Hi all
I'm in quite some trouble with this PDE problem, that is part of my assignment for wednesday.
First of all I got the PDE problem:
PDE: u_t - 2*u_xx = 0 , 0<x<1, t>0
BC: u(0,t) = t, u(1,t)=0
IC: u(x,0) = 0
I then have to show, that v(x,t)=u(x,t)-t(1-x) satisfies
PDE: v_t - 2*v_xx = -(1-x) , 0<x<1, t>0
BC: v(0,t) = 0, v(1,t)=0
IC: u(x,0) = 0
That is fairly easy.
Then I have to find a solution for X(x) to the ODE problem:
ODE: -2*X''(x) = -(1-x)
BC: X(0) = X(1) = 0
Again, this is simple, and my result is:
X(x) = x^2/4 - x^3/12 -x/6
But then i get in some trouble!
First I must derive a PDE problem for w(x,t)=v(x,t)-X(x), and then solve it. And after that, I have to find u(x,t)
To derive the PDE problem, it is just like I did before, and I end up with:
PDE: w_t - 2*w_xx = 0
BC: w(0,t) = 0, w(1,t) = 0
IC: w(x,0) + X(x) = 0
But how do I solve this? I know, that I should know, but I'm confused, and missed class when this was explaned.
I'm in desperate need of help.
Thanks!