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I know that the diffusion equation is a more general version of the heat equation. But what is the exact difference informally?

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  • $\begingroup$ We have the laplace operator in the diffusion equation, and the second derivative in $x$ in the heat equation? Is that the difference? So, the heat equation brings it down to a single dimension? $\endgroup$
    – onimoni
    Feb 19, 2015 at 15:42

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The difference is typically the diffusion coefficient: \begin{align} \frac{\partial \psi}{\partial t}&=\nabla\cdot\left(\kappa\nabla\psi\right)\tag{diffusion}\\ \frac{\partial \psi}{\partial t}&=\kappa\nabla^2\psi\tag{heat} \end{align} Under the diffusion equation, we typically take $\kappa$ to be a spatially-dependent variable whereas in the heat equation it is a uniform constant (allowing us to use the Laplacian on $\psi$).

However, the heat equation can have a spatially-dependent diffusion coefficient (consider the transfer of heat between two bars of different material adjacent to each other), in which case you need to solve the general diffusion equation.

There is no relation between the two equations and dimensionality. Both equations can be solved in one dimension, with a straight-forward substitution $\nabla\to\frac{\partial}{\partial x}$, or left in the multiple dimensions as I give above (which would likely require a numerical solver).

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  • $\begingroup$ I had to look it up because in all of my books I've ever used, there is no distinction made. Not with regards to linearity nor whether the coefficient is constant or variable in space and/or time. I'm... perplexed that there's actually a difference made by others. $\endgroup$
    – tpg2114
    Feb 19, 2015 at 15:57
  • $\begingroup$ LeVeque makes the distinction. Though he does say that the diffusion equation is the "more general" form of the heat equation. $\endgroup$
    – Kyle Kanos
    Feb 19, 2015 at 16:01
  • $\begingroup$ Fair enough. I have that book somewhere but can't find it so that happens to be the one I didn't look in. Go figure! $\endgroup$
    – tpg2114
    Feb 19, 2015 at 16:03
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There is no difference physical or mathematical .Heat equation is ONE application of the diffusion equation whether one,two or three dimensional and whether the diffusion coefficient is spatially uniform or not.No difference also between both in considering or accommodating the source/sink term.

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  • $\begingroup$ Basically every conservation equation takes the form of an advection-diffusion equation. $\endgroup$
    – 2b-t
    Nov 4, 2019 at 5:52
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Both diffusion and heat conduction lead to the same mathematical problem. The only difference are the physical dimensions of the quantities in the equation and their interpretation (temperature vs. concentration/probability).

The analogy can be seen beautifully in irreversible thermodynamics where you obtain the equations by combining a conservation equation for either energy or concentration with a linear relation between thermodynamic fluxes and forces, Fourier's law or Fick's law, respectively.

Mathematical terminology may vary between authors. For example, Evans (PDE) calls one and the same equation both "heat equation" and "diffusion equation".

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