We have implemented in Matlab a Gauss-like cubature formula over bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n− 1 using N ∼ cmn nodes, 1 ≤ c ≤ p, m being the total number of points given on the boundary. It does not need any decomposition of the domain,...

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n− 1 using N ∼ cmn2 nodes, 1 ≤ c ≤ p, m being the total number of points given on the boundary. It does not need any decomposition of ...

Abstract In this manuscript, we implement a spectral collocation method to find the solution of reaction–diffusion equation with some initial and boundary conditions. We approximate by using two-dimensional interpolating polynomial dependent Legendre–Gauss–Lobatto points. fully show that achieved solutions are convergent exact when number points increases. demonstrate capability efficiency prov...

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe the propagation of local errors in this method, and show that the global order of RK5GL3 is expected to be six, one better than the underlying RungeKutta m...

Abstract. It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex plane containing the interval of integration. In the present paper, a bound on the error of the Newton-Cotes quadrature formula for analytic functions is derived. Also the bounds on the Legendre polyno...

A number of conservative PDEs, like various wave equations, allow for a multi-symplectic formulation which can be viewed as a generalization of the symplectic structure of Hamiltonian ODEs. We show that Gauss-Legendre collocation in space and time leads to multi-symplectic integrators, i.e., to numerical methods that preserve a symplectic conservation law similar to the conservation of symplect...

The Padua points are the first known example of optimal points for total degree polynomial interpolation in two variables, with a Lebesgue constant increasing like log of the degree; cf. [1, 2, 3]. Moreover, they generate a nontensorial Clenshaw-Curtis-like cubature formula, which turns out to be competitive with the tensorial Gauss-Legendre formula and even with the few known minimal formulas ...

This paper proposes a new paradigm for solving systems of nonlinear equations through using Genetic Algorithm (GA) techniques. So, a great attention was presented to illustrate how genetic algorithms (GA) techniques can be used in finding the solution of a system described by nonlinear equations. To achieve this purpose, we apply Gauss–Legendre integration as a technique to solve the system of ...

Local projection methods which yield c'm_1) piecewise polynomials of order m + k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k f unctionals which implies breakpoint superconvergence (of...

Gauss–Legendre quadrature formulas have excellent convergence properties when applied to integrals ∫ 1 0 f(x) dx with f ∈ C∞[0, 1]. However, their performance deteriorates when the integrands f(x) are in C∞(0, 1) but are singular at x = 0 and/or x = 1. One way of improving the performance of Gauss–Legendre quadrature in such cases is by combining it with a suitable variable transformation such ...