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It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a t}} \ \exp\left(-\frac{x^2}{4at}\right)$$.

My question
I am looking for English or German references for an easy derivation of this particular solution which are comprehensible to undergraduates.

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4 Answers 4

up vote 8 down vote accepted

I think the term fundamental solution (at least sometimes) conventionally includes the integral around your $K$. I will assume this. If I recall correctly then the following argument is from "Partial Differential Equations" by Strauss.

A particularly simple solution follows from the self-similarity principle, i.e.

If $u(x,t)$ is a solution then so is $u(cx, a c^2t)$

This suggests looking for a particular solution of the form $K(x,t) = g(p)$, where $p = \frac{x}{\sqrt{4at}}$

Substituting $g$ into the heat equation leads to the differential equation

$$g''+\frac{p}{2}g' = 0 $$

Then the fundamental solution as above follows from solving this.

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Thank you. I don't think that you need the integral to call it a fundamental solution. Fundemantal in my opinion simply means that you can easily find further solution by convolution. –  vonjd Mar 5 '10 at 8:33
    
That's probably right, it's just that's how I have it in my notes and I think how it is defined in strauss. –  Q.Q.J. Mar 5 '10 at 8:41

I learned this fact from Folland's book Introduction to Partial Differential Equations; a proof is available here.

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I think you can find a solution of this from Elias Stein's book Fourier Analysis. Sorry I don't have the file available. But using fourier analysis to solve this may be like using a hammer to kill a fly.

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3  
Pencil, I hope you realize that Fourier invented his analysis in order to develop mathematical theory of heat, i.e. precisely in order to solve the heat equation! –  Victor Protsak Jul 28 '10 at 5:52
    
Hi, I believe all the examples in Stein's book are at least two dimensional, while this is clearly a one dimensional case. I don't remember Stein ever mentioned a one dimensional heat equation (which probably won't need fourier analysis). But thanks for reminding me this. Yes my memory could be wrong. –  Kerry Jul 28 '10 at 10:37

One approach is first to solve the heat equation with Heaviside initial data, taking into account the scaling property mentioned in Q.Q.J.'s answer, and then differentiate. This avoids an explicit definition of the Dirac delta, and is for example done in Logan's book.

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