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I have the 1-D heat equation

$\frac{\partial w}{\partial t} -\frac{\partial^{2} w}{\partial x^{2}} =0$
$-\pi < x < \pi$

With Initial and Boundary Conditions:
$w(x,0)=0$
$w(\pi,t)-w(-\pi,t) = 2\pi$
$w_{x}(\pi,t) - w_{x}(\pi,t)=0$

I've been using separation of variables where $w(x,t)=X(x)T(t)$ and getting solutions for all 3 scenarios where $\lambda >0 , \lambda <0 ,\lambda =0$

for $\frac{T'}{T} =\frac{X''}{X}=-\lambda$

I'm not sure if I'm missing a boundary condition I can derive or what to do with the 3 solutions I've obtained so far?

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Define a new equation in terms of $v = w(x,t)-x$. This function satisfies $$ v_t - v_{xx}=0 \\ v(x,0)=-x \\ v(\pi,t)-v(-\pi,t) = u(\pi,t)-u(-\pi,t)-\pi+(-\pi)=0 \\ v_x(\pi,t)-v_x(-\pi,t)= 0. $$ This is an equation with homogeneous boundary conditions; so separation of variables succeeds. The substitution moves the inhomogeneity to an initial condition, which is just fine.

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