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14. Solve the P.D.E.

14. Solve the P.D.E.

$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 ; u(x, 0)=u(0, y)=0, u(x, 1)=\pi, u(1, y)=2 \pi $$

$$ \text { 令 } u(x, y)=\phi(x, y)+\varphi(x, y) $$

$$ u(x, y)=\sum_{n=1}^{\infty} \frac{2}{n \sinh n \pi}\left[1-(-1)^{n}\right](\sinh n \pi y \sin n \pi x+2 \sinh n \pi x \sin n \pi y) $$

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