General Quadrature Formulae Research Articles

An approach for approximate solution of fractional-order smoking model with relapse class

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Grünwald–Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann–Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain transformations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model. A comparative study between Grünwald–Letnikov method and Runge–Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

APPROXIMATE INTEGRATION OF HIGHLY OSCILLATING FUNCTIONS

Elementary approximate formulae for numerical integration of functions containing oscillating factors of a special form with a parameter have been proposed in the paper. In this case general quadrature formulae can be used only at sufficiently small values of the parameter. Therefore, it is necessary to consider in advance presence of strongly oscillating factors in order to obtain formulae for numerical integration which are suitable in the case when the parameter is changing within wide limits. This can be done by taking into account such factors as weighting functions. Moreover, since the parameter can take values which cannot always be predicted in advance, approximate formulae for calculation of such integrals should be constructed in such a way that they contain this parameter in a letter format and they are suitable for calculation at any and particularly large values of the parameter. Computational rules with such properties are generally obtained by dividing an interval of integration into elementary while making successive approximation of the integral density at each elementary interval with polynomials of the first, second and third degrees and taking the oscillating factors as weighting functions. The paper considers the variant when density of the integrals at each elementary interval is approximated by a polynomial of zero degree that is a constant which is equal to the value of density in the middle of the interval. At the same time one approximate formula for calculation of an improper integral with infinite interval of the function with oscillating factor of a special type has been constructed in the paper. In this case it has been assumed that density of the improper integral rather quickly goes to zero when an argument module is increasing indefinitely. In other words it is considered as small to negligible outside some finite interval. Uniforms in parameter used for evaluation of errors in approximate formulae have been obtained in the paper and they make it possible to calculate integrals with the required accuracy.

Numerical Integration Schemes for Unequal Data Spacing

In this Paper, we have derived the numerical integration formula for unequal data spacing. For doing so we use Newton’s Divided difference formula to evaluate general quadrature formula and then generate Trapezoidal and Simpson’s rules for the unequal data spacing with error terms. Finally we use numerical examples to compare the numerical results with analytical results and the well-known Monte- Carlo integration results and find excellent agreement.

Radon transport model into a porous ground layer of finite capacity

The model of radon transfer is considered in a porous ground layer of finite power. With the help of the Laplace integral transformation, a numerical solution of this model is obtained which is based on the construction of a generalized quadrature formula of the highest degree of accuracy for the transition to the original - the function of solving this problem. The calculated curves are constructed and investigated depending on the diffusion and advection coefficients.The work was a mathematical model that describes the effect of the sliding attachment (stick-slip), taking into account hereditarity. This model can be regarded as a mechanical model of earthquake preparation. For such a model was proposed explicit finite- difference scheme, on which were built the waveform and phase trajectories hereditarity effect of stick-slip.

Schur-convexity and monotonicity of error of a general three-point quadrature formula

We study a general three-point quadrature formula and give necessary and sufficient conditions for this type of quadrature formula to be Schur-convex. A quadrature formula being Schur-convex implies the following: if we consider two intervals, one of which contains the other and such that share the same midpoint, then the error of this quadrature formula is smaller on a smaller interval.

Monotonicity of error and Schur-convexity of a general two-point quadrature formula

AbstractIn this work, the behaviour of the error of a general two-point quadrature formula is studied. It is established that the error in question has a monotonicity property and that this is equivalent to the fact that the function defined as the difference of the average value of the integral and the quadrature — that is, the error — is Schur-convex.

A sharp estimate for the Peano error representation

We provide a sharp estimate of the Peano error representation formula for linear functionals. As applications, we obtain sharp estimates for the remainder term in general quadrature formulae and expand some Ostrowski-like type inequalities to linear functionals. The new approach extends and unifies many earlier results on the subject (Dragomir and Wang, 1997; Cheng, 2001; Matić, 2003; Vong, 2011; Gonska et al., 2012).

Numerical Solution of Boole’s Rule in Numerical Integration by Using General Quadrature Formula

We have seen that definite integrals arise in many different areas and that the fundamental theorem of calculus is a powerful tool for evaluating definite integrals. This paper describes classical quadrature method for the numerical solution of Boole’s rule in numerical integration.

Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to spaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases.

Two types of interface defects

The solution of a plane problem in the theory of elasticity for a two-component body with an interface, a finite part of which is either weakly distorted or is a weakly curved crack is constructed using the perturbation method. In the first case, it is assumed that the discontinuities in the forces and displacements at the interface are known, and, in the second case, the non-equilibrium nature of the load in the crack is taken into account. General quadrature formulae are derived for the complex potentials, which enable any approximation to be obtained in terms of elementary functions in many important practical cases. An algorithm is indicated for calculating each approximation. Families of defects are studied, the form of which is determined by power functions. The effect of the amplitude of the distortion and the shape of the interface crack on the Cherepanov–Rice integral as well as the shape of the distorted part of the interface on the stress concentration is investigated in the first approximation. An analysis of the applicability of the oscillating solution for a distorted interface crack is carried out. The results of the calculations are shown in the form of graphical relations.

A Generalization of the Ostrowski Integral Inequality for Mappings Whose Derivatives Belong to Lp[a, b] and Applications in Numerical Integration

A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp[a,b], 1<p<∞, and applications for general quadrature formulae are given.

Convergence of Gaussian Quadrature Formulas

Convergence of a general Gaussian quadrature formula is shown and its rate of convergence is also given.

The Ostrowski integral inequality for mappings of bounded variation

A generalisation of the Ostrowski integral inequality for mappings of bounded variation and applications for general quadrature formulae are given.

Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen

Due to a theorem of Peano, the remainder term of Gregory's quadrature formula may be expressed by means of a polynomial spline-function. An explicite representation of this spline-function is given by a direct method, which does not make use of Peano's theorem. As an application the remainder term obtained is developed in terms of boundary derivations of the integrand, which leads to a general quadrature formula containing the Gregory formula and the Euler-MacLaurin summation formula as special cases.

A generalized quadrature formula for Cauchy integrals

A generalized quadrature formula for Cauchy integrals

Derivation of correction terms for general quadrature formulae

AbstractIn this paper, it will be shown how it is possible, using Bernoulli polynomials of the first kind, to derive correction terms for any quadrature approximation in the form of a weighted sum over a finite interval. The terms derived will involve either the difference or the sum of derivatives taken at the ends of the range of integration, and they will be described as being of either Type A or Type B respectively. There will also be some discussion of the uses of such terms.

The optimum addition of points to quadrature formulae

Methods are developed for the addition of points in an optimum manner to the Gauss, Lobatto and general quadrature formulae. A new set of n n -point formulae are derived of degree ( 3 n − 1 ) / 2 (3n - 1)/2 .