**2. Heat conduction equation**

The one-dimensional heat equation

$$\frac{\partial U}{\partial \theta} = \beta \frac{\partial^2 U}{\partial \mathbf{z}^2} \tag{1}$$

with boundary conditions

$$U(\mathbf{0}, \theta) = \mathbf{0}, U(\mathbf{1}, \theta) = \mathbf{0},\tag{2}$$

and initial condition

$$U(z,0) = h(z), 0 \le z \le 1. \tag{3}$$
