Heat equation
The heat equation is a partial differential equation, that is, it involves both derivatives of the time and of spatial variables.
$$\frac {\partial u} {\partial t} - \alpha \frac {\partial^2 u}{\partial x^2} = 0 $$
Where the parameter \(\alpha\) determines how much conductive the material is.
This is the case for one variable, and thus one dimension. You can imagine that as a long but very thin wire. Of course in reality there is no such thing as 1 Dimension material, but from physicists and mostly engineers we know that this can be approximated when the long side is much bigger than the radius of the wire.
In our simulation we set the right boundary’s temperature to be changing sinusoidally its temperature, while the left boundary’s temperature is defined by the user.
Numerical Method
The typical finite difference method was used in order to perform the simulation. The parameter \(\alpha\) was selected appropriately for the illustration purposes.
Simulation
Click here to experiment yourself