Solving the inverse problems of Laplace equation to determine the Robin coefficient/cracks' position inside a disk

CMES - Computer Modeling in Engineering and Sciences

Volumn 40, Issue 1, 2009, Pages 1-28

  Liu, Chein Shan ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Crack position; Fredholm integral equation; Inverse Cauchy problem; Laplace equation; Lavrentiev regularization; Robin coefficient

Indexed keywords

CRACK POSITION; FREDHOLM INTEGRAL EQUATION; INVERSE CAUCHY PROBLEM; LAVRENTIEV REGULARIZATION; ROBIN COEFFICIENT;

CRACK DETECTION; CRACKS; DIFFERENTIAL EQUATIONS; DISKS (STRUCTURAL COMPONENTS); FOURIER ANALYSIS; FOURIER SERIES; HARMONIC ANALYSIS; INTEGRAL EQUATIONS; INTEGRODIFFERENTIAL EQUATIONS; INVERSE PROBLEMS; LAPLACE TRANSFORMS;

LAPLACE EQUATION;

EID: 67049087093     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. Apostol, T. M. (1974) : Mathematical Analysis. Second Edition, Addison-Wesley, Reading, Mass.
2. Azaiez, M.; Ben Abda, A.; Ben Abdallah, J. (2005) : Revisiting the Dirichlet-to- Neumann solver for data completion and application to some inverse problems. Int. J. Appl. Math. Mech., vol. 1, pp. 106-121.
3. Ben Belgacem, F.; El Fekih, H. (2005) : On Cauchy's problem. I. a variational Steklov-Poincaré theory. Inv. Prob., vol. 21, pp. 1915-1936.
4. Berntsson, F.; Eldén, L. (2001) : Numerical solution of a Cauchy problem for the Laplace equation. Inverse Problems, vol. 17, pp. 839-854.
5. Brühl, M.; Hanke, M.; Pidcock, M. (2001) : Crack detection using electrostatic measurements. Math. Model. Numer. Anal., vol. 35, pp. 595-605.
6. Chaabane, S.; Jaoua, M. (1999) : Identification of Robin coefficients by the means of boundary measurements. Inverse Problems, vol. 15, pp. 1425-1438.
7. Chaabane, S.; Elhechmi, C.; Jaoua, M. (2004) : A stable method for the Robin inverse problem. Math. Comp. Simul., vol. 66, pp. 367-383.
8. Chang, J.-R.; Yeih, W,; Shieh, M.-H. (2001) : On the modified Tikhonov's reg- ularization method for the Cauchy problem of the Laplace equation. J. Mar. Sci. Tech., vol. 9, pp. 113-121.
9. Chapko, R.; Kress, R. (2005) : A hybrid method for inverse boundary value prob- lems in potential theory. J. Inv. Ill-Posed Problems, vol. 13, pp. 27-40.
10. Fasino, D.; Inglese, G. (1999) : An inverse Robin problem for Laplace's equation: theoretical results and numerical methods. Inverse Problems, vol. 15, pp. 41-48.
11. Jin, B.; Zheng, Y. (2006) : A meshless method for some inverse problems associ- ated with the Helmholtz equation. Comp. Meth. Appl. Mech. Engng., vol. 195, pp. 2270-2288.
12. Johnston, P. (2001) : Computational Inverse Problems in Electrocardiography, Ad- vances in Computational Bioengineering, Vol. 5, Johnston, P. Ed., WIT Press.
13. Kress, R. (2004) : Inverse Dirichlet problem and conformal mapping. Math. Comp. Simul., vol. 66, pp. 255-265.
14. Leblond, J.; Mahjoub, M.; Partington, J. R. (2006) : Analytic extensions and Cauchy-type inverse problems on annular domains: stability results. J. Inv. Ill- Posed Problems, vol. 14, to appear.
15. Lin, F.; Fang, W. (2005) : A linear integral equation approach to the Robin inverse problem. Inverse Problems, vol. 21, pp. 1757-1772.
16. Ling, L.; Takeuchi, T. (2008) : Boundary control for inverse Cauchy problems of the Laplace equations. CMES: Computer Modeling in Engineering and Sciences, vol. 29, pp. 45-54.
17. Liu, C.-S. (2007a) : Elastic torsion bar with arbitrary cross-section using the Fred- holm integral equations. CMC: Computers, Materials and Continua, vol. 5, pp. 31-42.
18. Liu, C.-S. (2007b) : A meshless regularized integral equation method for Laplace equation in arbitrary interior or exterior plane domains. CMES: Computer Model- ing in Engineering and Sciences, vol. 19, pp. 99-109.
19. Liu, C.-S. (2007c) : A MRIEM for solving the Laplace equation in the doubly- connected domain. CMES: Computer Modeling in Engineering and Sciences, vol. 19, pp. 145-161.
20. Liu, C.-S. (2008a) : A highly accurate MCTM for inverse Cauchy problems of Laplace equation in arbitrary plane domains. CMES: Computer Modeling in Engi- neering and Sciences, vol. 35, pp. 91-111.
21. Liu, C.-S. (2008b) : A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation. Eng. Anal. Bound. Elem., vol. 32, pp. 778-785.
22. Liu, C.-S. (2008c) : A highly accurate MCTM for direct and inverse problems of biharmonic equation in arbitrary plane domains. CMES: Computer Modeling in Engineering and Sciences, vol. 30, pp. 65-75.
23. Liu, C.-S.; Atluri, S. N. (2008a) : A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear op- timization. CMES: Computer Modeling in Engineering and Sciences, vol. 34, pp. 155-178.
24. Liu, C.-S.; Atluri, S. N. (2008b) : A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering and Sciences, vol. 31, pp. 71-83.
25. Liu, C.-S.; Chang, C. W.; Chiang, C. Y. (2008) : A regularized integral equa- tion method for the inverse geometry heat conduction problem. Int. J. Heat Mass Transfer, vol. 51, pp. 5380-5388.
26. Marin, L.; Lesnic, D. (2005) : The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equa- tion. Math. Comp. Model., vol. 42, pp. 261-278.
27. Mera, N. S.; Elliott, L.; Ingham, D. B.; Lesnic, D. (2000) : The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium. Int. J. Num. Meth. Eng., vol. 49, pp. 481-499.
28. Lavrentiev, M. M. (1967) : Some Improperly Posed Problems of Mathematical Physics. Springer, New York.
29. Saito, K.; Nakada, M.; Iijima, K.; Onish, K. (2005) : A quasi-spectral method for Cauchy problem of 2/D Laplace equation on an annulus. J. Phys. Conf. Ser., vol. 12, pp. 128-139.
30. Slodička, M.; Van Keer, R. (2004) : A numerical approach for the determination of a missing boundary data in elliptic problems. Appl. Math. Comp., vol. 147, pp. 569-580.
31. Tikhonov, A. N.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G. (1990) : Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic, London.