Math 6410 • ODE

Obtain the following symmetry properties for the Cauchy problem for the 1D heat
equation
\[
  u_t=ku_{xx}+f(x,t)\quad(-\infty<x<\infty,\ t>0),\qquad
  u(x,0)=\phi(x),\qquad u_t(x,0)=\psi(x).
\]
\begin{enumerate}
\item[(a)] If the data is even in \(x\) [\(\phi(x)=\phi(-x)\),
\(\psi(x)=\psi(-x)\), \(f(x,t)=f(-x,t)\)], then the solution is also even in \(x\)
[\(u(x,t)=u(-x,t)\)].
\item[(b)] If the data is odd in \(x\) [\(\phi(x)=-\phi(-x)\),
\(\psi(x)=-\psi(-x)\), \(f(x,t)=-f(-x,t)\)], then the solution is also odd in \(x\)
[\(u(x,t)=-u(-x,t)\)].
\item[(c)] If the data is periodic in \(x\) [\(\phi(x+A)=\phi(x)\),
\(\psi(x+A)=\psi(x)\), \(f(x+A,t)=f(x,t)\)], then the solution is also periodic in
\(x\) with the same period \(A\) [\(u(x+A,t)=u(x,t)\)].
\end{enumerate}