Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (3): 539-551.doi: 10.1016/S0252-9602(15)30002-3

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

Jun GUO1, Lili FAN2, Guozheng YAN3   

  1. 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).

Abstract:

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

CLC Number: 

  • 35D05
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