# Critic of heat equation

#### matematikawan

One critic of the Fourier heat equation

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

that I recently came across is that it gives rise to infinite speed of heat propagation.

I understand that the speed cannot be infinite because it contradict special relativity that no speed should be greater than the speed of light.

My question is how do they prove that the heat equation gives rise to infinite speed.

The paper also talk about Cattaneo's equation. Does anyone has any idea how to show that the propagation speed is infinite?

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#### starzero

I wondered about that as well.

Looking at the fundamental solution for the heat equation you will note that the value of this function is greater than zero for all x and for all t greater than zero.

This means that a rod of infinite length that has had one unit of heat added at a tiny spot for a brief instant of time has had that heat signal travel the entire length of the rod instantaneously.

This of course is paradoxical and few authors spend time explaining. One place where there is a brief explanation occurs in Widder’s book “The Heat Equation”. He explains that what the fundamental solution gives is a mathematical idealization.

Again looking at the graph of the fundamental solution you will not how quickly it decays towards zero.

Obviously in the real world, the heat cannot flow an infinite distance instantaneously.

#### matematikawan

Thanks starzero for that explaination. I will look into this further. Our university library has a copy of Widder's heat equation.

The paper: International Journal of Heat and Mass Transfer 51 (2008) 6024–6031
gives eight reasons why the Fourier’s law of heat conduction need to be modified.