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ON THE EXISTENCE OF SOLUTION FOR AN INVERSE PROBLEM

Year 2013, Volume 3, Issue 1, 33 - 45, 01.06.2013

Abstract

We consider a boundary detection problem. We present physical motivations. We formulate the problem as a shape optimization problem by introducing the Neumann condition of the accessible part in a cost functional to be minimized, which complicates the study of continuity state that requires more regularity of the free boundary. We show the existence of the optimal solution of the problem by the J. Haslinger and P. Neittaanm¨aki principle.

References

  • Chakib, A., Ghemires, T. and Nachaoui, A., (2003), A numerical study of Şltration problem in inhomogeneous dam with discontinuous permeability, Appl. Num. Math., 45, 123-138.
  • Cakoni, F. and Kress. R., (2007), Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems Imaging, 1, 229-245.
  • Chenais, D., (1975), on the existence of a solution in domain identiŞcation problem, J. of Math. Analysis and applications, 52, 189-219.
  • Chi, C.C., Yeih, W. and Liu C.S., (2009), A novel method for solving the Cauchy problem of Laplace equation using the Şctitious time integration method, Comput. Model. Eng. Sci., 47, 167-190.
  • Fasino, D., Inglese, G. and Mariani, F., (2007), Corrosion detection in conducting boundaries: II. Linearization, stability and discretization, Inverse Problems, 23, 1101-1114.
  • Haslinger, J. Neittaanm¨aki, (1988), Finite Element Approximation for Optimal shape design: Theory and applications, John, Wiley & Sons LTD. Chichester. New York. Brisbane. Toronto. Singapore.
  • Henrot, A. and Pierre, M., (2005), Variation et optimisation de formes: une analyse gomtrique, Springer, Berlin.
  • Hon, Y.C, Wu, Z., (2000), A numerical computation for inverse boundary determination problem, Eng. Anal. Bound. Elem., 24, 599-606.
  • Huang, C.H. and Shih, C.C., (2007), An inverse problem in estimating simultaneously the time- dependent applied force and moment of an Euler-Bernoulli beam, Comput. Model. Eng. Sci., 21, 239-254.
  • Kaup, P.G. and Santosa, F., (1995), Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestructive Eval., 14, 127-136.
  • Liu, C.S., (2008), A modiŞed collocation Trefftz method for the inverse Cauchy problem of Laplace equation, Eng. Anal. Bound. Elem., 32, 778-785.
  • Sladek, J., Sladek, V., Wen, P.H. and Hon, Y.C., (2009), The inverse problem of determining heat transfer coefficients by the meshless local Petrov-Galerkin method, Comput. Model. Eng. Sci., 48, 191-218.

Year 2013, Volume 3, Issue 1, 33 - 45, 01.06.2013

Abstract

References

  • Chakib, A., Ghemires, T. and Nachaoui, A., (2003), A numerical study of Şltration problem in inhomogeneous dam with discontinuous permeability, Appl. Num. Math., 45, 123-138.
  • Cakoni, F. and Kress. R., (2007), Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Problems Imaging, 1, 229-245.
  • Chenais, D., (1975), on the existence of a solution in domain identiŞcation problem, J. of Math. Analysis and applications, 52, 189-219.
  • Chi, C.C., Yeih, W. and Liu C.S., (2009), A novel method for solving the Cauchy problem of Laplace equation using the Şctitious time integration method, Comput. Model. Eng. Sci., 47, 167-190.
  • Fasino, D., Inglese, G. and Mariani, F., (2007), Corrosion detection in conducting boundaries: II. Linearization, stability and discretization, Inverse Problems, 23, 1101-1114.
  • Haslinger, J. Neittaanm¨aki, (1988), Finite Element Approximation for Optimal shape design: Theory and applications, John, Wiley & Sons LTD. Chichester. New York. Brisbane. Toronto. Singapore.
  • Henrot, A. and Pierre, M., (2005), Variation et optimisation de formes: une analyse gomtrique, Springer, Berlin.
  • Hon, Y.C, Wu, Z., (2000), A numerical computation for inverse boundary determination problem, Eng. Anal. Bound. Elem., 24, 599-606.
  • Huang, C.H. and Shih, C.C., (2007), An inverse problem in estimating simultaneously the time- dependent applied force and moment of an Euler-Bernoulli beam, Comput. Model. Eng. Sci., 21, 239-254.
  • Kaup, P.G. and Santosa, F., (1995), Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestructive Eval., 14, 127-136.
  • Liu, C.S., (2008), A modiŞed collocation Trefftz method for the inverse Cauchy problem of Laplace equation, Eng. Anal. Bound. Elem., 32, 778-785.
  • Sladek, J., Sladek, V., Wen, P.H. and Hon, Y.C., (2009), The inverse problem of determining heat transfer coefficients by the meshless local Petrov-Galerkin method, Comput. Model. Eng. Sci., 48, 191-218.

Details

Primary Language English
Journal Section Research Article
Authors

Chakir TAJANİ This is me
Ibn Tofail University, Faculty of Sciences, Department of Mathematics and Infirmatics, Kenitra, Morocco


Jaafar ABOUCHABAKA This is me
Ibn Tofail University, Faculty of Sciences, Department of Mathematics and Infirmatics, Kenitra, Morocco


Nadia SAMOUH This is me
Moulay Ismail University, Faculty of Sciences, Department of Mathematics, Meknes, Morocco

Publication Date June 1, 2013
Published in Issue Year 2013, Volume 3, Issue 1

Cite

Bibtex @ { twmsjaem761694, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {1}, pages = {33 - 45}, title = {ON THE EXISTENCE OF SOLUTION FOR AN INVERSE PROBLEM}, key = {cite}, author = {Tajani, Chakir and Abouchabaka, Jaafar and Samouh, Nadia} }