We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our construction of the discrete Lagrangian adopts Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula, and involves applying collocation...

s for MAIA 2007 The limits of bivariate Lagrange projectors Carl de Boor∗ and Boris Shekhtman Eastsound, Washington In a talk in Norway in 2003, the first author conjectured that any finite-rank ideal projector (i.e., finite-rank linear projector on the space of polynomials in d variables whose kernel is a polynomial ideal, i.e., a linear space also closed under pointwise multiplication by poly...

We study interpolation polynomials based on the points in [−1, 1]× [−1, 1] that are common zeros of quasi-orthogonal Chebyshev polynomials and nodes of near minimal degree cubature formula. With the help of the cubature formula we establish the mean convergence of the interpolation polynomials. 1991 Mathematics Subject Classification: Primary 41A05, 33C50.

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to constraint systems, the procedure of finding a Hamiltonian for a singular Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairaut transf...

In ‎the current study, a‎ general formulation of the discrete Chebyshev polynomials is given. ‎The operational matrix fractional integration for these also derived. ‎Then,‎ a numerical scheme based on and their has been developed to solve variational problems‎. this method, need using Lagrange multiplier during solution procedure eliminated.‎ The performance proposed validated through some illu...

The present paper considers a q-analogue of an operator defined by Erku?-Duman et al. (Calcolo, 45(1):53–67, 2008) involving q-Lagrange polynomials in several variables. Korovkin type theorems the settings deferred weighted A-statistical convergence and power series method are investigated.

The spline element method with constraints is a discretization method where the unknowns are expanded as polynomials on each element and Lagrange multipliers are used to enforce the interelement conditions, the boundary conditions and the constraints in numerical solution of partial differential equations. Spaces of piecewise polynomials with global smoothness conditions are known as multivaria...