2.2.4 Parabolic PDE: heat equation

The heat equation

$$\frac{\partial y}{\partial t} = K \frac{\partial^2 y}{{\partial x}^2}$$ (78)

is a simple parabolic PDE. With the initial values

$$y \left( x, t_0 \right) = y_0 \left( x \right)$$ (79)

and boundary values

$$y \left( x_1, t \right) = y_1 \left( t \right) \enspace , \enspace y \left( x_2, t \right) = y_2 \left( t \right)$$ (80)

it models e.g. heat flux or radiative energy transport in the (optically thick) stellar interior. In the following examples the conductivity $K$ is a constant.