Quadrature Formulas Research Articles

Quadrature Rules for the Fm-Transform Polynomial Components

The purpose of this paper is to reduce the complexity of computing the components of the integral Fm-transform, m≥0, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the Fm-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the Fm-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the Fm-transformation for 0≤m≤3. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.

On an Optimal Quadrature Formula in a Hilbert Space of Periodic Functions

The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals ∫02πeiωxφ(x)dx in the Sobolev space H˜2m. Here, H˜2m is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is O(hm) for |ω|<N and O(|ω|−m) for |ω|≥N. Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the exp(iωx) weight in the case m=2 and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas.

Numerical Analysis of Alternating Direction Implicit Orthogonal Spline Collocation Scheme for the Hyperbolic Integrodifferential Equation with a Weakly Singular Kernel

This paper studies an alternating direction implicit orthogonal spline collocation (ADIOSC) technique for calculating the numerical solution of the hyperbolic integrodifferential problem with a weakly singular kernel in the two-dimensional domain. The integral term is approximated with the help of the second-order fractional quadrature formula introduced by Lubich. The stability and convergence analysis of the proposed strategy are proven in L2-norm. Numerical results highlight the high accuracy and efficiency of the proposed strategy and clarify the theoretical prediction.

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.

Memory response in dual-phase-lag thermoelastic medium due to instantaneous heat source

The present work deals with thermoelastic interactions in an isotropic infinite medium with memory-dependent derivative under a dual-phase-lag model. The governing equations are expressed in vector–matrix differential equation (VMDE) form in the Laplace–Fourier transform domain and solved using the eigenvalue approach. Numerical estimations of different thermophysical quantities such as displacement, temperature, and stresses are presented graphically for a coper material by employing the Gaussian quadrature formula and Honig–Hirdes method. Several remarkable points have been mentioned in the graphical representation of the field variables for various kernel functions, time-delay parameters, and magnitude of instantaneous heat source.

A discretization method for nonlocal diffusion type equations

In this paper we consider a numerical scheme for the treatment of an integro-differential equation. The latter represents the formulation of a nonlocal diffusion type equation. The discretization procedure relies on the application of the line method. However, quadrature formulae are needed for the evaluation of the integral operator. They are based on generalized Bernstein polynomials. Numerical evidence shows that the proposed method is a suitable and reliable approach for the problem.

New Fractional Estimates of Simpson-Mercer Type for Twice Differentiable Mappings Pertaining to Mittag-Leffler Kernel

The main motivation of this study is to introduce a novel auxiliary result of Simpson’s formula by employing the Mercer scheme for twice differentiable functions involving the Atangana-Baleanu (AB) fractional integral operator concerned with the Mittag-Leffler as a nonsingular or nonlocal kernel. Thus, by employing Mercer’s convexity on twice differentiable mappings along with Hölder’s and power-mean inequalities, one can develop a variety of new Simpson’s error estimates. Lastly, some applications to q -digamma function and modified Bessel functions are presented. Furthermore, the graphical illustrations described the efficiency and applicability of the proposed technique with success. We make links between our findings and a number of well-known discoveries in the literature. It is hoped that the proposed methodology will provide a new venue in the numerical techniques for calculating the quadrature formulae.

Gaussian conditional random fields for classification

Gaussian conditional random fields (GCRF) are a well-known structured model for continuous outputs that uses multiple unstructured predictors to form its features and at the same time exploits dependence structure among outputs, which is provided by a similarity measure. In this paper, a Gaussian conditional random field model for structured binary classification (GCRFBC) is proposed. The model is applicable to classification problems with undirected graphs, intractable for standard classification CRFs. The model representation of GCRFBC is extended by latent variables which yield some appealing properties. Thanks to the GCRF latent structure, the model becomes tractable, efficient and open to improvements previously applied to GCRF regression models. In addition, the model allows for reduction of noise, that might appear if structures were defined directly between discrete outputs. Two different forms of the algorithm are presented: GCRFBCb (GCRGBC — Bayesian) and GCRFBCnb (GCRFBC — non-Bayesian). The extended method of local variational approximation of sigmoid function is used for solving empirical Bayes in Bayesian GCRFBCb variant, whereas MAP value of latent variables is the basis for learning and inference in the GCRFBCnb variant. The inference in GCRFBCb is solved by Newton–Cotes formulas for one-dimensional integration. Both models are evaluated on synthetic data and real-world data. We show that both models achieve better prediction performance than unstructured predictors. Furthermore, computational and memory complexity is evaluated. Advantages and disadvantages of the proposed GCRFBCb and GCRFBCnb are discussed in detail.

Graph signal interpolation with positive definite graph basis functions

For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in Euclidean spaces or spherical basis functions. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner-type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF's and to study GBF interpolation in more detail. We are able to characterize the native spaces of the interpolants. We provide explicit estimates for the interpolation error and obtain bounds for the numerical stability. As a final application, we show how GBF interpolation can be used to get quadrature formulas on graphs.

A Collocation Method for Mixed Volterra–Fredholm Integral Equations of the Hammerstein Type

This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear interpolation on triangles. The resulting quadrature formula has a higher degree of precision than expected, leading to a collocation method that is superconvergent at the collocation nodes. The convergence of the method is established, as well as the rate of convergence. Numerical examples are considered, showing the applicability of the proposed scheme and the agreement with the theoretical results.

Optimal quadrature formulas for oscillatory integrals in the Sobolev space

This work studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space L_{2}^{(m)}(0,1) for numerical calculation of Fourier coefficients. Using Sobolev’s method, we obtain new sine and cosine weighted optimal quadrature formulas of such type for N + 1geq m, where N + 1 is the number of nodes. Then, explicit formulas for the optimal coefficients of optimal quadrature formulas are obtained. The obtained optimal quadrature formulas in L_{2}^{(m)}(0,1) space are exact for algebraic polynomials of degree (m-1).

Anti-Gaussian quadrature formulae of Chebyshev type

We prove that there is no positive measure d σ d\sigma on the interval [ a , b ] [a,b] such that the corresponding anti-Gaussian quadrature formula is also a Chebyshev quadrature formula. We also show that the only positive and even measure d σ ( t ) = d σ ( − t ) d\sigma (t)=d\sigma (-t) on the symmetric interval [ − a , a ] [-a,a] , for which the anti-Gaussian formula has the form ∫ − a a f ( t ) d σ ( t ) = μ 0 2 [ f ( a ) + f ( − a ) ] + R 2 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=\frac {\mu _{0}}{2}[f(a)+f(-a)]+R_{2}^{AG}(f) for n = 1 n=1 and ∫ − a a f ( t ) d σ ( t ) = w 1 f ( a ) + w ∑ μ = 2 n f ( t μ ) + w 1 f ( − a ) + R n + 1 A G ( f ) \int _{-a}^{a}f(t)d\sigma (t)=w_{1}f(a)+w\sum _{\mu =2}^{n}f(t_{\mu })+w_{1}f(-a)+R_{n+1}^{AG}(f) for all n ≥ 2 n\geq 2 , is the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt . It turns out that the formula for n ≥ 2 n\geq 2 is the ( n − 1 ) (n-1) -point Gauss-Lobatto quadrature formula for the measure d σ ( t ) = ( a 2 − t 2 ) − 1 / 2 d t d\sigma (t)=(a^{2}-t^{2})^{-1/2}dt , which is a generalization of what happens in the case of the Chebyshev measure of the first kind. Moreover, we compute the anti-Gaussian formulae for any one of the four Chebyshev measures.

Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

Quantum algorithms based on the block-encoding framework for matrix functions by contour integrals

he matrix functions can be defined by Cauchy's integral formula and can be approximated by the linear combination of inverses of shifted matrices using a quadrature formula. In this paper, we propose a quantum algorithm for matrix functions based on a procedure to implement the linear combination of the inverses on quantum computers. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., \textbf{20}, 1\&amp;2, 14--36, (Feb. 2020)] that proposed a quantum algorithm to compute a quantum state for the matrix function based on the circular contour centered at the origin, the quantum algorithm in the present paper can be applied to a more general contour. Moreover, the algorithm is described by the block-encoding framework. Similarly to the previous study, the algorithm can be applied even if the input matrix is not a Hermitian or normal matrix. This is an advantage compared with quantum singular value transformation.

Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions

In this paper, we introduce a new finite class of non-polynomial basis functions for the spectral tau approximation of time-fractional partial differential equations on a semi-infinite interval. Different from many other spectral tau approaches, the singular basis functions are based on a finite class of Romanovski–Jacobi polynomials through a fractional coordinate transform. As such, the singularity of the new basis can be tailored to suit that of the singular solutions to a class of time-fractional partial differential equations, leading to spectrally accurate approximations. We introduce the fractional Romanovski-Jacobi-Gauss-type quadrature formulae along with basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We discuss the relationship between such kind of finite orthogonal functions and other classes of infinite orthogonal functions such as fractional Laguerre functions and fractional modified generalized Laguerre functions. Moreover, we derive a new formula expressing explicitly the fractional integral of the fractional Romanovski-Jacobi functions in terms of the same functions themselves. This family of orthogonal systems offers great flexibility to match a wide range of time-fractional differential equations with Robin boundary conditions and with nonsmooth solutions.

Variational Physics Informed Neural Networks: the Role of Quadratures and Test Functions

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.

Constrained mock-Chebyshev least squares quadrature

In this paper we use the constrained mock-Chebyshev least squares interpolation to obtain stable quadrature formulas with high degree of exactness and accuracy from equispaced nodes. We numerically prove the effectiveness of the proposed algorithm by several examples.

3D thermoelastoplastic study of FML structure under the action of a moving Gaussian-distributed laser heat source

Fiber metal laminated (FML) structures are used widely, however, studies on the three-dimensional thermoelastoplastic behaviors of FML structure subjected to a moving Gaussian-distributed laser heat source have not yet been developed. In present work, the semi-analytical solution of three-dimensional transient heat transfer equation for FML structure under the action of a moving Gaussian-distributed laser heat source is obtained through the separate variable method (SVM) and the Newton Cotes method (NCM). A yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of the orthotropic FML structure. Based on classical nonlinear laminated plate theory, the incremental nonlinear governing equations are obtained, and the equations are solved by the combination of finite difference method, Newmark method and iterative method. Numerical results show that the temperature field, moving velocity of laser, number of FML layers, thickness ratio and boundary condition have great influences on thermoelastoplastic behaviors of the FML structure.

Spectral Collocation Algorithm for Solving Fractional Volterra-Fredholm Integro-Differential Equations via Generalized Fibonacci Polynomials

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

A new operational vector approach for time‐fractional subdiffusion equations of distributed order based on hybrid functions

This research study deals with the numerical solutions of linear and nonlinear time‐fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block‐pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method.