The equation is a standard SHM equation, with solutions , where . Notice that to find you need to find the square root of . For that, remember that , which has a square root .
1. Show that the heat equation has complex-valued solutions of the form
F(x) exp(iwt) provided that kF′′ = iwF.
Done This
2. Find F if F(x) -> 0 as
x -> infinity and F(0) = A, where A is a real constant. Let T(x, t) be the
real part of F(x) exp(iwt). Verify that
T(x, t) = Aexp(−((w/2k)^1/2)x)cos(wt − ((w/2k)^1/2)x)
and that T is a solution of the heat equation for which T(x, t) --> 0 as
x --> inifinity and T(0, t) = Acoswt. Can you think of a physical situation
to which T corresponds?
I am not sure how to start this second bit, any hints would be much appreciated
many thanks