Quadrature rules for rational functions

2000 ◽  
Vol 86 (4) ◽  
pp. 617-633 ◽  
Author(s):  
Walter Gautschi ◽  
Laura Gori ◽  
M. Laura Lo Cascio
Keyword(s):  
Rational Functions ◽  
Quadrature Rules

Applications

2004 ◽  
Author(s):  
Walter Gautschi
Keyword(s):  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Rational Functions ◽  
Quadrature Rule ◽  
Gauss Quadrature ◽  
Quadrature Rules ◽  
Condition Numbers ◽  
Computational Aspects ◽  
Close Relatives ◽  
Gauss Quadrature Rules

The connection between orthogonal polynomials and quadrature rules has already been elucidated in §1.4. The centerpieces were the Gaussian quadrature rule and its close relatives—the Gauss–Radau and Gauss–Lobatto rules (§1.4.2). There are, however, a number of further extensions of Gauss’s approach to numerical quadrature. Principal among them are Kronrod’s idea of extending an n-point Gauss rule to a (2n + 1)-point rule by inserting n + 1 additional nodes and choosing all weights in such a way as to maximize the degree of exactness (cf. Definition 1.44), and Turán’s extension of the Gauss quadrature rule allowing not only function values, but also derivative values, to appear in the quadrature sum. More recent extensions relate to the concept of accuracy, requiring exactness not only for polynomials of a certain degree, but also for rational functions with prescribed poles. Gauss quadrature can also be adapted to deal with Cauchy principal value integrals, and there are other applications of Gauss’s ideas, for example, in combination with Stieltjes’s procedure or the modified Chebyshev algorithm, to generate polynomials orthogonal on several intervals, or, in comnbination with Lanczos’s algorithm, to estimate matrix functionals. The present section is to discuss these questions in turn, with computational aspects foremost in our mind. We have previously seen in Chapter 2 how Gauss quadrature rules can be effectively employed in the context of computational methods; for example, in computing the absolute and relative condition numbers of moment maps (§2.1.5), or as a means of discretizing measures in the multiple-component discretization method for orthogonal polynomials (§2.2.5) and in Stieltjes-type methods for Sobolev orthogonal polynomials (§2.5.2). It is time now to discuss the actual computation of these, and related, quadrature rules.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Padé approximants and noise: rational functions

1999 ◽  
Vol 105 (1-2) ◽  
pp. 285-297 ◽  
Author(s):  
Jacek Gilewicz ◽  
Maciej Pindor
Keyword(s):  
Rational Functions ◽  
Padé Approximants ◽  
Pade Approximants

scholarly journals Global Well-Posedness of the Ocean Primitive Equations with Nonlinear Thermodynamics

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Peter Korn
Keyword(s):  
Equation Of State ◽  
Equations Of State ◽  
Boussinesq Equations ◽  
Climate Science ◽  
Rational Functions ◽  
Ocean Dynamics ◽  
Global Ocean ◽  
Primitive Equations ◽  
Well Posedness ◽  
Nonlinear Thermodynamics

AbstractWe consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space $${{\mathcal {H}}^{1}}$$ H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

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