Heat transfer equations
Heat transfer equation $$ \begin{equation} \frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2} \end{equation} $$ subject to the boundary conditions: at $x = 0$ $$ \begin{equation} T = \alpha\quad\text{ at }\quad x = 0 \end{equation} $$ and $$ \begin{equation} T \to \beta\quad\text{ as }\quad x\to \infty \end{equation} $$