We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods we prove that adaptive Monte Carlo methods are much better. Abstract. We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deter-ministic ...

In some applications of Galerkin boundary element methods one has to compute integrals which, after proper normalization, are of the form ∫ b a ∫ 1 −1 f(x, y) x− y dxdy, where (a, b) ≡ (−1, 1), or (a, b) ≡ (a,−1), or (a, b) ≡ (1, b), and f(x, y) is a smooth function. In this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p−, or h...

We reformulate the discretization of the Johnson–Nedelec method [11] of coupling boundary elements and finite elements for an exterior bidimensional Laplacian. This new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix. We show that if the parameter of discretization is sufficiently small, the fully discr...

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions w(x) ≡ w(α,β)(x) = (1− x)α(1 + x)β (α, β > −1) we give a necessary and sufficient ...

A numerical method to solve Abel-type integral equations of first kind is given. In this paper we suggest the research of a numerical solution for Abel-type integral equations of the first kind, by using a collocation method employing an interpolatory product-quadrature formula with a trigonometric polynomial of the first order. Some results of numerical examples are reported.

A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent to the discontinuous Galerkin method. Test problems include both ...

First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial tn which generates a positive qf, a weight function (depending on...

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained replacing x with xα, positive α. Fractional derivatives are in Caputo sense. With help incomplete beta functions, able to build exactly Riemann–Liouville fractional integral operator associated FOHBPB. This operator, together Newton–Cotes ...

In this paper, we found the error bounds for one of open Newton–Cotes formulas, namely Milne’s formula differentiable convex functions in framework fractional and classical calculus. We also give some mathematical examples to show that newly established are valid formula.