Google Scholar

[11]

Springer-Verlag, Berlin, 1976.  Google Scholar

[12]

Elsevier, Amsterdam, 2006.  Google Scholar

[13]

Waves and Stability in Continuous Media, (Edited by S. Rionero and T. Ruggeri), World Scientific, Singapore, 1994, 246-251.  Google Scholar

[14]

Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[15]

Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 291-348.  Google Scholar

[16]

Phys. Stat. Sol. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

Academic Press, New York, 1974.  Google Scholar

[18]

Academic Press, San Diego, 1999.  Google Scholar

[19]

Marcel Dekker, New York, 2000.  Google Scholar

[20]

Birkhäuser, Basel, 1993.  Google Scholar

[21]

J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[22]

Ph.D. Thesis, Graduate School of Mathematical Sciences, The University of Tokyo, 2010. Google Scholar

[23]

J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[1]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227

[2]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

[3]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[4]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001

[5]

Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022

[6]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[8]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057

[9]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382

[10]

Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008

[11]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[12]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[13]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098

[14]

Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005

[15]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004

[16]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018

[17]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[18]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[19]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

[20]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (146)
  • HTML views (0)
  • Cited by (43)

Other articles
by authors

[Back to Top]