Numerical Quadrature Formulas Research Articles

A highly efficient and accurate numerical method for the electromagnetic scattering problem with rectangular cavities

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted.

Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable.

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.

Spectrally accurate numerical quadrature formulas for a class of algebraically singular periodic Hadamard finite part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integrals Hσ(t;u)=⨎0T|sinπ(x−t)T|σu(x)dx,0<t<T,σ<−1,σ⁄∈Z,u(x) being sufficiently differentiable and T-periodic on R. Thus σ=−(m+δ), m∈{1,2,3,…}, 0<δ<1. For each such σ, we regularize Hσ(t;u), and show that Hσ(t;u)=Hσ+2r(t;Uσ),r=⌊(m+1)/2⌋, where Uσ(x)=∑k=0raku(2k)(x) for some constants ak, Hσ+2r(t;Uσ) being a regular integral. We then propose to approximate Hσ(t;u) by the quadrature formula Qσ,n(t;u)≡Hσ(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qσ,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qσ,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R).We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals Hσ(t;u) with σ<−1 and σ⁄∈Z apply, with no changes, to the regular singular integrals Hσ(t;u) with σ>−1 and σ⁄∈Z.We illustrate the effectiveness of Qσ,n(t;u) with numerical examples both for σ<−1 and σ>−1.Finally, we show that the HFP or regular integral ⨎0Tf(x)dx of any T-periodic integrand f(x) that has algebraic singularities of the form |x−t+kT|σ, 0<t<T, k=0,±1,±2,…, with σ⁄∈Z, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed as Hσ(t;u), where u(x) is a T-periodic and sufficiently differentiable function of x on R that can be computed from f(x). Therefore, ⨎0Tf(x)dx can be computed efficiently using our new numerical quadrature formulas Qσ,n(t;u) on the individual Hσ(t;u). Again, only 2n function evaluations, namely, u(xn,k), k=0,1,…,2n−1, are needed for the whole process.

A new second order Taylor-like theorem with an optimized reduced remainder

In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function f defined on the interval [a,b], this formula is derived by introducing a linear combination of f′ computed at n+1 equally spaced points in [a,b], together with f′′(a) and f′′(b). We then consider two classical applications of this Taylor-like expansion: the interpolation error and the numerical quadrature formula. We show that using this approach improves both the Lagrange P2- interpolation error estimate and the error bound of the Simpson rule in numerical integration.

Behavior of Lagrange‐Galerkin solutions to the Navier‐Stokes problem for small time increment

AbstractWe consider two kinds of numerical quadrature formulas of Gauss type and Newton‐Cotes type, which are required in the real computation of Lagrange–Galerkin scheme for the Navier–Stokes problem. The Lagrange–Galerkin scheme with numerical quadrature, which has been used practically but not fully analyzed, is proved to be convergent at least for Gauss type quadrature under a condition on the time increment. As for the scheme with Newton‐Cotes type quadrature, it has more smooth convergent property than that of Gauss type, whose reason is discussed.

Stability inequalities and numerical solution for neutral Volterra delay integro-differential equation

The aim of the present paper is to introduce a new numerical method for solving Volterra integro-differential equation involving delay and neutral types. Using the numerical quadrature formula we form a finite difference scheme on a uniform mesh. The presented numerical method obtains a second-order convergence in discrete maximum norm. Furthermore, we illustrate the efficacy of the proposed method by constructing examples.

Parameters Estimation in Non-Negative Integer-Valued Time Series: Approach Based on Probability Generating Functions

This manuscript deals with a parameter estimation of a non-negative integer-valued (NNIV) time series based on the so-called probability generating function (PGF) method. The theoretical background of the PGF estimation technique for a very general, stationary class of NNIV time series is described, as well as the asymptotic properties of the obtained estimates. After that, a particular emphasis is given to PGF estimators of independent identical distributed (IID) and integer-valued non-negative autoregressive (INAR) series. A Monte Carlo study of the thus obtained PGF estimates, based on a numerical integration of the appropriate objective function, is also presented. For this purpose, numerical quadrature formulas were computed using Gegenbauer orthogonal polynomials. Finally, the application of the PGF estimators in the dynamic analysis of some actual data is given.

Spectrally accurate numerical quadrature formulas for a class of periodic Hadamard Finite Part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integralsKm(t;u)=⨎0TSm(π(x−t)T)u(x)dx,0<t<T,m∈{1,2,…}, where u(x) is T-periodic and sufficiently differentiable andS2r−1(y)=cos⁡ysin2r−1⁡y,S2r(y)=1sin2r⁡y,r=1,2,3,…. For each m, we regularize the HFP integral Km(t;u) and show thatKm(t;u)=K0(t;Um)≡∫0T(log⁡|sin⁡π(x−t)T|)Um(x)dx,Um(x) being some linear combination of the first m derivatives of u(x). We then propose to approximate Km(t;u) by the quadrature formula Qm,n(t;u)≡Km(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qm,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qm,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R). We illustrate the effectiveness of Qm,n(t;u) with numerical examples for m=0,1,…,5.We also show that the HFP integral ⨎0Tf(x,t)dx of any T-periodic integrand f(x,t) that has mth order poles at x=t+kT, k=0,±1,±2,…, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed in terms of the Ks(t;u(⋅,t)), where u(x,t) is a T-periodic and sufficiently differentiable function in x on R that can be computed from f(x,t). Therefore, ⨎0Tf(x,t)dx can be computed efficiently using our new numerical quadrature formulas Qs,n(t;u(⋅,t)) on the individual Ks(t;u(⋅,t)). Again, only 2n function evaluations, namely, u(xn,k,t), k=0,1,…,2n−1, are needed for the whole process.

Discontinuous Galerkin approximations to second-kind Volterra integral equations with weakly singular kernel

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results.

Some evaluations of the fractional $ p $-Laplace operator on radial functions

&lt;abstract&gt;&lt;p&gt;We face a rigidity problem for the fractional $ p $-Laplace operator to extend to this new framework some tools useful for the linear case. It is known that $ (-\Delta)^s(1-|x|^{2})^s_+ $ and $ -\Delta_p(1-|x|^{\frac{p}{p-1}}) $ are constant functions in $ (-1, 1) $ for fixed $ p $ and $ s $. We evaluated $ (-\Delta_p)^s(1-|x|^{\frac{p}{p-1}})^s_+ $ proving that it is not constant in $ (-1, 1) $ for some $ p\in (1, +\infty) $ and $ s\in (0, 1) $. This conclusion is obtained numerically thanks to the use of very accurate Gaussian numerical quadrature formulas.&lt;/p&gt;&lt;/abstract&gt;

Correction to: Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions

Correction to: Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions

Continuous stage stochastic Runge\u2013Kutta methods

In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.

An alternative method to construct a consistent second-order theory on the equilibrium figures of rotating celestial bodies

The main objective of this work is to construct a new method to develop a consistent second-order amplitudes theory to evaluate the potential of a rotating deformable celestial body when the hydrostatic system equilibrium has been achieved. In this case, we have: ∇⃗P=ρ∇⃗Ψ, △Ψ=−4πGρ+2ω2, where P is the pressure, ρ is the density, Ψ is the total potential, △ is Laplace operator, G is the gravitational constant and ω⃗ is the angular velocity of the system. To integrate these equations in a general case of mass distribution a state equation relating pressure and density is needed.To assess the full potential, Ψ, it is necessary to calculate the self-gravitational potential, Ω, and the centrifugal potential, Vc. The equilibrium configuration involves the hydrostatic equilibrium, it is, the rigid rotation of the system corresponding to the minimum potential and, according to Kopal, this state involves the identification of equipotential, isobaric, isothermal and isopycnic surfaces.To study the structure of the body we define a coordinate system OXYZ where O is the center of mass of the component, OX is an axis fixed in an arbitrary point of the body equator, OZ an axis parallel to angular velocity ω→ and OY defining a direct trihedron. For an arbitrary point P in the rotating body the Clairaut coordinates are given by (a,θ,λ) where a is the radius of the sphere that contains the same mass that the equipotential surface that contains P and (θ,λ) are the angular spherical coordinates of P.This problem has been solved in the first order in ω2 following two techniques: the first one is based on the asymptotic properties of the numerical quadrature formulae. The second is similar to the one used by Laplace to develop the inverse of the distance between two planets. The second-order theory based on the first method has been developed by the authors in a recent paper. In this work we develop a consistent second-order theory about the equilibrium figures of rotating celestial bodies based on the second method.Finally, to show the performance of the method it is interesting to study a numerical example based on a convective star.

Kernel functions‐based approach for distributed order diffusion equations

AbstractIn this work, we solve distributed order diffusion equations (DODEs) by applying the theory on reproducing kernel functions (RKFs). The classical numerical quadrature formulae is used to approximate the DODE to a multi‐term Caputo fractional order diffusion equation (FDE). The Mittag‐Leffler RKF is introduced to estimate fractional derivatives of Caputo. And a space–time RKFs collocation scheme is derived for the multi‐term Caputo time FDEs. The accuracy of the present numerical technique is indicated by employing several experiments.

Richardson extrapolation for the discrete iterated modified projection solution

Approximate solutions of Urysohn integral equations using projection methods involve integrals which need to be evaluated using a numerical quadrature formula. It gives rise to the discrete versions of the projection methods. For $r \geq 1,$ a space of piece-wise polynomials of degree $\leq r - 1$ with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at $r$ Gauss points. Asymptotic expansion for the iterated modified projection solution is available in literature. In this paper, we obtain an asymptotic expansion for the discrete iterated modified projection solution and use Richardson extrapolation to improve the order of convergence. Our results indicate a choice of a numerical quadrature which preserves the order of convergence in the continuous case.

Discrete superconvergent degenerate kernel method for Fredholm integral equations

Approximate solutions of integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we propose the discrete version of the superconvergent degenerate kernel method for solving Fredholm integral equation of the second kind with a smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain optimal convergence rates for both approximated solution and iterated discrete solution. Numerical results are presented to illustrate the theoretical estimates for the error of this method.

High-Order Variational Time Integrators for Particle Dynamics

Abstract The general family of Galerkin variational integrators are analyzed and a complete classification of such methods is proposed. This classification is based upon the type of basis function chosen to approximate the trajectories of material points and the numerical quadrature formula used in time. This approach leads to the definition of arbitrarily high order method in time. The proposed methodology is applied to the simulation of brownout phenomena occurring in helicopter take-off and landing.

Vectorial Slepian Functions on the Ball

Due to the uncertainty principle, a function cannot be simultaneously limited in space as well as in frequency. The idea of Slepian functions, in general, is to find functions that are at least optimally spatio-spectrally localized. Here, we are looking for Slepian functions which are suitable for the representation of real-valued vector fields on a three-dimensional ball. We work with diverse vectorial bases on the ball which all consist of Jacobi polynomials and vector spherical harmonics. Such basis functions occur in the singular value decomposition of some tomographic inverse problems in geophysics and medical imaging. Our aim is to find band-limited vector fields that are well-localized in a part of a cone whose apex is situated in the origin. Following the original approach towards Slepian functions, the optimization problem can be transformed into a finite-dimensional algebraic eigenvalue problem. The entries of the corresponding matrix are treated analytically as far as possible. For the remaining integrals, numerical quadrature formulae have to be applied. The eigenvalue problem decouples into a normal and a tangential problem. The number of well-localized vector fields can be estimated by a Shannon number which mainly depends on the maximal radial and angular degree of the basis functions as well as the size of the localization region. We show numerical examples of vectorial Slepian functions on the ball, which demonstrate the good localization of these functions and the accurate estimate of the Shannon number.

A Modified Precise Integration Method for Long-Time Duration Dynamic Analysis

This paper presents a modified Precise Integration Method (PIM) for long-time duration dynamic analysis. The fundamental solution and loading operator matrices in the first time substep are numerically computed employing a known unconditionally stable method (referred to as original method in this paper). By using efficient recursive algorithms to evaluate these matrices in the first time-step, the same numerical results as those using the original method with small time-step are obtained. The proposed method avoids the need of matrix inversion and numerical quadrature formulae, while the particular solution obtained has the same accuracy as that of the homogeneous solution. Through setting a proper value of the time substep, satisfactory accuracy and numerical dissipation can be achieved.

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.