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Volume 19, Issue 5
The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs

Wei-Jun Tang, Hong-Yuan Fu & Long-Jun Shen

J. Comp. Math., 19 (2001), pp. 489-500.

Published online: 2001-10

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  • Abstract

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region $R^2$\Γ with Γ a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.  

  • Keywords

Helmholtz equation, Quadrature method.

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-19-489, author = {Tang , Wei-JunFu , Hong-Yuan and Shen , Long-Jun}, title = {The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs}, journal = {Journal of Computational Mathematics}, year = {2001}, volume = {19}, number = {5}, pages = {489--500}, abstract = {

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region $R^2$\Γ with Γ a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.  

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9001.html} }
TY - JOUR T1 - The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs AU - Tang , Wei-Jun AU - Fu , Hong-Yuan AU - Shen , Long-Jun JO - Journal of Computational Mathematics VL - 5 SP - 489 EP - 500 PY - 2001 DA - 2001/10 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9001.html KW - Helmholtz equation, Quadrature method. AB -

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region $R^2$\Γ with Γ a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.  

Tang , Wei-JunFu , Hong-Yuan and Shen , Long-Jun. (2001). The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs. Journal of Computational Mathematics. 19 (5). 489-500. doi:
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