I have the PDE for the homogeneous heat equation (partial derivative of u with respect to time equals the second partial derivative of u with respect to x) with the following Dirichlet boundary and initial conditions:
$$ \\ u(0,t) = t\\ u(1,t) = 0 \\ u(x,0) = x(1-x)$$
I have non-homogeneous boundary conditions, so we propose $ u(x,t) = w(x,t) + U(x)$, with $U(x)$ the stationary solution (as time approaches infinity, the solution $u(x,t)$ approaches $U(x)$, so $w(x,t)$ is the transient part of my solution.
Separating the problem in two others, for $U(x)$ and $w(x,t)$, I get quite confused as in what should be the boundary condition for $U(0)$, since $u(0,t) = t$ is time-dependent.
If I solve for $U(x)$ with conditions $U(0) = t$ and $U(1) = 0$, I get $U(x) = Ax+B$, so we conclude that $U(x) = -tx + t$; and then, solving for $w(x,t)$, we have from $u(x,t) = w(x,t) + U(x)$: $$w(x,0) = x(1-x) - (-tx+t) = (x-t)(1-x) $$
The problem is that when I find the transient solution (for $w(x,t)$), to determine the coefficients for its series representation, I need to apply our initial condition $w(x,0) = (x-t)(1-x)$ to find the Fourier series and find the complete solution for the problem.
Note: I am not sure if that initial condition makes sense, since it is time dependent, even though $t=0$. Apart from that, I have no idea how to construct the periodic extension to find a Fourier series that matches my condition, since it's multivariable.