THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

doi:10.1016/S0252-9602(15)30002-3

数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (3): 539-551.doi: 10.1016/S0252-9602(15)30002-3

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

郭军¹, 范丽丽², 严国政³

1. School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China;
2. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China;
3. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2013-06-17 修回日期:2014-03-02 出版日期:2015-05-01 发布日期:2015-05-01
通讯作者: Lili FAN E-mail:fll810@live.cn
基金资助:
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).

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

Jun GUO¹, Lili FAN², Guozheng YAN³

1. School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China;
2. School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China;
3. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2013-06-17 Revised:2014-03-02 Online:2015-05-01 Published:2015-05-01
Contact: Lili FAN E-mail:fll810@live.cn
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).

摘要/Abstract

摘要：

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.

关键词: Boundary integral equation, Helmholtz equation, Fredholm theory

Abstract:

Key words: Boundary integral equation, Helmholtz equation, Fredholm theory

中图分类号:

35D05

郭军, 范丽丽, 严国政. THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN[J]. 数学物理学报(英文版), 2015, 35(3): 539-551.

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.

参考文献

[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

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

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