Dirichlet problem for a rectangle

**vams42** · October 21st 2010, 10:28 PM

Hello,

am new to the dirichlet topic, and am unable to understand how to solve it, neither the approach to solve the problem. could you tell me how to solve this problem, if so it wil be helpful in solving other problems also

the problem is

$\nabla^{2}= 0 0<x<1 0<y<1 u(x,0) = x(x-1) u(x,1) = 0 u(0,y) = 0 u(1,y) = 0$

REPLY ASAP

**HallsofIvy** · October 22nd 2010, 07:28 AM

I presume you mean $\nabla^2 u= 0$ with boundary conditions $u(x,0)= x(x-1)$ , $u(x, 1)= 0$ , $u(0,y)= 0$ , and $u(1, y)= 0$ .

There are a variety of different methods you could try and since you have shown no attempt to solve it, we don't know which would be appropriate for you.

For example, since the boundary conditions in y are 0 at both x= 0 and x= 1, you could try a sine series solution:
$u(x,y)= \sum_{n=1}^\infty A_n(x)sin(\pi y)$

**vams42** · October 23rd 2010, 04:42 PM

thnx....i could solve such problems by directly substituting the values in the formula, but for other cases like

$\nabla^{2}=0 and boundary conditions are 0<x<1, 0<y<1 and given u(x,0)=0 , u(x,1)=sin Πx , u(0,y)=0 , u(1,y)=0$

for such kind of problems, i was told i need to derive the formula, which i am unable to derive, so could you help me in explaining how to derive it, n use it to solve the problem

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