演習問題

1. 次の境界値問題

$$\begin{array}{ll} u_{xx} + u_{yy} = 0 , \ 0 < x < L, \ 0 < y ... ...< y < K\\ u(x,0) = 0, \ u(x,K) = f(x), \ 0 < x < L \end{array}$$

の解は次の無限級数で与えられることを示せ．

$$u(x,y) = \sum_{n=1}^{\infty}u_{n}(x,y)$$

$$= \sum_{n=1}^{\infty}C_{n}\sin{\frac{n\pi x}{L}}\sinh{\frac{n\pi y}{L}}$$

ただし, $C_{n}\sinh{\frac{n\pi K}{L}} = \frac{2}{L}\int_{0}^{L}f(x)\sin{\frac{n\pi x}{L}}dx$.
2. Laplace方程式を次の条件の元で解け．
(a)

$$\begin{array}{l} u(0,y) = 0 = u(L,y), \ 0 < y < K\\ u_{y}(x,0) = 0, \ u(x,K) = f(x), \ 0 < x < L \end{array}$$

(b)

$$\begin{array}{l} u(0,y) = g(y), \ u(L,y) = 0, \ 0 < y < K\\ u(x,0) = 0 = u(x,K), \ 0 < x < L \end{array}$$

3. Laplace方程式の境界条件が

$$\begin{array}{l} u_{x}(0,y) = 0 = u_{x}(L,y), \ 0 < y < K\\ u(x,0) = 0, \ u_{y}(x,K) = f(x) \ 0 < x < L \end{array}$$

のとき

$$u(x,y) = \sum_{n=0}^{\infty}u_{n}(x,y)$$

$$= D_{0}y + \sum_{n=1}^{\infty}D_{n}\cos{\frac{n\pi x}{L}}\sinh{\frac{n\pi y}{L}},$$

ただし,

$$D_{0} = \frac{1}{L}\int_{0}^{L}f(x)dx$$

$$D_{n}\frac{nx}{L}\cosh{\frac{n\pi K}{L}} = \frac{2}{L}\int_{0}^{L}f(x)\cos{\frac{n\pi x}{L}} dx, \ n \geq 1$$

で与えられることを示せ．