Some integral inequalities for functions with (n−1)st derivatives of bounded variation

AbstractIn this paper, we generalize Cerone’s results, and a unified treatment of error estimates for a general inequality satisfying f(n−1) being of bounded variation is presented. We derive the estimates for the remainder terms of the mid-point, trapezoid, and Simpson formulas. All constants of the errors are sharp. Applications in numerical integration are also given

On generalizations of Ostrowski inequality via Euler harmonic identities

Copyright © 2002 L. J. Dedić et al. This work is licensed under a Creative Commons License.Some generalizations of Ostrowski inequality are given, by using some Euler identities involving harmonic sequences of polynomials.L. J. Dedić, M. Matić, J. Pečarić, and A. Vukeli

Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level m

This article deals with some properties -which are, to the best of our knowledge, new- of the generalized Euler polynomials of level m. These properties include a new recurrence relation satisfied by these polynomials and quadrature formulae of Euler-Maclaurin type based on them. Numerical examples are also given.The work of the first author (YQ) has been supported by Decanato de Investigación y Desarrollo, Universidad Simón Bolívar, grant DID-USB (S1-IC-CB-003-16). Also, the first author thanks the hospitality of Coordinación de Matemáticas of Universidad del Atlántico during her visit for the Fall Semester 2016. The work of the second author (AU) has been supported by Universidad del Atlántico, Colombia, grant Impacto Caribe (IC-002627-2015)

Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle”

AbstractAn explicit modulus of Hölder continuity is given for the flow associated to the canonic Brownian motion on the diffeomorphism group of the circle

Integral Error Representation of Hermite Interpolating Polynomial and Related Inequalities for Quadrature Formulae

We consider integral error representation related to the Hermite interpolating polynomial and derive some new estimations of the remainder in quadrature formulae of Hermite type, using Holder’s inequality and some inequalities for the Čebyšev functional. As a special case, generalizations for the zeros of orthogonal polynomials are considered

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in L2-norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm. © 2023, The Author(s) under exclusive licence to Istituto di Informatica e Telematica (IIT).CX20220454; Russian Science Foundation, RSF: 22-21-00075The authors are grateful for helpful comments and suggestions from the reviewers. This work was supported by Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20220454). Ahmed S. Hendy wishes to acknowledge the support of the RSF grant, project 22-21-00075