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.

翻译：给定函数$f$和$g$的Fourier-Legendre展开及其满足的温和条件，我们基于对应的Fourier-Legendre系数推导出两者乘积的Fourier-Legendre展开。通过此方法，可进一步获得$f$整数幂次项的展开。我们建立了收敛速度的上界估计，并利用这类展开对一类具有二次多项式非线性的非线性偏微分方程进行半解析求解。数值结果展示了基于Fourier-Legendre展开的求解方法在解决此类重要非线性偏微分方程时的效率和计算精度。