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 ...

We present and study a novel numerical algorithm to approximate the action of T := L where L is a symmetric and positive definite unbounded operator on a Hilbert space H0. The numerical method is based on a representation formula for T in terms of Bochner integrals involving (I + tL) for t ∈ (0,∞). To develop an approximation to T , we introduce a finite element approximation Lh to L and base o...

The solutions of certain elliptic PDEs can be expressed as contour integrals of Dunford type. In this paper efficient contours and quadrature rules for the approximation of such integrals are proposed. The trapezoidal and midpoint rules are used in combination with a conformal mapping that fully exploits the analyticity of the integrand, leading to rapidly converging quadrature formulas of doub...

in this paper, we use the continuous legendre wavelets on the interval [0,1] constructed by razzaghi m. and yousefi s. [6] to solve the linear second kind integral equations. we use quadrature formula for the calculation of the products of any functions, which are required in the approximation for the integral equations. then we reduced the integral equation to the solution of linear algebraic ...

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

in this paper, we consider the second-kind chebyshev polynomials (skcps) for the numerical solution of the fractional optimal control problems (focps). firstly, an introduction of the fractional calculus and properties of the shifted skcps are given and then operational matrix of fractional integration is introduced. next, these properties are used together with the legendre-gauss quadrature fo...

In this paper we solve the problem about optimal interval quadrature formula for the class WrF of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearran...

The requirement that a Chebyshev quadrature formula have distinct real nodes is not always compatible with the requirement that the degree of precision of an npoint formula be at least equal to n. This condition may be expressed as | \d\ \p = 0, 1 g p, where d (dx, ■ ■ ■ , d„) with Mo(w) ~ , -IT dj = 2w A iM ; = 1, 2, • • ■ , z!, ZJ ,_, Pj(io), j = 0, 1, • • • , are the moments of the weight fu...

Let D be a real function such that D(z) is analytic and D(z) ± 0 for \z\ < 1. Furthermore, put W(x) = \J\ x2\D(e'v)\2 , x = costp , tp e [0, 71 ], and denote by pn(', rV) the polynomial which is orthogonal on [-1, +1] to Pn_[ (P„_! denotes the set of polynomials of degree at most n 1 ) with respect to W . In this paper it is shown that for each sufficiently large n the polynomial En+X(-, W) (ca...