Transforming Non-Homogeneous Boundary Conditions in 2D PDEs

[sigh1342]

Homework Statement

now I have a PDE
$$u_{xx}+u_{yy}=0,for 0<x,y<1$$
$$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$
Then I want to know whether there are some method to make the PDE become homogeneous boundary condition.
$$i.e. u|_{\partialΩ}=0$$

 

[LCKurtz]

Homework Statement

now I have a PDE
$$u_{xx}+u_{yy}=0,for 0<x,y<1$$
$$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$
Then I want to know whether there are some method to make the PDE become homogeneous boundary condition.
$$i.e. u|_{\partialΩ}=0$$

Could you solve the PDE if the boundary conditions had zero along three sides and a function along the fourth side? For example $u(x,0)=x,u(0,y)=0,u(x,1)=0,u(1,y)=0$? Say you do that and call the solution $u_1(x,y)$. Treat the other sides similarly. Then if you sum your solutions they will solve the homogeneous DE and the sum will satisfy all your BC's.

 

[sigh1342]

Actually the method I want to use need the boundary condition be all zero.
like the 1D case , $$u_{xx}=0 ,a<x<b, u(a)=\alpha, u(b)=\beta$$
the we can use $$u'=u-\beta+\frac{(x-b)(\beta-\alpha)}{a-b}$$
so $$u'_{xx}=0 , a<x<b , u'(a)=0, u'(b)=0$$
I would want to find whether there are similar method can work in 2D case.
Actually my prof. has told us that we can use some linear transform to do it.
But I have no idea.
Thank you :)