Quadrature Formulas Research Articles

A Two-Step Iterative Method for Absolute Value Equations

This paper presents a new numerical iteration method for solving the absolute value equations. The proposed method uses the generalized Newton method as a predictor step, and the five-point open Newton–Cotes formula is considered the corrector step. The convergence of the proposed method is studied in detail. The proposed method solves large systems effectively due to its simplicity and effectiveness. In this paper, we have solved the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solved it using the proposed method. Several numerical examples were provided to demonstrate the accuracy and effectiveness of the new approach. In addition, the novel approach solves absolute value equations with greater accuracy and precision than other existing methods.

A meshless approach based on fractional interpolation theory and improved neural network bases for solving non-smooth solution of 2D fractional reaction–diffusion equation with distributed order

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

The multi-parameterized integral inequalities for multiplicative Riemann–Liouville fractional integrals

Using the multiplicative Riemann–Liouville fractional integrals, for ⁎⁎differentiable functions, we present a fractional integral identity together with multi-parameters. Relying on it, and the fact that the function f⁎⁎ is multiplicatively convex or (lnf⁎⁎)q is convex for q>1, in particular considering the case 0<q≤1, a series of three-point Newton–Cotes type inequalities are obtained in this article. To enable the readers to comprehend the results better, we provide three examples to support the findings obtained here. Some applications in special means are offered as well.

Optimal quadrature formulas for approximating strongly oscillating integrals in the Hilbert space [formula omitted] of periodic functions

The present paper is dedicated to a variational method for the construction of optimal quadrature formulas in the sense of Sard in the Hilbert space W˜2(m,m−1) of complex-valued and periodic functions. In this, the coefficients of the optimal quadrature formula are found separately in the case ωh is integer and non-integer cases. In addition, using the constructed optimal quadrature formula, the numerical results of exponentially weighted integrals of certain functions in the case m=2 is presented. The numerical results show that the order of convergence of the optimal quadrature formula is O1N+|ω|2 in the space W˜2(2,1).

Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity

Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation f(x)=0. The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity m>1. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.

The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures

Abstract For an analytic integrand, the error term in the Gaussian quadrature can be represented as a contour integral, where the contour is commonly taken to be an ellipse. Thus, finding its upper bound can be reduced to finding the maximum of the modulus of the kernel on the ellipse. The location of this maximum was investigated in many special cases, particularly, for the Gaussian quadrature with respect to the Chebyshev measures modified by a quadratic divisor (known as the Bernstein–Szeg̋ measures). Here, for the Gaussian quadratures with respect to the Chebyshev measures modified by a linear over linear rational factor, we examine the kernel and describe sufficient conditions for the maximum to occur on the real axis. Furthermore, an assessment of the kernel is made in each case, since in some cases the true maximum is hard to reach. Hence, we derive the error bounds for these quadrature formulas. The results are illustrated by the numerical examples. An alternative approach for estimating the error of the Gaussian quadrature with respect to the same measure can be found in [Djukić, D. L., Djukić, R. M. M., Reichel, L. &amp; Spalević, M. M. (2023, Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. Appl. Numer. Math., ISSN 0168-9274)].

Gaussian quadrature for certain two-dimensional hypersingular integrals

In this paper, the Gaussian quadrature formula is used to approximately calculate two-dimensional hypersingular integrals of the form ∬Sf(x,y)r3dS on the circular domain. The two-dimensional hypersingular integrals are changed into iterated integrals by polar transformation. The inner layer of the integral has singularity and Gaussian quadrature formula is constructed for this part. The outer layer without singularity is calculated by Gauss–Legendre quadrature formula of Riemann integral. The corresponding theoretical analysis is given. Furthermore, this type of two-dimensional hypersingular integral is extended to the form ∬Sf(x,y)rp+1dS(p>2) and an error estimate is also derived. Several numerical examples are presented to verify the effectiveness of the proposed algorithm, no matter the singularity is located at the center or near the boundary of the circular domain.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

An efficient spline-based DQ method for 2D/3D Riesz space-fractional convection–diffusion equations

This paper proposes an efficient spline-based DQ method for the 2D and 3D convection–diffusion equations (CDEs) with Riesz fractional derivative in space, which have been widely used to describe the anomalous solute transport in complex media. Firstly, a spline-based differential quadrature (DQ) formula is developed to approximate the Riesz derivative by using cubic B-splines as trial functions, which allows us to approximate the fractional derivatives with high accuracy and small computational cost. We then utilize it to discretize the fractional derivatives in the governing equation and a cubic B-spline DQ scheme is further established by applying the finite difference (FD) scheme to the resulting system of ordinary differential equations. A brief implementation of the proposed DQ method is also presented. To examine the effectiveness of this spline-based DQ method, numerical tests are finally done on some benchmark problems and the simulation of rotating Gaussian hill in convection-dominated flow governed by fractional derivatives. The advantages in computational accuracy and efficiency are illustrated by comparing the results with the other algorithms in open literature.

An expanded analysis of local fractional integral inequalities via generalized (s,P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(s,P)$\\end{document}-convexity

This research aims to scrutinize specific parametrized integral inequalities linked to 1, 2, 3, and 4-point Newton-Cotes rules applicable to local fractional differentiable generalized (s,P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(s,P)$\\end{document}-convex functions. To accomplish this objective, we introduce a novel integral identity and deduce multiple integral inequalities tailored to mappings within the aforementioned function class. Furthermore, we present an illustrative example featuring graphical representations and potential practical applications.

Weighted least squares collocation methods

We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a O(h) term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method.

New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature.

Distinct-positivity-preserving methods for fifth-order finite volume ALW-MR-WENO schemes with a bigger sufficient CFL number for extreme problems

In this paper, the new distinct-positivity-preserving (DPP) methods are proposed for fifth-order finite volume multi-resolution WENO schemes with adaptive linear weights (which are termed as the ALW-MR-WENO schemes) [J. Comput. Phys., 493 (2023), 112471] for solving some extreme problems on structured meshes. Associated WENO spatial reconstruction procedures only require two hierarchical unequal-sized central spatial stencils for achieving fifth-order accuracy in smooth regions and keeping the essentially non-oscillatory property in non-smooth regions. One redefines five cell averages after performing such spatial reconstructions and then designs one quartic polynomial and two cubic reconstruction polynomials based on them in one dimension. After that, a new detective process is used to examine the positivity of the point values of such cubic reconstruction polynomials at some checking points inside the target cell. If the negativity happens, a new over-determined compression limiter is carried out to ensure that the compressed polynomials can achieve the positivity in the whole target cell instead of only at some discrete points inside it. The quartic polynomial could use its four quadrature point values by using a four-point Gauss-Lobatto quadrature formula when performing numerical integrations. Since it is easy to rewrite the point values of one quartic polynomial with the point values of two cubic polynomials timing with associated linear weights, both cubic polynomials could use a four-point Gauss-Lobatto quadrature formula or a three-point Simpson's quadrature formula without losing any algebra precisions. By doing this novelty switching methodology between two quadrature formulas, it is the first time to theoretically prove and increase a sufficient CFL number from 1/12 to 1/7.2 for the fifth-order WENO schemes in one dimension. This methodology can be easily extended to multi-dimensions. Finally, the novelty DPP methods for fifth-order ALW-MR-WENO schemes with a larger practical CFL number of 0.6, instead of the sufficient CFL number of 1/7.2, are also simple and robust enough for some extreme problems without timely halving the CFL number to avoid the appearance of negative density and negative pressure.

Galerkin approximation for multi-term time-fractional differential equations

Fractional differential equations (FDEs) are utilized as a precise model for describing a wide range of biological and physical processes, benefiting from the inherent symmetry feature in natural phenomena. The introduction of multi-term time-fractional order equations serves to tackle intricate phenomena spanning various physical domains. This study solves multi-term time-fractional DEs using the quadratic B-spline Galerkin method. In Galerkin approach, the quadratic B-spline is utilized as both a test and a basis function. The fractional portion of time is evaluated using the Caputo definition. In addition, the Gauss quadrature formula is used to evaluate the integration of complex function which enhance the accuracy. Lax-Richtmyer's stability criterion is employed to evaluate the stability of the proposed scheme. The E2 and E∞ norms are computed and displayed in the tables to illustrate the robustness and efficiency of the method. We compute simulation for various nodal points as well as various mesh sizes to analyze the solution behavior. The obtained results are compared with the exact and available results, indicating that the proposed scheme is more efficient. The graphical solution of each example is also demonstrated which verifies that exact and approximate solutions graphs closed with each other.

A new efficient two-step iterative method for solving absolute value equations

PurposeThis study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method.Design/methodology/approachWe present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method.FindingsThe paper provides empirical insights into how to solve a system of absolute value equations.Originality/valueThis paper fulfills an identified need to study absolute value equations.

A generalization of formulas for the discriminants of quasi-orthogonal polynomials with applications to hypergeometric polynomials

In this article, we extend the classical framework for computing discriminants of special quasi-orthogonal polynomials from Schur’s resultant formula, and establish a framework for computing discriminants of a sufficiently broader class of polynomials from the resultant formulas that are proven by Ulas and Turaj. More precisely, we derive a formula for the discriminant of a sequence {rA,n+crA,n-1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{r_{A,n}+c r_{A,n-1}\\}$$\\end{document} of polynomials. Here, c is an element of a field K and {rA,n}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{r_{A,n}\\}$$\\end{document} is a sequence of polynomials satisfying a certain recurrence relation. There are several works computing the discriminants of given polynomials. For example, Kaneko–Niiho and Mahlburg–Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to j-invariants of supersingular elliptic curves. Sawa–Uchida proved the formula for the discriminants of quasi-Jacobi polynomials and applied it to prove the nonexistence of certain rational quadrature formulas. Our main theorem presents a uniform way to prove a vast generalization of the above formulas for the discriminants.

A numerical calculation of arc length and area using some spline quasi-interpolants

In this paper, we propose two methods to approach numerically the length of curves and the area of surface of revolution created by rotating a curve around an axis. The first is based on an approximation of functions by quadratic spline discrete quasi-interpolant and calculating its exact length. The second method consists toapproximate the values of the first derivatives by those of cubic spline discrete quasiinterpolant. Those values are used to provide a quadrature formula to calculate the integral giving the length. In both methods, we prove that the order of convergence is O(h^4). The theoretical results given in this work are justified by some numerical examples.

Spectral decomposition of [formula omitted] and Poincaré inequality on a compact interval — Application to kernel quadrature

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫abf(x)dμ(x)=∑i=1nwif(xi) where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]⊂R and ∑i=1nwiδxi is a discrete probability distribution on [a,b]. We show that H1(μ) is a reproducing kernel Hilbert space with a continuous kernel K, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a T-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature.We derive several results for the Poincaré quadrature weights and the associated worst-case error. When μ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as b−a23n−1 for large n. By comparison with known results for H1(0,1), this shows that the Poincaré quadrature is asymptotically optimal. For a general μ, we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately O(n−1) for large n.

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog Dm(hβ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D_{m}(h\\beta )$\\end{document} of the differential operator d2mdx2m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{d^{2m}}{dx^{2m}}+1$\\end{document} designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space is demonstrated. The errors of the optimal quadrature formula in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space and in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space.