Undergraduate Derivation of Fundamental Solution to Heat Equation - MathOverflow

Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-03-05T07:10:27Z

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.

Answer by Q.Q.J. for Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-03-05T08:06:36Z

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.

Answer by Akhil Mathew for Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-03-05T15:21:34Z

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

Answer by Changwei Zhou for Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-07-28T05:17:50Z

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.

Answer by timur for Undergraduate Derivation of Fundamental Solution to Heat Equation

2011-07-09T17:16:52Z

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.