Question about separation of variables

Question

This is for the heat equation, where

$$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} (3,t)=0.$$

I am trying to solve for $U(x,t)$ but am currently stuck with factoring dealing with the "$+1$" in the separation of variables.

I started with $U(x,t)=F(x) G(t)$ then put it into the heat equation and set it equal to a constant -$\lambda^2$. To deal with the $+1$, I moved it to the other side with the lambda but now I am can't seem to get the sine or exponential expression I need.

Can you solve the equation for the new function $v = u - t$? — Kaster, Apr 21 '14 at 23:00
Have you been asked to use the separation of variables techniques? — Mhenni Benghorbal, Apr 21 '14 at 23:20
@mvgwpemb what do you mean? What I've suggested is pretty much the same as Alex posted. — Kaster, Apr 21 '14 at 23:50

score 3 · Answer 1 · answered Apr 21 '14 at 23:19

3

Let $V=U-t$. Then

$$V_t-kV_{xx}=0.$$

Now do separation of variables (work out the appropriate boundary conditions for $V$ as well).

answered Apr 21 '14 at 23:19

Alex R.

33,289

Actually, your hint leads nowhere since for the new variable you get time dependent boundary conditions. – Artem Apr 23 '14 at 19:06
@Artem: why is that bad? – Alex R. Apr 23 '14 at 20:36
Because separation of variables works if and only if the boundary conditions are homogeneous (zero) – Artem Apr 23 '14 at 21:12
@Artem: one can generalize separation of variables for time dependent boundaries. See for example this: http://www.sciencedirect.com/science/article/pii/0017931062900637 (Sorry for the paywall) – Alex R. Apr 24 '14 at 03:11
This is a very nice confirmation of my original statement: Your hint leads nowhere. Thanks. – Artem Apr 24 '14 at 05:24

score 0 · Answer 2 · edited Apr 13 '17 at 12:20

0

First assume the solution has the form

$$ u ( x,t ) = F_1 ( x ) F_2( t)+c_1+c_2x-\frac{1}{2k}\,x^2, $$

then advance with separation of variables technique.

edited Apr 13 '17 at 12:20

Community

1

answered Apr 23 '14 at 19:00

Mhenni Benghorbal

48,099

Question about separation of variables

2 Answers2