We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly and unevanly spaced data. First, we study the new interpolation formula which generalizes both Newton's and Lagrange's interpolation formula with the new divided difference table for unevanly spaced points and using this; we derive other interpolation formulas, in terms of differences for evanly spaced data. Second, we study two new different methods of numerical differentiation for both evanly and unevanly spaced points without differentiating the interpolating polynomials or the use of operators. Third, we derive new numerical integration formulas using new differentiation formulas and Taylor formula for both evanly and unevanly spaced data. Basic computer algorithms for few new formulas are given. In Comparison to former polynomial interpolation, numerical differentiation and numerical integration formuals, these new formulas have some featured advantages for approximating functional values, numerical derivatives of higher order and approximate integral values for evanly and unevanly spaced data.
A MATLAB program for computing differentiation matrices for arbitrary one-dimensional meshes is presented in this manuscript. The differentiation matrices for a mesh of N arbitrarily spaced points are formed from those obtained using Lagrangian interpolation on stencils of a fixed but arbitrary number M<=N of contiguous mesh points. For the particular case M=N and meshes with Chebyshev or Legendre distributions of points, the program yields the well known spectral differentiation matrices. For M<N and M odd, the differentiation matrices coincide, for the special case of an evenly spaced mesh, …
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.
Fractional order differentiation by integration with Jacobi polynomials
King Abdullah university of science and technology（阿卜杜拉国王科技大学）；Arts et Métiers ParisTech（巴黎高等工艺学院）；École Centrale de Lille（里尔中央理工学院）；INRIA Lille-Nord Europe（法国国家信息与自动化研究所北欧分部）
Differentiation by integration with Jacobi polynomials
INRIA Lille-Nord Europe（法国国家信息与自动化研究所里尔北欧分部）；Université de Lille 1（里尔第一大学）；Arts et Métiers ParisTech centre de Lille（巴黎国立高等工艺美术学院里尔中心）；École Centrale de Lille（里尔中央理工学院）
A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials
La Laguna University, Spain（拉戈哈大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas ha…
A general recurrence relation for the weight-functions in Mühlbach-Neville-Aitken representions with application to WENO interpolation and differentiation
Geometrically-exact time-integration mesh-free schemes for advection-diffusion problems derived from optimal transportation theory and their connection with particle methods
University of Pennsylvania（宾夕法尼亚大学）；Politecnico di Milano（米兰理工大学）；California Institute of Technology（加州理工学院）
This work studies the parallelization and empirical convergence of two finite difference acoustic wave propagation methods on 2-D rectangular grids, that use the same alternating direction implicit (ADI) time integration. This ADI integration is based on a second-order implicit Crank-Nicolson temporal discretization that is factored out by a Peaceman-Rachford decomposition of the time and space equation terms. In space, these methods highly diverge and apply different fourth-order accurate differentiation techniques. The first method uses compact finite differences (CFD) on nodal meshes that r…
Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局（A*STAR））；Nanyang Technological University（南洋理工大学）
In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and their derivative terms which are obtained by automatic differentiation (AD), are proposed to allow efficient training with improved accuracy. The computation of differential operators required for PINNs loss evaluation at collocation points are conventionally obtained via AD. Although AD has the advantage of being able to compute the exact gradients at any point, such PINNs can only achieve high accuracies with large numbers of collocation points, otherwise they are prone to optimizi…
