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]

R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.  Google Scholar

[2]

M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.  Google Scholar

[3]

M. Choulli and M. Yamamoto, Generic well-posedness of an inverse parabolic problem--the Hölder-space approach, Inverse Problems, 12 (1996), 195-205. doi: 10.1088/0266-5611/12/3/002.  Google Scholar

[4]

M. Choulli and M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameters, J. Inverse Ill-Posed Problems, 7 (1999), 241-254. doi: 10.1515/jiip.1999.7.3.241.  Google Scholar

[5]

M. Ginoa, S. Gerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), 449-453. doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[6]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 223-276.  Google Scholar

[7]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lec. Notes in Math. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[8]

K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem, in "Inverse Problems in Mathematical Physics," 49-54, Lecture Notes in Phys. 422, Springer-Verlag, Berlin, 1993.  Google Scholar

[9]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.  Google Scholar

[10]

F. John, "Partial Differential Equations," Springer-Verlag, Berlin, 1982. Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1976.  Google Scholar

[12]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006.  Google Scholar

[13]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Waves and Stability in Continuous Media, (Edited by S. Rionero and T. Ruggeri), World Scientific, Singapore, 1994, 246-251.  Google Scholar

[14]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[15]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 291-348.  Google Scholar

[16]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

K. B. Oldham and J. Spanier, "The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order," Academic Press, New York, 1974.  Google Scholar

[18]

I. Podlubny, "Fractional Differential Equations," Academic Press, San Diego, 1999.  Google Scholar

[19]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, New York, 2000.  Google Scholar

[20]

J. Prüss, "Evolutionary Integral Equations and Applications," Birkhäuser, Basel, 1993.  Google Scholar

[21]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[22]

K. Sakamoto, "Inverse Source Problems for Diffusion Equations," Ph.D. Thesis, Graduate School of Mathematical Sciences, The University of Tokyo, 2010. Google Scholar

[23]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.  Google Scholar

[2]

M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems, 13 (1997), 19-27. doi: 10.1088/0266-5611/13/1/003.  Google Scholar

[3]

M. Choulli and M. Yamamoto, Generic well-posedness of an inverse parabolic problem--the Hölder-space approach, Inverse Problems, 12 (1996), 195-205. doi: 10.1088/0266-5611/12/3/002.  Google Scholar

[4]

M. Choulli and M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameters, J. Inverse Ill-Posed Problems, 7 (1999), 241-254. doi: 10.1515/jiip.1999.7.3.241.  Google Scholar

[5]

M. Ginoa, S. Gerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), 449-453. doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[6]

R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 223-276.  Google Scholar

[7]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lec. Notes in Math. 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[8]

K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem, in "Inverse Problems in Mathematical Physics," 49-54, Lecture Notes in Phys. 422, Springer-Verlag, Berlin, 1993.  Google Scholar

[9]

V. Isakov, Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185-209. doi: 10.1002/cpa.3160440203.  Google Scholar

[10]

F. John, "Partial Differential Equations," Springer-Verlag, Berlin, 1982. Google Scholar

[11]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, Berlin, 1976.  Google Scholar

[12]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006.  Google Scholar

[13]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Waves and Stability in Continuous Media, (Edited by S. Rionero and T. Ruggeri), World Scientific, Singapore, 1994, 246-251.  Google Scholar

[14]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), 23-28. doi: 10.1016/0893-9659(96)00089-4.  Google Scholar

[15]

F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, F. Mainardi), Springer-Verlag, New York, 1997, 291-348.  Google Scholar

[16]

R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

K. B. Oldham and J. Spanier, "The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order," Academic Press, New York, 1974.  Google Scholar

[18]

I. Podlubny, "Fractional Differential Equations," Academic Press, San Diego, 1999.  Google Scholar

[19]

A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics," Marcel Dekker, New York, 2000.  Google Scholar

[20]

J. Prüss, "Evolutionary Integral Equations and Applications," Birkhäuser, Basel, 1993.  Google Scholar

[21]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[22]

K. Sakamoto, "Inverse Source Problems for Diffusion Equations," Ph.D. Thesis, Graduate School of Mathematical Sciences, The University of Tokyo, 2010. Google Scholar

[23]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[1]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[2]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[3]

Li Li. An inverse problem for a fractional diffusion equation with fractional power type nonlinearities. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021064

[4]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007

[5]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021055

[6]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052

[7]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007

[8]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006

[9]

Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605

[10]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029

[11]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[12]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

[13]

Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems & Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053

[14]

Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435

[15]

Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781

[16]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[17]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[18]

I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323

[19]

Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems & Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008

[20]

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

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (215)
  • HTML views (0)
  • Cited by (43)

Other articles
by authors

[Back to Top]