# Potential equation problem

1. Apr 12, 2008

### buzzmath

1. The problem statement, all variables and given/known data
solve the potential equation on the rectangle 0<x<a, 0<y<b and boundary conditions u(x,b)=100 u(0,y)=0 u(a,y)=100 du/dy(x,0)=0

2. Relevant equations
potential equation is d^2u/dx^2+d^2u/dy^2=0 that is the second partial of u with respect to x plus the second partial of u with respect to y equals zero

3. The attempt at a solution
I first defined the polynomial v(x,y)=100x/a this has the conditions v(x,b)=100x/b v(0,y)=0 v(a,y)=100 dv/dy(x,0)=0 so I can now write u(x,y)=v(x,y) + w(x,y) where w(x,y) satisfies the conditions of satisfying the potential equation and has boundary conditions w(x,b)=100(1-x/a) w(0,y)=0 w(a,y)=0 dw/dy(x,0)=0
now all that is needed is to solve w(w,y) I will use seperation of variables and write w(x,y)=X(x)Y(y) then by taking the second partials and adding them together I get X''/X=-Y''/Y since w(x,y) satisfies the potential equation since X and Y have variable independent of one another we can write X''/X=Y''/Y=-L^2 a constant
so now I have the differential equations X''+XL^2=0 and Y''-YL^2=0
from the boundary conditions we have X(0)=X(a)=0 and the general solution for X is X=Acos(Lx)+Bsin(Lx) but X(0)=0 so A=0 and we have X=sin(Lx) and since X(a)=0 we define Ln as n(pi)/a and Xn=sin(Lnx) and the general solution of Y=ancosh(Lny)+bnsinh(Lny) and we have Y'(0)=0 so bnLn=0 if Ln=0 then X=0 so we can disregard this and assume bn=0 now I use the principle of superposition and write w(x,Y)=(sum of n=1 to infinity)ancosh(Lny)sin(Lnx) now we use the final boundary condition to find an's w(x,b)=(sum of n=1 to infinity)ancosh(Lnb)sin(Lnx)=100-100x/a
I multiply each side by sin(mx) and integrate from -a to a since this is the period but if n doesn't equal m then the left hand side is zero by orthonogality and a if m=n so we have ancosh(Lny)*a=integral from -a to a of (100-100x/a)sin(nx)dx from this we see the an values and the answer is w(x,Y)=(sum from n=1 to infinity)ancosh(Lny)sin(Lnx) and u(x,y)=v(x)+w(x,y)
I'm not sure if this is right or not. is how I found Y look right and did I find the right values of an's and bn's? Thanks for the help and sorry if it looks kind of messy