# Undergraduate Derivation of Fundamental Solution to Heat Equation

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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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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