The heat diffusion equation is $$\frac{\partial T}{\partial t}=D\frac{\partial^2 T}{\partial x^2}$$
To obtain de energy current density we can start off from Fourier's law in 1D:
$$q = -k\frac{dT}{dx},$$
where $k$ is the thermal conductivity. Therefore,
$$\frac{\partial q}{\partial x} = -k \frac{\partial^2 T}{\partial x^2}$$
How would this derivation follow in order to obtain the energy current density $j$? Please, could you give me any hint?
Thanks.