The diffusion problem is given by the following partial differential equation (PDE) and boundary conditions:
$$
\begin{cases}
\frac{\partial u}{\partial t} - \nabla^2 u = f(x, t) & \text{in } \Omega \times (0, T) \\
u(x, t) = g(x, t) & \text{on } \partial \Omega
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