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：鉴于Fourier-Legendre的扩大为美元和G美元,以及美元和G美元的温和条件,我们以相应的Fourier-Legendre系数计算得出Fourier-Legendre的产品扩大为Fourier-Legendre系数,这样就可以获得全倍的美元功率的扩大。我们对趋同率设定了上限。然后,我们利用这些扩大来用多元非线性非线性非线性2级来解决一类非线性PDE。 获得的数字结果表明了基于Fourier-Legendre的解决方案方法在解决重要的非线性PDE中的效率和性能准确性。