American Institute of Mathematical Sciences

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, (1975).   Google Scholar [2] M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition,, Inverse Problems, 13 (1997), 19.  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.  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.  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.  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, (1997), 223.   Google Scholar [7] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lec. Notes in Math. 840, (1981).   Google Scholar [8] K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem,, in, 422 (1993), 49.   Google Scholar [9] V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185.  doi: 10.1002/cpa.3160440203.  Google Scholar [10] F. John, "Partial Differential Equations,", Springer-Verlag, (1982).   Google Scholar [11] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976).   Google Scholar [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations,", Elsevier, (2006).   Google Scholar [13] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation,, Waves and Stability in Continuous Media, (1994), 246.   Google Scholar [14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Appl. Math. Lett., 9 (1996), 23.  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, (1997), 291.   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.  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, (1974).   Google Scholar [18] I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999).   Google Scholar [19] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000).   Google Scholar [20] J. Prüss, "Evolutionary Integral Equations and Applications,", Birkhäuser, (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.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar [22] K. Sakamoto, "Inverse Source Problems for Diffusion Equations,", Ph.D. Thesis, (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.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

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References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1975).   Google Scholar [2] M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition,, Inverse Problems, 13 (1997), 19.  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.  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.  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.  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, (1997), 223.   Google Scholar [7] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lec. Notes in Math. 840, (1981).   Google Scholar [8] K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem,, in, 422 (1993), 49.   Google Scholar [9] V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185.  doi: 10.1002/cpa.3160440203.  Google Scholar [10] F. John, "Partial Differential Equations,", Springer-Verlag, (1982).   Google Scholar [11] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976).   Google Scholar [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations,", Elsevier, (2006).   Google Scholar [13] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation,, Waves and Stability in Continuous Media, (1994), 246.   Google Scholar [14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Appl. Math. Lett., 9 (1996), 23.  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, (1997), 291.   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.  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, (1974).   Google Scholar [18] I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999).   Google Scholar [19] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000).   Google Scholar [20] J. Prüss, "Evolutionary Integral Equations and Applications,", Birkhäuser, (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.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar [22] K. Sakamoto, "Inverse Source Problems for Diffusion Equations,", Ph.D. Thesis, (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.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar
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