scholarly journals Expectation Identity of the Hypergeometric Distribution and Its Application in the Calculations of High-Order Origin Moments

2021 ◽  
Author(s):  
Yuan-Quan Wang ◽  
Ying-Ying Zhang ◽  
Jia-Lei Liu
Keyword(s):  
Stirling Numbers ◽  
High Order ◽  
Hypergeometric Distribution ◽  
Novel Method ◽  
Theoretical Results

Abstract We provide a novel method to analytically calculate the high-order origin moments of a hypergeometric distribution, that is, the expectation identity method. First, the expectation identity of the hypergeometric distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments of the hypergeometric distribution by using the expectation identity. Furthermore, we analytically calculate the general kth (k=1,2,…) origin moment of the hypergeometric distribution by using the expectation identity, and the results are summarized in a theorem. Moreover, we use the general kth origin moment to validate the first four origin moments of the hypergeometric distribution. Next, the coefficients of the first ten origin moments of the hypergeometric distribution are summarized in a table containing Stirling numbers of the second kind. Moreover, the general kth origin moment of the hypergeometric distribution by using the expectation identity is restated by another theorem involving Stirling numbers of the second kind. Finally, we provide some numerical and theoretical results.

scholarly journals A Hybrid High-Order method for Kirchhoff–Love plate bending problems

2018 ◽  
Vol 52 (2) ◽  
pp. 393-421 ◽  
Author(s):  
Francesco Bonaldi ◽  
Daniele A. Di Pietro ◽  
Giuseppe Geymonat ◽  
Françoise Krasucki
Keyword(s):  
Sharp Estimate ◽  
High Order ◽  
Plate Bending ◽  
Mechanical Modeling ◽  
High Order Method ◽  
Local Polynomial ◽  
Bending Behavior ◽  
Bending Problems ◽  
Norm Of The Error ◽  
Theoretical Results

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in hk+3 is also derived for the L2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

scholarly journals New Results on Pulsating OB Stars

1995 ◽  
Vol 155 ◽  
pp. 44-55 ◽  
Author(s):  
Paweł Moskalik
Keyword(s):  
Data Model ◽  
Physical Mechanism ◽  
High Order ◽  
Early Type ◽  
Instability Strip ◽  
Model Calculations ◽  
B Stars ◽  
Ob Stars ◽  
Early Type Stars ◽  
Theoretical Results

AbstractUntil very recently the physical mechanism driving oscillations in β Cep and other early type stars has been a mystery. The breakthrough came with the publication of new OPAL and OP opacity data. Model calculations with the new opacities have demonstrated that the pulsations are driven by the familiar K-mechanism, acting in the metal opacity bump at T ≈ 2 × 105K. The mechanism excites the low order p- and g-modes in the upper part of the instability strip and the high order g-modes in the lower part of the strip. The theoretical instability domains agree well with the observed domains of the β Cep and the SPB stars. In this review I present these recent theoretical results and discuss their consequences for our understanding of B stars pulsations.

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

scholarly journals Global exponential stability of positive periodic solutions for a cholera model with saturated treatment

2018 ◽  
Vol 23 (5) ◽  
pp. 619-641
Author(s):  
Hongzheng Quana ◽  
Xueyong Zhoub ◽  
Jianzhou Liua
Keyword(s):  
Exponential Stability ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Incidence Rate ◽  
Global Exponential Stability ◽  
Positive Periodic Solutions ◽  
Treatment Function ◽  
Novel Method ◽  
Saturated Treatment ◽  
Theoretical Results

In this paper, we consider a cholera model with periodic incidence rate and saturated treatment function. Under certain conditions, we establish a criterion on the global exponential stability of positive periodic solutions for this model by using a novel method. We illustrate our theoretical results with numerical simulations by using Matlab.

scholarly journals Numerical Solution of Stieltjes Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1571
Author(s):  
Francisco J. Fernández ◽  
F. Adrián F. Tojo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Explicit Solution ◽  
Numerical Scheme ◽  
Linear Ordinary Differential Equation ◽  
Stieltjes Integral ◽  
Novel Method ◽  
Theoretical Results

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators.

scholarly journals Optimal High-Order Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Artidiello ◽  
A. Cordero ◽  
Juan R. Torregrosa ◽  
M. P. Vassileva
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
High Order ◽  
Function Technique ◽  
Numerical Tests ◽  
New Methods ◽  
Theoretical Results ◽  
Made In

A class of optimal iterative methods for solving nonlinear equations is extended up to sixteenth-order of convergence. We design them by using the weight function technique, with functions of three variables. Some numerical tests are made in order to confirm the theoretical results and to compare the new methods with other known ones.

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

2021 ◽  
Author(s):  
Murat Polat ◽  
Ömer Oruç
Keyword(s):  
Conservation Laws ◽  
Lie Group ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
High Order ◽  
Basis Functions ◽  
Numerical Integrator ◽  
The Kdv Equation ◽  
De Vries ◽  
Novel Method

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

Sign in / Sign up

Export Citation Format

Share Document