[PDE] Transforming Nonhomogeneous BCs into Homogeneous Ones

[Arkuski]

So there's this problem in my text that's pretty challenging. I can't seem to work out the answer that is given in the back of the book, and then I found a solution manual online that contains yet another solution.

The problem is a the heat equation as follows:

PDE: $u_{t} = α^2u_{xx}$

BCs: $u(0,t) = 1$
$u_x(1,t)+hu(1,t) = 1$

IC: $u(x,0) = \displaystyle\sin (πx)+x$

Also, $0<x<1$.

We assume that $u(x,t) = S(x,t)+U(x,t)$ and that $S(x,t) = A(t)[1-x]+B(t)[x]$. By substituting these into the BCs, I get $A(t) = 1$ and $B(t) = \frac{2}{1+h}$. Now we have $S(x,t) = 1-x+\frac{2x}{1+h} = 1+x\frac{1-h}{1+h}$. With the steady state solution in place, we construct our homogenous problem as follows:

PDE: $U_{t} = α^2U_{xx}$

BCs: $U(0,t) = 0$
$U_x(1,t)+hU(1,t) = 0$

IC: $U(x,0) = \displaystyle\sin (πx)+x\frac{2h}{1+h}-1$

If I try to solve this one, it turns into an eigenvalue problem which isn't covered until the next section and the IC is a nightmare. Anyways, the book gives me the following answer:

$x+e^{-(πα)^2t}\displaystyle\sin (πx)$

Moreover, the solution manual I found on the top of pg 16 looks as if it's solving an entirely different problem. Any help on this problem would be greatly appreciated.

 

[LCKurtz]

It looks to me like all your work is correct as far as you went. The book's answer$$
x+e^{-(πα)^2t}\displaystyle\sin (πx)$$doesn't solve the original problem's boundary conditions, so it isn't the answer to the problem. After all, you would expect the answer to depend on $h$.