円盤領域における波動方程式


  $\displaystyle \frac{1}{c^2}u_{tt}$ $\displaystyle =$ $\displaystyle \Laplacian u$   $\displaystyle \mbox{(in $\Omega\times(0,\infty)$)}$$\displaystyle ,$
  $\displaystyle u(x,t)$ $\displaystyle =$ 0   $\displaystyle \mbox{(on $\rd\Omega\times(0,\infty)$)}$$\displaystyle ,$
  $\displaystyle u(x,0)$ $\displaystyle =$ $\displaystyle \phi(x)$   $\displaystyle \mbox{($x\in\overline\Omega$)}$$\displaystyle ,$
  $\displaystyle u_t(x,0)$ $\displaystyle =$ $\displaystyle \psi(x)$   $\displaystyle \mbox{($x\in\overline\Omega$)}$

の解は
  $\displaystyle u(r,\theta,t)$ $\displaystyle =$ $\displaystyle \frac{1}{2}\sum_{m=1}^\infty
A_{0m}J_0(\mu_{0m}r)\cos(\mu_{0m}t)
...
..._n(\mu_{nm}r)\cos(\mu_{mn}t)
\left(A_{nm}\cos n\theta+B_{nm}\sin n\theta\right)$
      $\displaystyle +
\frac{1}{2}\sum_{m=1}^\infty
C_{0m}J_0(\mu_{0m}r)\frac{\sin(\mu...
...{\sin(\mu_{nm}t)}{\mu_{nm}}
\left(C_{nm}\cos n\theta+D_{nm}\sin n\theta\right).$

桂田 祐史
2017-11-20