Acta mathematica scientia, Series B >
THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN
Received date: 2013-06-17
Revised date: 2014-03-02
Online published: 2015-05-01
Supported by
The second author is supported by the grant from the National Natural Science Foundation of China (11301405). The third author is supported by the grants from the National Natural Science Foundation of China (11171127 and 10871080).
We consider a kind of scattering problem by a crack Γ that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Γ. Both sides of the crack Γ are given Dirichlet-impedance boundary conditions, and different boundary condition (Dirichlet, Neumann or Impedance boundary condition) is set on the boundary of D. Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.
Key words： Boundary integral equation; Helmholtz equation; Fredholm theory
Jun GUO , Lili FAN , Guozheng YAN . THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN[J]. Acta mathematica scientia, Series B, 2015 , 35(3) : 539 -551 . DOI: 10.1016/S0252-9602(15)30002-3
[1] Ammari H, Bao G, Wood A W. An integral equation method for the electromagnetic scattering from cavities. Math Meth Appl Sci, 2000, 23: 1057-1072
[2] Colton D L, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer-Verlag, 1998
[3] Colton D L, Kress R. Integral Equation Methods in Scattering Theory. Wiley, New York: Springer-Verlag, 1983
[4] Cakoni F, Colton D L. The linear sampling method for cracks. Inverse Problem, 2003, 19: 279-295
[5] Cakoni F, Colton D L. Qualitative Method in Inverse Scattering Theory. Berlin: Springer, 2006
[6] Hsiao G, Wendland W L. On integral equation method for the plane mixed boundary value problem for the Laplacian. Math Methods Appl Sci, 1979: 265-321
[7] Kress R. Acoustic Scattering: Special Theoretical Tools. London: Acamedia, 2001
[8] Kress R. Linear Integral Equations (second ed.). Berlin: Springer-verlag, 1999
[9] Kress R. Frechet differentiability of the far field operator for scattering from a crack. J Inverse Ill-posed Probl, 1995, 3: 305-313
[10] Kirsch A, Ritter R. The linear sampling method for inverse scattering from an open arc. Inverse Problems, 2000, 16: 85-109
[11] Lee K M. Inverse scattering problem for an impedance crack. Wave Motion, 2008, 45: 254-263
[12] Mclean W. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge: Cambridge University Press, 2000
[13] Lions J, Magenes E. Non-homogeneous Boundary Value Problems and Applications. New York: Springer, 1972
[14] Monch L. On the inverse acoustic scattering problem by an open arc: On the sound-hard case. Inverse Problems, 1997, 13: 1379-1392
[15] Peter J, Potthast R. Testing the integrity of some cavity-the Cauchy problem and the range test. Applied Numer Math, 2008, 58: 899-914
[16] Qin H H, Colton D L. The inverse scattering problem for cavities. Applied Numer Math, 2012, 62: 699-708
[17] Qin H H, Colton D L. The inverse scattering problem for cavities with impedance boundary condition. Adv Comput Math, 2012, 36: 157-174
[18] Yan G Z, Ye J G. Boundary integral methods for scattering problems with cracks buried in a piecewise homogeneous medium. Math Meth Appl Sci, 2012, 35: 84-96
[19] Yan G Z, Yao M. Mathematical basis of scattering problems from penetrable obstacles and cracks. J Math Phys, 2010, 51: 123520
[20] Yan G Z, Yao M. The Method of boundary integral equations for a mixed scattering problem. J Differential Equations, 2009, 246: 4618-4631
/
| 〈 |
|
〉 |