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