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I've gotten to a point in this problem but I'm not certain how to continue...

$u_t=u_{xx}$, $00$

Boundary conditions $u(0,t)=2$ and $u(\pi,t)=-1$

Initial condition $u(x, 0)=1$ for $0

First, I translated the BCs and got

$u(0,t)=2=F(t)G(0)$

$u(\pi,t)=-1=F(t)G(\pi)$

I made the assumption that $u(x,t)=w(x,t)+v(x)$ which substituted into the PDE as $w_t=w_{xx}+v''$ and chose $v''=0$.

Thus, $w(x,t)$ satisfies the following homogenous PDE: $w_t=w_{xx}$, $w(0,t)=0$ and $w(\pi,t)=0$

$v(0)=2$ and $ v(\pi)=-1$ imply $w(0,t)=0$ and $w(\pi, t)=0$

Using the initial condition, $u(x,0)=w(x,0)+v(x)$, we know $v(x)=Ax+B$, which I solved for (using $v(0)=2$ and $v(\pi)=-1$) and $v(x)=-3x/\pi+2$, so:

$w(x, 0)=f(x)-(-3x/\pi +2)$

But now I'm stuck and have no idea which direction to go from here. Any advice is greatly appreciated.

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    Is there a typo in your initial condition? In any case, you are seeing a familiar difficulty that occurs when using separation of variables in inhomogeneous problems: often the initial condition does not satisfy the BC. In these cases you have to be a bit careful to ensure that your solution makes sense... – 2017-02-14
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    Typo noted. Does this mean that I'm doing it the wrong way? – 2017-02-14
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    Well no, but think for a moment: if your IC doesn't satisfy your BC, then your true solution has to be discontinuous in time at the boundary at the initial time: that is, $u(0,t)$ will jump from being $1$ to being $0$ instantly, and $u(\pi,t)$ will jump from being $1$ to being $0$ instantly. That's a bit of a problem from the classical perspective, do you see what I mean? This suggests that the Fourier series method *might* give the wrong answer; you need to pay careful attention to what it actually does give you. – 2017-02-14

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Start by finding a solution $u_0$ to the pde satisfying the boundary conditions (but not the initial condition). It can be steady-state, i.e. a function of $x$ alone.

Then $u(x,t) = u_0(x) + v(x,t)$ where $v(x,t)$ satisfies the pde with boundary conditions $v(0,t) = v(\pi,t) = 0$ and initial condition $v(x,0) = 1 - u_0(x)$. You find this with the usual separation of variables method and a Fourier series.