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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

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  • $\begingroup$ 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. $\endgroup$ Feb 22, 2018 at 10:59
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    $\begingroup$ Yes ************ $\endgroup$
    – user65203
    Feb 22, 2018 at 11:01
  • $\begingroup$ thanks folks*** $\endgroup$
    – pkj
    Feb 22, 2018 at 11:02
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    $\begingroup$ 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). $\endgroup$
    – MrYouMath
    Feb 22, 2018 at 11:04
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    $\begingroup$ 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 $\endgroup$
    – Yuriy S
    Feb 22, 2018 at 11:14

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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.

These equations describe both heat or thermal diffusion and diffusion of some concentration, because both processes can be thought of as being dispersed by the random movements of many particles. All of these can therefore be connected to the theory of stochastic processes because the Fokker-Planck (forward Kolomogorov) equation of a stochastic Ito diffusion process is a convection-diffusion PDE.

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