# Diffusion equation in terms of heat equation

I am solving a problem which goes like this:

Solve the diffusion equation:$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$ for $$t>0$$ and $$x \in > \mathbb{R}$$, with initial condition $$u(x,0)=e^{-x}$$. Interpret the result in terms of heat conduction.

I have solved the equation and the solution is simply $$u(x,t)=e^{t-x}$$, but I haven't managed to interpret it in terms of heat conduction. I understand that it corresponds to the heat equation in an infinite bar with conduction coefficient $$\kappa=1$$, but, how it is possible that the solution be, not only divergent but also that it increases with time?

## 1 Answer

It is a result of the boundary condition. If for $$t = 0$$, the function has a positive second derivative, (it is concave upwards as $$e^{-x}$$) it will increases with time.

It could be a long rod, while one of the ends is being heated (since t < 0) at a rate greater than the flow of heat by conduction through it.