円盤領域における熱方程式

単位円盤 $ \Omega$ における

  $\displaystyle u_t$ $\displaystyle =$ $\displaystyle \kappa\Laplacian u$   $\displaystyle \mbox{(in $\Omega\times(0,\infty)$)}$$\displaystyle ,$
  $\displaystyle u(x,t)$ $\displaystyle =$ 0   $\displaystyle \mbox{(on $\rd\Omega\times(0,\infty)$)}$$\displaystyle ,$
  $\displaystyle u(x,0)$ $\displaystyle =$ $\displaystyle f(x)$   $\displaystyle \mbox{($x\in\overline\Omega$)}$

の解は

$\displaystyle u(r,\theta,t)
=\frac{1}{2}\sum_{m=1}^\infty
A_{0m}J_0(\mu_{0m}r)
...
...ppa\mu_{nm}^2 t\right)
J_n(\mu_{nm}r)
(A_{nm}\cos n\theta+B_{nm}\sin n\theta),
$

$\displaystyle A_{nj}=\frac{2}{\pi J_{n+1}(\mu_{nj})^2}
\int_0^1\left(\int_0^{2\pi}f(r\cos\theta,r\sin\theta)
J_n(\mu_{nj}r)\cos n\theta \D\theta\right) \D r,
$

$\displaystyle B_{nj}=\frac{2}{\pi J_{n+1}(\mu_{nj})^2}
\int_0^1\left(\int_0^{2\pi}f(r\cos\theta,r\sin\theta)
J_n(\mu_{nj}r)\sin n\theta \D\theta\right) \D r.
$



Subsections
桂田 祐史
2017-11-20