Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations
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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.
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