DUAL BOUNDARY INTEGRAL EQUATIONS FOR HELMHOLTZ EQUATION AT

DUAL BOUNDARY INTEGRAL EQUATIONS FOR HELMHOLTZ EQUATION AT A CORNER USING CONTOUR APPROACH AROUND SINGULARITY NTOU HRE Report: C. E. Lin Number: M 98520025 Adviser: J. T. Chen Date: Jan. 07. 2010

Outlines 1. Dual Integral formulation of BEM for Helmholtz equation with a corner 2. Discussions on the Laplace and Helmholtz equations at a corner 3. Conclusions NTOU HRE 2

Dual Integral formulation of BEM for Helmholtz equation with a corner NTOU HRE 3

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Single layer potential: Double layer potential: Normal derivative of single layer potential: Normal derivative of double layer potential: Tangent derivative of single layer potential: Tangent derivative of double layer potential: 5

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Discussions on the Laplace and Helmholtz equations at a corner ⋯ Wave equation ⋯ Helmholtz equation “k” is very small and can be negligible ⋯ Laplace equation NTOU HRE 7

Conclusions • The free terms of the six kernel functions in the dual integral equation for the Helmholtz equation at a corner have been examined • It is discovered that employing the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, respectively, which differs from the limiting process from an interior point to a boundary point where the jump term is descended from the L kernel only. NTOU HRE 8

• Laplace equation is a special case of the Helmholtz equation when the value of w-ave number approaches zero. NTOU HRE 9

THANKS FOR YOUR KIND OF ATTENTION NTOU HRE 10