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Uniqueness for an illposed reactiondispersion model. Application to organic pollution in streamwaters
1.  LMAC, EA 2222, Université de Technologie de Compiègne, BP 20529, 60205 COMPIEGNE Cedex, France 
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,'' Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] 
M. Andrle, F. Ben Belgacem and A. El Badia, Identification of moving pointwise sources in an advectiondispersionreaction equation, Inverse Problems, 27 (2011), 025007, 21 pp. doi: 10.1088/02665611/27/2/025007. 
[3] 
F. Ben Belgacem, Uniqueness for an illposed parabolic system, Comptes Rendus Mathématique Acad. Sci. Paris, 349 (2011), 11611165. doi: 10.1016/j.crma.2011.10.006. 
[4] 
A. Bermúdez, Mathematical techniques for some environmental problems related to water pollution control, in "Mathematics, Climate and Environment" (eds. Díaz and Lions) (Madrid, 1991), RMA Res. Notes Appl. Math., 27, Masson, Paris, (1993), 1227. 
[5] 
C. Bernardi, C. Canuto and Y. Maday, Generalized infsup condition for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal., 25 (1988), 12371271. doi: 10.1137/0725070. 
[6] 
F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,'' Springer Series in Computational Mathematics, 15, SpringerVerlag, New York, 1991. 
[7] 
L. C. Brown and T. O. Barnwell, "The Enhanced Stream Water Quality Models QUAL2E and QUAL2EUNCAS: Documentation and User Manual,'' Environmental Protection Agency, 1987. 
[8] 
S. C. Chapra, "Applied Numerical Methods with MATLAB,'' McGrawHill, 2004. 
[9] 
J. Cousteix and J. Mauss, "Analyse Asymptotique et Couche Limite,'' Mathématiques et Applications, 56, SpringerVerlag, Berlin, Heidelberg, 2006. 
[10] 
R. Dautray and J.L. Lions, "Mathematical Analyis and Numerical Methods for Science and Technology,'' Vol. 5, Evolution Problems. I, with the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig, SpringerVerlag, Berlin, 1992. 
[11] 
W. Eckhaus, "Asymptotic Analysis of Singular Perturbations,'' Studies in Mathematics and its Applications, 9, NorthHolland Publishing Co., AmsterdamNew York, 1979. 
[12] 
A. El Badia, T. HaDuong and A. Hamdi, Identification of a point source in a linear advectiondispersionreaction equation: Application to a pollution source problem, Inverse Problems, 21 (2005), 11211136. doi: 10.1088/02665611/21/3/020. 
[13] 
A. El Badia and A. Hamdi, Inverse source problem in an advectiondispersionreaction system: Application to water pollution, Inverse Problems, 23 (2007), 21032120. doi: 10.1088/02665611/23/5/017. 
[14] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, SpringerVerlag, Berlin, 1983. 
[15] 
G. Gripenberg, S.O. London and O. Steffans, "Volterra Integral and Functional Equations,'' Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. 
[16] 
A. Hamdi, Identification of a timevarying point source in a system of two coupled linear diffusionadvectionreaction equations: Application to surface water pollution, Inverse Problems, 25 (2009), 115009, 21 pp. doi: 10.1088/02665611/25/11/115009. 
[17] 
A. Hamdi, The recovery of a timedependent point source in a linear transport equation: Application to surface water pollution, Inverse Problems, 25 (2009), 075006, 18 pp. doi: 10.1088/02665611/25/7/075006. 
[18] 
G. Jolánkai, "Basic River Water Quality Models: Computer Aided Learning (CAL) Programme on Water Quality Modelling (WQMCAL),'' International Hydrological Programme, UNESCO document 121363, Paris, 1997. 
[19] 
J.L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' Dunod, Paris, 1968. 
[20] 
G. I. Marchuk, "Mathematical Models in Environmental Problems,'' Encyclopedia of Mathematics and its Applications, 34, NorthHolland Publishing Co., Amsterdam, 1986. 
[21] 
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'' An extended version of the Japanese edition, Ecology and diffusion, Translated by G. N. Parker, Biomathematics, 10, SpringerVerlag, BerlinNew York, 1980. 
[22] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Applied Mathematical Sciences, 44, SpringerVerlag, New York, 1983. 
[23] 
J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, Journal of Differential Equations, 66 (1987), 118139. doi: 10.1016/00220396(87)90043X. 
[24] 
C. N. Sawyer, P. L. McCarty and G. F. Parkin, "Chemistry for Environmental Engineering and Science,'' Unpublished proceedings of the Delaware Conference on Ill Posed Inverse Problems, TR 595, University of Wisconsin, Madison, 1980. 
[25] 
G. Wahba, "Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems With Noisy Data,'' McGrawHill, 2003. 
show all references
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,'' Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] 
M. Andrle, F. Ben Belgacem and A. El Badia, Identification of moving pointwise sources in an advectiondispersionreaction equation, Inverse Problems, 27 (2011), 025007, 21 pp. doi: 10.1088/02665611/27/2/025007. 
[3] 
F. Ben Belgacem, Uniqueness for an illposed parabolic system, Comptes Rendus Mathématique Acad. Sci. Paris, 349 (2011), 11611165. doi: 10.1016/j.crma.2011.10.006. 
[4] 
A. Bermúdez, Mathematical techniques for some environmental problems related to water pollution control, in "Mathematics, Climate and Environment" (eds. Díaz and Lions) (Madrid, 1991), RMA Res. Notes Appl. Math., 27, Masson, Paris, (1993), 1227. 
[5] 
C. Bernardi, C. Canuto and Y. Maday, Generalized infsup condition for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal., 25 (1988), 12371271. doi: 10.1137/0725070. 
[6] 
F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods,'' Springer Series in Computational Mathematics, 15, SpringerVerlag, New York, 1991. 
[7] 
L. C. Brown and T. O. Barnwell, "The Enhanced Stream Water Quality Models QUAL2E and QUAL2EUNCAS: Documentation and User Manual,'' Environmental Protection Agency, 1987. 
[8] 
S. C. Chapra, "Applied Numerical Methods with MATLAB,'' McGrawHill, 2004. 
[9] 
J. Cousteix and J. Mauss, "Analyse Asymptotique et Couche Limite,'' Mathématiques et Applications, 56, SpringerVerlag, Berlin, Heidelberg, 2006. 
[10] 
R. Dautray and J.L. Lions, "Mathematical Analyis and Numerical Methods for Science and Technology,'' Vol. 5, Evolution Problems. I, with the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig, SpringerVerlag, Berlin, 1992. 
[11] 
W. Eckhaus, "Asymptotic Analysis of Singular Perturbations,'' Studies in Mathematics and its Applications, 9, NorthHolland Publishing Co., AmsterdamNew York, 1979. 
[12] 
A. El Badia, T. HaDuong and A. Hamdi, Identification of a point source in a linear advectiondispersionreaction equation: Application to a pollution source problem, Inverse Problems, 21 (2005), 11211136. doi: 10.1088/02665611/21/3/020. 
[13] 
A. El Badia and A. Hamdi, Inverse source problem in an advectiondispersionreaction system: Application to water pollution, Inverse Problems, 23 (2007), 21032120. doi: 10.1088/02665611/23/5/017. 
[14] 
D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224, SpringerVerlag, Berlin, 1983. 
[15] 
G. Gripenberg, S.O. London and O. Steffans, "Volterra Integral and Functional Equations,'' Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990. 
[16] 
A. Hamdi, Identification of a timevarying point source in a system of two coupled linear diffusionadvectionreaction equations: Application to surface water pollution, Inverse Problems, 25 (2009), 115009, 21 pp. doi: 10.1088/02665611/25/11/115009. 
[17] 
A. Hamdi, The recovery of a timedependent point source in a linear transport equation: Application to surface water pollution, Inverse Problems, 25 (2009), 075006, 18 pp. doi: 10.1088/02665611/25/7/075006. 
[18] 
G. Jolánkai, "Basic River Water Quality Models: Computer Aided Learning (CAL) Programme on Water Quality Modelling (WQMCAL),'' International Hydrological Programme, UNESCO document 121363, Paris, 1997. 
[19] 
J.L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,'' Dunod, Paris, 1968. 
[20] 
G. I. Marchuk, "Mathematical Models in Environmental Problems,'' Encyclopedia of Mathematics and its Applications, 34, NorthHolland Publishing Co., Amsterdam, 1986. 
[21] 
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'' An extended version of the Japanese edition, Ecology and diffusion, Translated by G. N. Parker, Biomathematics, 10, SpringerVerlag, BerlinNew York, 1980. 
[22] 
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Applied Mathematical Sciences, 44, SpringerVerlag, New York, 1983. 
[23] 
J.C. Saut and B. Scheurer, Unique continuation for some evolution equations, Journal of Differential Equations, 66 (1987), 118139. doi: 10.1016/00220396(87)90043X. 
[24] 
C. N. Sawyer, P. L. McCarty and G. F. Parkin, "Chemistry for Environmental Engineering and Science,'' Unpublished proceedings of the Delaware Conference on Ill Posed Inverse Problems, TR 595, University of Wisconsin, Madison, 1980. 
[25] 
G. Wahba, "Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems With Noisy Data,'' McGrawHill, 2003. 
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