(a) Why does the heat equation imply infinite propagation speed? (b) This is clearly nonphysical. Nevertheless...

(a) Why does the heat equation imply infinite propagation speed?

(b) This is clearly nonphysical. Nevertheless the heat equation is widely used. Why is the infinite propagation speed not a problem?

Expert Solution

b): While an infinite speed of propagation is very non physical,
it can be rationalized by claiming that to a reasonable degree of approximation, the value of
u is zero outside a bounded set so for practical purposes, the speed of propagation is
finite.
The two properties of instantaneous smoothing of the data and infinite speed of
propagation are characteristic of solutions of the heat equation and could be referred to as
”diffusionlike evolution” . These properties are, in fact, characteristic of solutions of a
large class of partial differential equations called parabolic equations. It is generally the
case that properties that pertain to solutions of the heat equation are true for solutions to all
parabolic equations.

genius_generous answered 2 years ago

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