Let us put the system on a mathematical grid !
- We place the discretised rod in a coordinate system.
- The position of a mass point is denoted by \(x\) and its temperature by \(T\).
- Temperature of any 3 consecutive points at time \(t\) are denoted by \(T (x + dx, t)\), \(T(x,t)\) and \(T (x - dx, t)\)
- The variation of temperature \(T\) of a mass point at \(x\) with time \(t\) is proportional to the difference in temperature with its neighbouring points,
$$ (\frac{\partial T}{\partial t})_{t=t_{i}} \propto [T(x_{i} + dx, t_{i}) - T(x_{i}, t_{i})] + [T(x_{i} - dx, t_{i}) - T(x_{i}, t_{i})] $$
Drag slider to see variation in Temperature, \(T\), with time, \(t\)!