$u_t - k u_{xx} = f(x,t) ; u(0,t) = 0 ; u(x,0) = \phi(x) \;\;\; x \in (0,\infty)$
I am trying to solve the diffusion equation with Dirichlet condition $u(0,t) = 0$ on the half-line, by the method of extending the domain to $x \in ( -\infty,\infty)$. I know that I can odd-extend the initial data function $\phi$, but I don't know how to extend the function$ f(x,t)$. It seems like $f$ also has to be extended by the odd extension, but I can't figure out why (why not by even extension?).
How should I extend the function $f$?
Thanks!