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\G with G 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

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@Article{JCM-19-489, author = {}, 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\G with G 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 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 KW - Quadrature method AB - Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R~2\G with G 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.
Wei Jun Tang, Hong Yuan Fu & Long Jun Shen. (1970). 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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