Illustration of the Diffusion Equation

We illustrate the behaviour of the diffusion equation

$$\frac{\partial T}{\partial t} = K\nabla^{2}T,$$

where $T(x,y,t)$ is, for example, a temperature distribution. $K$ in this case is the heat diffusivity. We consider the one-dimensional case, so

$$\nabla^2 = \frac{\partial^{2}}{\partial x^{2}},$$

and $T = T(x,t)$. We consider a Gaussian tenperature distribution as initial condition, which is given by the equation

$$T(x,0) = \exp\left(-\frac{x^{2}\sigma^{2}}{2}\right)$$

This initial value problem can be solved by using the Fourier transform. Details of the solution are omitted and left as an exercise. The temperature distribution at an instant $t$ is given by the equation

$$T(x,t) = \frac{1}{\sqrt{2Kt\sigma^{2}-1}}\exp\left(-\frac{x^{2}\sigma^{2}}{2(2Kt\sigma^{2}-1)}\right).$$

We illustrate this solution by using a java applet. Just click on the equation above. The parameter $\sigma$ controls the shape of the initial temperature distribution. The parameter $K$ controls how fast the temperature difuses. They can assume very small values, so we use the log scale in this parameter slider bar.