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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 24, Issue 6


Inverse source problem based on two dimensionless dispersion-current functions in 2D evolution transport equations

Adel Hamdi
  • Corresponding author
  • Laboratoire de Mathématiques LMI, Institut National des Sciences Appliquées de Rouen, Avenue de l’Université, 76801 Saint-Etienne-du-Rouvray, Cedex, France
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/ Imed Mahfoudhi
  • Laboratoire de Mathématiques LMI, Institut National des Sciences Appliquées de Rouen, Avenue de l’Université, 76801 Saint-Etienne-du-Rouvray, Cedex, France
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Published Online: 2016-03-20 | DOI: https://doi.org/10.1515/jiip-2014-0051


The paper deals with the nonlinear inverse source problem of identifying an unknown time-dependent point source occurring in a two-dimensional evolution advection-dispersion-reaction equation with spatially varying velocity field and dispersion tensor. The originality of this study consists in establishing a constructive identifiability theorem that leads to develop an identification method using only significant boundary observations and operating other than the classic optimization approach. To this end, we derive two dispersion-current functions that have the main property to be of orthogonal gradients which yield identifiability of the elements defining the involved unknown source from some boundary observations related to the associated state. Provided the velocity field fulfills the so-called no-slipping condition, the required boundary observations are reduced to only recording the state on the outflow boundary and its flux on the inflow boundary of the monitored domain. We establish an identification method that uses those boundary records (1) to localize the sought source position as the unique solution of a nonlinear system defined by the two dispersion-current functions, (2) to give lower and upper bounds of the total amount loaded by the unknown time-dependent source intensity function, (3) to transform the identification of this latest into solving a deconvolution problem. Some numerical experiments on a variant of the surface water BOD pollution model are presented.

Keywords: Nonlinear inverse source problem; boundary null controllability; optimization,advection-dispersion-reaction equation; application to surface water pollution

MSC 2010: 35R30; 76B75; 93B05


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About the article

Received: 2014-07-20

Revised: 2015-04-29

Accepted: 2016-02-01

Published Online: 2016-03-20

Published in Print: 2016-12-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 24, Issue 6, Pages 663–685, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0051.

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