Adrien-Marie Legendre (1752–1833) made great contributions to analysis, number theory, celestial mechanics, and practical science. His name is attached to the Legendre differential equation, Legendre polynomials, the Legendre transformation, the Legendre symbol in number theory, the Legendre conditions in calculus of variations, the Legendre relation for elliptic integrals, the Legendre duplica...

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...

Corresponding Author: Simon Liao The University of Winnipeg, Winnipeg, Manitoba, Canada, R3B 2E9, Canada Email: [email protected] Abstract: In this research, a numerical integration method is proposed to improve the computational accuracy of Legendre moments. To clarify the improved computation scheme, image reconstructions from higher order of Legendre moments, up to 240, are conducted. Wi...

Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...

This paper considers the numerical simulation of incompressible viscous fluid flow in an infinite strip. A mixed spectral method is proposed using the Legendre approximation in one direction and the Legendre rational approximation in another direction. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Legendre-Legendre rational approximation are establishe...

In this paper we present a construction of interpolatory Hermite multiwavelets for functions that take values in nonlinear geometries such as Riemannian manifolds or Lie groups. We rely on the strong connection between wavelets and subdivision schemes to define prediction-correction approach based operate manifold-valued data. The main result concerns decay wavelet coefficients: show our essent...

We consider an approximation scheme using Haar wavelets for solving a class of infinite horizon optimal control problems (OCP's) of nonlinear interconnected large-scale dynamic systems. A computational method based on Haar wavelets in the time-domain is proposed for solving the optimal control problem. Haar wavelets integral operational matrix and direct collocation method are utilized to find ...