Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation

Inverse Problems

Volumn 25, Issue 11, 2009, Pages

  Cheng, Jin ^a,b   Nakagawa, Junichi ^c   Yamamoto, Masahiro ^d   Yamazaki, Tomohiro ^d  

^a FUDAN UNIVERSITY   (China)

^b Shanghai Institute of Immunology   (China)

^c Technical Development Bureau   (Japan)

^d UNIVERSITY OF TOKYO   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

ANOMALOUS DIFFUSION; CAPUTO DERIVATIVES; DIRAC DELTA FUNCTION; EIGENFUNCTION EXPANSIONS; FRACTIONAL DIFFUSION EQUATION; INITIAL VALUES; NEUMANN BOUNDARY CONDITION; VALUE PROBLEMS; WEAK SOLUTION;

EIGENVALUES AND EIGENFUNCTIONS; INVERSE PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS;

DIFFUSION;

EID: 70450198851     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/25/11/115002     Document Type: Article

References (42)

1. Adams E E and Gelhar L W 1992 Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis Water Resour. Res. 28 3293-307
2. Bazhlekova E 1998 The abstract Cauchy problem for the fractional evolution equation Fractional Calculus Appl. Anal. 1 255-70
3. Berkowitz B, Cortis A, Dentz M and Scher H 2006 Modeling non-Fickian transport in geological formations as a continuous time random walk Rev. Geophys. 44 RG2003/2006, 2005RG000178
4. Berkowitz B, Scher H and Silliman S E 2000 Anomalous transport in laboratory-scale, heterogeneous porus media Water Resour. Res. 36 149-58
5. Brezis H 1983 Analyse Fonctionnelle (Paris: Masson)
6. Eidelman S D and Kochubei A N 2004 Cauchy problem for fractional diffusion equations J. Diff. Eqns 199 211-55
7. Freiling G and Yurko V 2001 Inverse Sturm-Liouville Problems and Their Applications (New York: Nova Science Publishers)
8. Giona M, Gerbelli S and Roman H E 1992 Fractional diffusion equation and relaxation in complex viscoelastic materials Physica A 191 449-53
9. Giona M and Roman H E 1992 A theory of transport phenomena in disordered systems Chem. Eng. J. 49 1-10
10. Gorenflo R, Luchko Y and Zabrejko P P 1999 On solvability of linear fractional differential equations in Banach spaces Fractional Calculus Appl. Anal. 2 163-76
11. Hatano Y and Hatano N 1998 Dispersive transport of ions in column experiments: an explanation of long-tailed profiles Water Resour. Res. 34 1027-33
12. Hanyga A 2002 Multidimensional solutions of time-fractional diffusion-wave equations Proc. R. Soc. Lond. A 458 933-57
13. Isakov V 2006 Inverse Problems for Partial Differential Equations (Berlin: Springer)
14. Isakov V and Kindermann S 2000 Identification of the diffusion coefficient in a one-dimensional parabolic equation Inverse Problems 16 665-80
15. Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier)
16. Klibanov M V and Timonov A A 2004 Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Utrecht: VSP)
17. Lavrentev M M, Romanov V G and Shishatskii S P 1986 Ill-posed Problems of Mathematical Physics and Analysis (Providence, RI: American Mathematical Society)
18. Levitan B M and Sargsjan I S 1975 Introduction to Spectral Theory (Providence, RI: American Mathematical Society)
19. Luchko Y 2009 Maximum principle for the generalized time-fractional diffusion equation J. Math. Anal. Appl. 351 218-23
20. Luchko Y 2009 Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation Comput. Math. Appl. (at press)
21. Mainardi F 1994 On the initial value problem for the fractional diffusion-wave equation Waves and Stability in Continuous Media (Singapore: World Scientific) pp 246-51
22. Mainardi F 1995 The time fractional diffusion-wave equation Radiofizika 38 20-36
23. Mainardi F 1996 The fundamental solutions for the fractional diffusion-wave equation Appl. Math. Lett. 9 23-8
24. Marchenko V A 1952 Some problems in the theory of linear differential operators Trudy Moskov. Mat. Obshch. 1 327-420
25. Metzler R, Glöckle W G and Nonnenmacher T F 1994 Fractional model equation for anomalous diffusion Physica A 211 13-24
26. Metzler R and Klafter J 2000 The random walk's guide to anomalous diffusion: a fractional dynamics approach Phys. Rep. 339 1-77
27. Metzler R and Klafter J 2000 Boundary value problems for fractional diffusion equations Physica A 278 107-25
28. Murayama R 1981 The Gel'fand-Levitan theory and certain inverse problems for the parabolic equation J. Fac. Sci. Univ. Tokyo IA 28 317-30
29. Nigmatullin R R 1986 The realization of the generalized transfer equation in a medium with fractal geometry Phys. Status Solidi b 133 425-30
30. Pierce A 1979 Unique identification of eigenvalues and coefficients in a parabolic problem SIAM J. Control Optim. 17 494-9
31. Podlubny I 1999 Fractional Differential Equations (San Diego: Academic)
32. Prüss J 1993 Evolutionary Integral Equations and Applications (Basel: Birkhäuser Verlag)
33. Roman H E 1995 Structures of random fractals and the probability distribution of random walks Phys. Rev. E 51 5422-25
34. Roman H E and Alemany P A 1994 Continuous-time random walks and the fractional diffusion equation J. Phys. A: Math. Gen. 27 3407-10
35. Romanov V G 1987 Inverse Problems of Mathematical Physics (Utrecht: VNU Science Press)
36. Sokolov I M, Klafter J and Blumen A 2002 Fractional kinetics Phys. Today 55 48-54
37. Suzuki T 1985 Gel'fand-Levitan's theory, deformation formulas and inverse problems J. Fac. Sci. Univ. Tokyo IA 32 223-71
38. Suzuki T 1986 Inverse problems for heat equations on compact intervals and on circles: I J. Math. Soc. Japan 38 39-65
39. Suzuki T and Murayama R 1980 A uniqueness theorem in an identification problem for coefficients of parabolic equations Proc. Japan Acad. A 56 259-63
40. Xiong Y, Huang G and Huang Q 2006 Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk J. Contaminant Hydrol. 86 163-75
41. Yosida K 1964 Functional Analysis (Berlin: Springer)
42. Zhou L and Selim H M 2003 Application of the fractional advection-dispersion equations in porous media Soli. Sci. Soc. Am. J. 67 1079-84