  A forward diffusion equation describing the evolution of the allele frequency
spectrum is presented. The influx of mutations is accounted for by imposing a
suitable boundary condition. For a Wright-Fisher diffusion with or without
selection and varying population size, the boundary condition is $\lim_{x
\downarrow 0} x f(x,t)=\theta \rho(t)$, where $f(\cdot,t)$ is the frequency
spectrum of derived alleles at independent loci at time $t$ and $\rho(t)$ is
the relative population size at time $t$. When population size and selection
intensity are independent of time, the forward equation is equivalent to the
backwards diffusion usually used to derive the frequency spectrum, but the
forward equation allows computation of the time dependence of the spectrum both
before an equilibrium is attained and when population size and selection
intensity vary with time. From the diffusion equation, we derive a set of
ordinary differential equations for the moments of $f(\cdot,t)$ and express the
expected spectrum of a finite sample in terms of those moments. We illustrate
the use of the forward equation by considering neutral and selected alleles in
a highly simplified model of human history. For example, we show that
approximately 30% of the expected heterozygosity of neutral loci is
attributable to mutations that arose since the onset of population growth in
roughly the last $150,000$ years.
