Undergraduate Derivation of Fundamental Solution to Heat Equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:57:08Z http://mathoverflow.net/feeds/question/17166 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17166/undergraduate-derivation-of-fundamental-solution-to-heat-equation Undergraduate Derivation of Fundamental Solution to Heat Equation vonjd 2010-03-05T07:10:27Z 2011-07-09T17:16:52Z <p>It is well known that the 1-dimensional <a href="http://en.wikipedia.org/wiki/Heat_equation" rel="nofollow">heat equation</a> $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a t}} \ \exp\left(-\frac{x^2}{4at}\right)$$.</p> <p><strong>My question</strong><br> I am looking for English or German references for an easy derivation of this particular solution which are comprehensible to undergraduates.</p> http://mathoverflow.net/questions/17166/undergraduate-derivation-of-fundamental-solution-to-heat-equation/17167#17167 Answer by Q.Q.J. for Undergraduate Derivation of Fundamental Solution to Heat Equation Q.Q.J. 2010-03-05T08:06:36Z 2010-03-05T08:06:36Z <p>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.</p> <p>A particularly simple solution follows from the self-similarity principle, i.e. </p> <p>If $u(x,t)$ is a solution then so is $u(cx, a c^2t)$</p> <p>This suggests looking for a particular solution of the form $K(x,t) = g(p)$, where $p = \frac{x}{\sqrt{4at}}$</p> <p>Substituting $g$ into the heat equation leads to the differential equation</p> <p>$$g''+\frac{p}{2}g' = 0$$</p> <p>Then the fundamental solution as above follows from solving this.</p> http://mathoverflow.net/questions/17166/undergraduate-derivation-of-fundamental-solution-to-heat-equation/17186#17186 Answer by Akhil Mathew for Undergraduate Derivation of Fundamental Solution to Heat Equation Akhil Mathew 2010-03-05T15:21:34Z 2010-03-05T15:21:34Z <p>I learned this fact from Folland's book <em>Introduction to Partial Differential Equations;</em> a proof is available <a href="http://amathew.wordpress.com/2010/01/16/the-fourier-transform-the-heat-equation-and-fundamental-solutions/" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/17166/undergraduate-derivation-of-fundamental-solution-to-heat-equation/33618#33618 Answer by Changwei Zhou for Undergraduate Derivation of Fundamental Solution to Heat Equation Changwei Zhou 2010-07-28T05:17:50Z 2010-07-28T05:17:50Z <p>I think you can find a solution of this from <em>Elias Stein</em>'s book <em>Fourier Analysis</em>. 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. </p> http://mathoverflow.net/questions/17166/undergraduate-derivation-of-fundamental-solution-to-heat-equation/69890#69890 Answer by timur for Undergraduate Derivation of Fundamental Solution to Heat Equation timur 2011-07-09T17:16:52Z 2011-07-09T17:16:52Z <p>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 <a href="http://books.google.com/books?id=zHvMVnzMYaMC" rel="nofollow">Logan's book</a>.</p>