0

I want to show that \begin{gather} u(x,t) = \int_0^\infty \frac{f(t+s)}{\sqrt{2\pi s^3}}\sum_{n=-\infty}^\infty \left[(4n+1-x)exp\left(-\frac{(4n+1-x)^2}{2s}\right)+(4n+1+x)exp\left(-\frac{(4n+1+x)^2}{2s}\right)\right]ds\\ \text{with}\quad u(1,t)=f(t). \end{gather} fulfills the heat equation $u_t+\frac{1}{2}u_{xx}=0$ for $t>0$ and $0<x<1$.

$f(t)$ is finite and continuous.

Milli Onaire
  • 33
  • I have looked up solutions to the heat equation on wikipedia. They look similar, but I don't know where to go from there. Especially since the boundary condition is not at $x=0$, but $x=1$ and I don't know anything about the value at $t=0$. – Milli Onaire Jun 22 '17 at 18:51
  • I suggest to start calculating the derivatives w.r.t. $t$ and $x$ – supinf Jun 23 '17 at 00:09
  • x is no problem, but I don't have an explicit representation of f. – Milli Onaire Jun 23 '17 at 14:04

0 Answers0