C. A. Balanis, Antenna theory: Analysis and design, 2005.

G. Caloz, M. Costabel, M. Dauge, and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer, Asymptotic Analysis, vol.50, issue.12, pp.121-173, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00001555

K. R. Carver and J. W. Mink, Microstrip antenna technology, IEEE Transactions on Antennas and Propagation, vol.29, issue.1, pp.2-24, 1981.
DOI : 10.1109/TAP.1981.1142523

D. C. Chang and E. F. Kuester, Total and partial reflection from the end of a parallel-plate waveguide with an extended dielectric slab, Radio Science, vol.19, issue.504-510, pp.1-13, 1981.
DOI : 10.1029/RS016i001p00001

J. Chazarain and A. Piriou, Introduction to the theory of partial differential equations, 1982.

C. Chi and N. G. , An efficient numerical approach for modeling microstrip-type antennas, IEEE Transactions on Antennas and Propagation, vol.38, issue.9, pp.1399-1404, 1990.
DOI : 10.1109/8.56991

J. Coustex and J. Mauss, Asymptotic analysis and Boundary Layers, 2007.

M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol.1341, 1988.
DOI : 10.1007/BFb0086682

R. W. Dearnley and A. R. Barel, A comparison of models to determine the resonant frequencies of a rectangular microstrip antenna, IEEE Transactions on Antennas and Propagation, vol.37, issue.1, pp.114-118, 1989.
DOI : 10.1109/8.192173

M. Van-dyke, Perturbation Method in Fluid Mechanics, Journal of Applied Mechanics, vol.43, issue.1, 1975.
DOI : 10.1115/1.3423785

W. Eckhaus, Matched Asymptotic Expansions and Singular Perturbations, North-Holland Mathematics Studies, vol.6, 1973.

J. Jin, Z. Lou, Y. Li, N. W. Riley, and D. J. Riley, Finite Element Analysis of Complex Antennas and Arrays, IEEE Transactions on Antennas and Propagation, vol.56, issue.8, pp.2222-2240, 2008.
DOI : 10.1109/TAP.2008.926776

J. Jin, The finite element method in electromagnetics, second edition, 2002.

P. Joly and S. Tordeux, Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots, ESAIM: Mathematical Modelling and Numerical Analysis, vol.40, issue.1, pp.63-97, 2006.
DOI : 10.1051/m2an:2006008
URL : https://hal.archives-ouvertes.fr/inria-00527590

E. F. Kuester, R. T. Johnk, and D. C. Chang, The thin-substrate approximation for reflection from the end of a slab-loaded parallel-plate waveguide with application to microstrip patch antennas, IEEE Transactions on Antennas and Propagation, vol.30, issue.5, pp.30-910, 1982.
DOI : 10.1109/TAP.1982.1142907

A. Makhlouf, Justification et Amélioration de Modèles d'Antennes Patch par la Méthode des Développements Asymptotiques Raccordés, 2008.

T. M. Martinson and E. F. Kuester, Accurate analysis of arbitrarily ShapedPatch resonators on thin substrates The edge admittance of a wide microstrip patch as seen by an obliquely incident wave A generalized edge boundary condition for open microstrip structures, IEEE Transactions on Microwave Theory and Technique IEEE Transactions on Atennas and Propagation Journal of Electromagnetic Waves and Applications, vol.36, issue.4 4, pp.324-331, 1988.

V. Maz-'ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary-value problems in singularly perturbed domains, 2000.

P. Mciver and A. D. Rawlins, TWO-DIMENSIONAL WAVE-SCATTERING PROBLEMS INVOLVING PARALLEL-WALLED DUCTS, The Quarterly Journal of Mechanics and Applied Mathematics, vol.46, issue.1, pp.89-116, 1993.
DOI : 10.1093/qjmam/46.1.89

J. R. Mosig and F. E. Gardiol, General integral equation formulation for microstrip antennas and scatterers, Proc. IEE, pp.424-432, 1985.
DOI : 10.1049/ip-h-2.1985.0076

S. A. Nazarov and B. A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol.13, 1994.
DOI : 10.1515/9783110848915

J. Nédélec, Acoustic and electromagnetic equations: Integral representations for harmonic problems, 2001.

E. Newman and P. Tulyathan, Analysis of microstrip antennas using moment methods, IEEE Transactions on Antennas and Propagation, vol.29, issue.1, pp.47-53, 1981.
DOI : 10.1109/TAP.1981.1142532

B. Noble, Wiener-Hopf Technique for the solution of partial differential equations, 1958.

M. Taylor, Partial differential equations I, basic theory, 1996.

S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes, 2004.

G. N. Watson, A treatise on the theory of Bessel functions. Reprint of the second (1944) edition, Cambridge Mathematical Library, 1995.

C. H. Wilcox, Toulouse cedex 1 (France) E-mail address: abendali@insa-toulouse, UMR CNRS 5219, Département de Génie Mathématique Toulouse cedex 1 (France) University of Toulouse cedex 1 (France) E-mail address: stordeux@insa-toulouse.fr, 1975.