円盤領域における熱方程式

単位円盤 $\Omega$ における

$$u_t = \kappa\Laplacian u \mbox{(in $\Omega\times(0,\infty)$)} ,$$

$$u(x,t) = \mbox{(on $\rd\Omega\times(0,\infty)$)} ,$$

$$u(x,0) = f(x) \mbox{($x\in\overline\Omega$)}$$

の解は

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

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

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