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2.2.4 Parabolic PDE: heat equation

The heat equation

\frac{\partial y}{\partial t} = K \frac{\partial^2 y}{{\partial x}^2}
\end{displaymath} (78)
is a simple parabolic PDE. With the initial values
y \left( x, t_0 \right) = y_0 \left( x \right)
\end{displaymath} (79)
and boundary values
y \left( x_1, t \right) = y_1 \left( t \right)
\enspace , \enspace
y \left( x_2, t \right) = y_2 \left( t \right)
\end{displaymath} (80)
it models e.g. heat flux or radiative energy transport in the (optically thick) stellar interior. In the following examples the conductivity \bgroup\color{DEFcolor}$K$\egroup is a constant.