The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k + t in n-space such that the number of evaluation points is O ((2n)k/k t), n ~ . Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 1 t but these are not presented here.

(March 5, 1969) In a previous paper, a general theorem was investigated for the s tability of numerical integration methods for the so lution of systems of diffe rentia l equation s. In thi s paper, further theore ms are developed as suffi c ie nt conditions for the instability of nume rica l integration methods. Applying these theore ms , the ins:abiljt.y of known formula s are checked eas ily...

A high-accuracy prescription (quadrature rule) is the set of points Xj and weights Wj such that the error E (n) is a small and rapidly decreasing function of n, the number of mesh points used. Now suppose we want to integrate a function F(X l ,x2'''''Xd) = F(x) over the d-dimensional cube. The direct product technique would be to use the rule (1) d timesi ". n2 "d L L '" L Wj, Wh ... wjdF (Xj" ...

We present an overview of intrinsic integration schemes for di erential equations evolving on manifolds, paying particular attention to homogeneous spaces. Various examples of applications are introduced, showing the generality of the methods. Finally we discuss abstract Runge{Kutta methods. We argue that homogeneous spaces are the natural structures for the study and the analysis of these meth...

stefan problem with kinetics is reduced to a system of nonlinear volterra integral equations of second kind and newton's method is applied to linearize it. product integration solution of the linear form is found and sufficient conditions for convergence of the numerical method are given. an example is provided to illustrated the applicability of the method.

‎In this paper‎, ‎we study the numerical solution of Convection-Diffusion equation with a memory term subject to initial boundary value conditions‎. ‎Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc collocation method is employed in space‎. ‎The accuracy and error analysis of the method are discussed‎. ‎Numeric...

Micropolar plasticity provides the capability to carry out post-failure simulations of geo-structures due to microstructural considerations and embedded length scale in its formulation. An essential part of the numerical implementation of a micropolar plasticity model is the integration of the rate constitutive equations. Efficiency and robustness of the implementation hinge on the type of int...

The most general approach for the solution of the dynamic response of structural systems is the direct numerical integration of the dynamic equilibrium equations. This involves, after the solution is defined at time zero, the attempt to satisfy dynamic equilibrium at discrete points in time. Most methods use equal time intervals at ∆ ∆ ∆ ∆ t t t N t , , ........ 2 3 . Many different numerical t...

we develop and apply the product integration method to a large class of linear weakly singular volterra systems. we show that under certain sufficient conditions this method converges. numerical implementation of the method is illustrated by a benchmark problem originated from heat conduction.