Runge-Kutta collocation methods for rigid body Lie-Poisson equations

The rigid body Lie-Poisson structure in three dimensions is considered. We show that the symplectic collocation type Runge-Kutta methods preserve the one-form of the underlying system. The linear error growth, energy and momentum conservation properties of the numerical solutions are discussed for Euler top equation

FORMULATION OF SOME GAUSSIAN INTEGRALS OVER R(N) VIA GENERATING-FUNCTIONS

This paper deals with the analytic formulation of the integrals over R(n) with the weight function exp(x(T)Cx) where the integrand is the product of quadratic forms x(T) A(j)x, j = 1,...,p, for arbitrary n x n symmetric matrices A(j). The technique is based on generating functions. First some functions are defined to generate these integrals for the special case A(j) = A(j) = A...A, and then practical formulas for the general case are derived

Order of convergence of evolution operator method

In this paper the order of convergence of the evolution operator method used to solve a nonlinear autonomous system in ODE's [2] is investigated. The order is found, to be 2N+1 where N comes from the [N+1,N] Pade' approximation used in the method. The order is independent of the choice of the weight function

2d polynomial interpolation: A symbolic approach with mathematica

This paper extends a previous work done by the same authors on teaching 1d polynomial interpolation using Mathematica [1] to higher dimensions. In this work, it is intended to simplify the the theoretical discussions in presenting multidimensional interpolation in a classroom environment by employing Mathematica's symbolic properties. In addition to symbolic derivations, some numerical tests are provided to show the interesting properties of the higher dimensional interpolation problem. Runge's phenomenon was displayed for 2d polynomial interpolation

Romberg integration: A symbolic approach with mathematica

Higher order approximations of an integral can be obtained from lower order ones in a systematic way. For 1-D integrals Romberg Integration is an example which is based upon the composite trapezoidal rule and the well-known Euler-Maclaurin expansion of the error. In this work, Mathematica is utilized to illustrate the method and the underlying theory in a symbolic fashion. This approach seems plausible for discussing integration in a numerical computing laboratory environment

Periodic solutions of the hybrid system with small parameter

In this paper we investigate the existence and stability of the periodic solutions of a quasilinear differential equation with piecewise constant argument. The continuous and differentiable dependence of the solutions on the parameter and the initial value is considered. A new Gronwall-Bellman type lemma is proved. Appropriate examples are constructed

Symbolic polynomial interpolation using Mathematica

This paper discusses teaching polynomial interpolation with the help of Mathematica. The symbolic power of Mathematica is utilized to prove a theorem for the error term in Lagrange interpolating formula. Derivation of the Lagrange formula is provided symbolically and numerically. Runge phenomenon is also illustrated. A simple and efficient symbolic derivation of cubic splines is also provided

Approximation of semilinear Cauchy problem for the second order equations in Banach spaces

this paper to consider the property of compactness and under the compactness condition consider the existence and approximation of the mild solution of (3). The following theorem ideologically is taken from [7] and reformulate the existence result for the second order case