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Regularisation of BIE in three-dimensional Laplace's equation

We regularise the hypersingular integral equation (3.15) using (L.2).
 
$\displaystyle -\frac{\partial u^{\infty}(\mbox{\boldmath$ x $ })}{\partial n_x}$ = $\displaystyle \mbox{p.f.}\int_{S} \frac{\partial^2 G(\mbox{\boldmath$ x $ }-\mb...
...tial n_y} \phi(\mbox{\boldmath$ y $ }) dS_y,\quad (\mbox{\boldmath$ x $ }\in S)$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) \mbox{p.f.}\int_{S} n_i(\mbox{\boldma...
...boldmath$ y $ })}
{\partial x_j \partial y_i} \phi(\mbox{\boldmath$ y $ }) dS_y$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) \mbox{p.f.}\int_{S} n_i(\mbox{\boldma...
...\boldmath$ y $ })}{\partial x_q \partial y_p}
\phi(\mbox{\boldmath$ y $ }) dS_y$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) \mbox{p.f.}\int_{S} n_i(\mbox{\boldma...
...\boldmath$ y $ })}{\partial x_q \partial y_p}
\phi(\mbox{\boldmath$ y $ }) dS_y$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) \mbox{p.f.}\int_{S} n_i(\mbox{\boldma...
... G =0, \ \frac{\partial}{\partial x_i}
= - \frac{\partial}{\partial y_i}\right)$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) e_{lpj}
\mbox{p.f.}\int_{S} n_i(\mbox...
...ial x_p}\frac{\partial \phi(\mbox{\boldmath$ y $ })}{\partial y_q}
\right) dS_y$  
  = $\displaystyle n_j(\mbox{\boldmath$ x $ }) e_{lpj}
\oint_{\partial S} \frac{\par...
...y $ })}_{0}
dy_l \ (\raisebox{1ex}{.}.\raisebox{1ex}{.} \mbox{Stokes' theorem})$ (L.3)
    $\displaystyle -
n_j(\mbox{\boldmath$ x $ }) e_{lpj}
\mbox{v.p.}\int_{S} n_i(\mb...
...}{\partial x_p}
\frac{\partial \phi(\mbox{\boldmath$ y $ })}{\partial y_q} dS_y$  
  = $\displaystyle - n_j(\mbox{\boldmath$ x $ }) e_{jpl} \mbox{v.p.}\int_{S} n_i(\mb...
...\partial y_p}
\frac{\partial \phi(\mbox{\boldmath$ y $ })}{\partial y_q} dS_y .$  



Ken-ichi Yoshida
2001-07-28