Question

Uxx + Uyy =0 convert the situation equation into its Canonical form and find out its general solution

Answer 1

this is the canonical form of elliptic equation, 2-D Laplace equation:

$u_{xx} + u_{yy}=0$

we have $u_{xx} =- u_{yy}=a$

where a is constant

general solution:

$u(x, y) = \frac{a}{4}(x^2-y^2)$

solution for rectangle with sides L and H:

for boundary conditions:

$u(0,y)=g(y),u(L,y)=0,u(x,0)=0,u(x,H)=0$

$u(x,y)=\sum B_n sinh(\frac{n\pi (x-L)}{H})sin(\frac{n\pi y}{H})$

$B_n=\frac{2}{Hsinh(\frac{n\pi (-L)}{H})}\int^H_0 g(y)sin(\frac{n\pi y}{H})$

for boundary conditions:

$u(0,y)=0,u(L,y)=0,u(x,0)=0,u(x,H)=f(x)$

$u(x,y)=\sum B_n sinh(\frac{n\pi y}{L})sin(\frac{n\pi x}{L})$

$B_n=\frac{2}{Lsinh(\frac{n\pi H}{L})}\int^L_0 f(x)sin(\frac{n\pi x}{L})$