It is well known that the 1-dimensional [heat equation](http://en.wikipedia.org/wiki/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.



Undergraduate Derivation of Fundamental Solution to Heat Equation
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.


