# Is heat equation is also called as thermal diffusion equation?

Is the following equation known as heat equation, also referred as thermal diffusion equation? Yes/No.

$${\displaystyle {\frac {\partial u}{\partial t}}-\alpha \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)=0}$$ Thanks

• If $\alpha$ scalar (or in other words $\alpha{\bf I}$), then that one is for isotropic diffusion. You can also have anisotropic diffusion where the rate is different in different orientations. Feb 22, 2018 at 10:59
• Yes ************
– user65203
Feb 22, 2018 at 11:01
• thanks folks***
– pkj
Feb 22, 2018 at 11:02
• It is called heat equation, Fourier's heat equation, thermal diffusion equation, Fick's second law (if $u$ is the concentration and $\alpha$ is the diffusion coefficient) and diffusion equation (if $u$ is the concentration and $\alpha$ is the diffusion coefficients). Feb 22, 2018 at 11:04
• To me the diffusion origins of this equation seem more fundamental (even though it originated as heat equation) because this equation can be very clearly derived from the random walk of many particles. Then we can use a similar model for the temperature distribution through random collisions. Which is why the equation descrives both the phenomena. Though I have never heard the terms "thermal diffusion" I guess it makes sense in the context Feb 22, 2018 at 11:14

A bit late but just so this question has an answer, the more general equation is the Convection-Diffusion equation (or drift-diffusion equation): $$\partial_t u = \nabla\cdot(D \,\nabla u) - \nabla\cdot(v u) + S$$ where $$D$$ is the diffusivity, $$u$$ is some variable (temperature, concentration, etc...), $$v$$ is some veolocity field (e.g. if the water is moving while the substance is diffusing), and $$S$$ is a source or sink term of $$u$$.
A special case is the anisotropic diffusion equation where $$v=0$$ and $$S=0$$: $$\partial_t u = \nabla\cdot[D \,\nabla u]$$ so the substance $$u$$ is diffusing "outward" from its starting distribution, but it movies differently in different directions. A special case of this is the heat equation (also called the isotropic diffusion equation), where $$D=\alpha I$$, so that $$\partial_t u = \alpha \Delta u = \alpha\left( \partial_{xx} u + \partial_{yy} u + \partial_{zz} u \right)$$ where (at least from what I've seen) this last equation is only called thermal in the the context of physics, and just heat equation or (isotropic) diffusion equation elsewhere.