  Given the Fourier-Legendre expansions of $f$ and $g$, and mild conditions on
$f$ and $g$, we derive the Fourier-Legendre expansion of their product in terms
of their corresponding Fourier-Legendre coefficients. In this way, expansions
of whole number powers of $f$ may be obtained. We establish upper bounds on
rates of convergence. We then employ these expansions to solve
semi-analytically a class of nonlinear PDEs with a polynomial nonlinearity of
degree 2. The obtained numerical results illustrate the efficiency and
performance accuracy of this Fourier-Legendre based solution methodology for
solving an important class of nonlinear PDEs.
