Transformation/change of variables in differential equation
Maybe my background is just weak...I was thinking about this for almost 1.5 hours already, but I still end up totally confused. Perhaps this is because I was never able to understand the ideas of a function and change of variables completely...perhaps I have a serious conceptual flaw.
Consider the following partial differential equation with initial values(IV) and boundary conditions(BC):
u_t - k u_xx = x + 2t, 1<x<7, t>0
BC: u(1,t) = u(7,t) = 0
IV: u(x,0) = x+5
Our goal is to transform the above to the interval [0,6].
Let w = x-1.
Transform the whole problem to the interval w E [0,6]. (write in terms of w)
u_x = u_w dw/dx = (u_w) (1) = u_w
u_xx = ...(apply chain rule again) = u_ww
[On the left side, think of u as u(x,t). On the right side, think of u as u(w,t)]
x=1 <=> w=0
x=7 <=> w=6
So the boundary conditions get transformed to u(0,t)=u(6,t)=0 [here think of u as u(w,t)]
We know u(x,0) = x+5
=> u(w,0) = w+5
[I believe the logic in this step cannot be wrong, consider e.g. f(4z)=cos(4z), now how do we find f(4z-y)? Of course, f(4z-y)=cos(4z-y). How do we find f(z)? Surely, f(z)=cos(z). Right??]
So my final answer is: [here think of u as u(w,t)]
u_t - k u_ww = w+1+2t, 0<w<6, t>0
BC: u(0,t) = u(6,t) = 0
IV: u(w,0) = w+5
However, I really have some bad feeling that the result u(w,0) = w+5 is wrong, but I don't know where the mistake is.
I tried to calculate it in a different way and the answer is the same.
= u(x-1,0) = (x-1)+5
=> u(w,0) = w+5
Can someone please kindly explain why and where my mistake is? What is the correct answer?
Any help is greatly appreciated!
[also under discussion in sos math cyberboard]