We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing ...

We present and analyze strategies which can be used for the parallel computation of large numbers of integrals which may be of diierent levels of diiculty. Paralleliza-tion on the integral level, which is generally used for large numbers of integrals, is combined with parallelization on the subregion level, which enables handling local integration diiculties within individual problems. This res...

We study numerical integration of H older{type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As...

Calculations of observables in Quantum Chromodynamics are typically performed using a method that combines numerical integrations over the momenta of final state particles with analytical integrations over the momenta of virtual particles. I discuss a method for performing all of the integrations

‎In this article‎, we use a finite difference technique‎ ‎to solve variable-order fractional integro-differential equations‎ ‎(VOFIDEs‎, ‎for short)‎. ‎In these equations‎, ‎the variable-order fractional integration(VOFI) and‎ ‎variable-order fractional derivative (VOFD) are described in the‎ ‎Riemann-Liouville's and Caputo's sense,respectively‎. ‎Numerical experiments‎, ‎consisting of two exam...

In this paper, an effective numerical method is introduced for the treatment of nonlinear two-dimensional Volterra-Fredholm integro-differential equations. Here, we use the so-called two-dimensional block-pulse functions.First, the two-dimensional block-pulse operational matrix of integration and differentiation has been presented. Then, by using this matrices, the nonlinear two-dimensional Vol...

Geometric Numerical Integration is a subfield of the numerical treatment of differential equations. It deals with the design and analysis of algorithms that preserve the structure of the analytic flow. The present review discusses numerical integrators, which nearly preserve the energy of Hamiltonian systems over long times. Backward error analysis gives important insight in the situation, wher...

where x is a vector of displacements of structural coordinates, M is a positive definite mass matrix, C is a non-negative definite damping matrix, and K is a non-negative definite stiffness matrix. The nonlinear restoring forces are given in R(x, ẋ) and f ext(t) is a vector of external dynamic loads. At any point in time, t = ti+1 = (i+ 1)h, we may solve for the accelerations in terms of the di...