scholarly journals Linear Rational Finite Differences from Derivatives of Barycentric Rational Interpolants

2012 ◽  
Vol 50 (2) ◽  
pp. 643-656 ◽  
Author(s):  
Georges Klein ◽  
Jean-Paul Berrut
Keyword(s):  
Finite Differences ◽  
Rational Interpolants ◽  
Derivatives Of

An Introduction to Finite Differencing

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Finite Difference ◽  
Finite Differences ◽  
Single Point ◽  
Finite Differencing ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Student Reading ◽  
Higher Derivatives ◽  
Derivatives Of

This chapter on finite differencing appears oddly placed in the early part of a text on spectral modeling. Finite differences are still traditionally used for vertical differencing and for time differencing. Therefore, we feel that an introduction to finite-differencing methods is quite useful. Furthermore, the student reading this chapter has the opportunity to compare these methods with the spectral method which will be developed in later chapters. One may use Taylor’s expansion of a given function about a single point to approximate the derivative(s) at that point. Derivatives in the equation involving a function are replaced by finite difference approximations. The values of the function are known at discrete points in both space and time. The resulting equation is then solved algebraically with appropriate restrictions. Suppose u is a function of x possessing derivatives of all orders in the interval (x — n∆x, x + n∆x). Then we can obtain the values of u at points x ± n∆ x, where n is any integer, in terms of the value of the function and its derivatives at point x, that is, u(x) and its higher derivatives.

Exact discretization of non-commutative space-time

2020 ◽  
Vol 35 (16) ◽  
pp. 2050135
Author(s):  
V. E. Tarasov
Keyword(s):  
Crystal Lattice ◽  
Long Range ◽  
Finite Differences ◽  
Star Product ◽  
Space Time ◽  
Range Interaction ◽  
Long Range Interaction ◽  
Commutative Space ◽  
Discrete Operators ◽  
Derivatives Of

Non-commutative space-time is discussed for discrete case. An exact discretization of the non-commutative space-time is proposed. New discrete operators, which satisfy the same algebraic relations as standard derivatives of integer orders, are used. Exact discretization of the non-commutative space-time is suggested by using generalization of the star-product (the Moyal–Weyl product) based on the recently proposed exact finite differences. The suggested discrete non-commutative space-time corresponds to the standard “continuous” non-commutative space-time without any approximations. The suggested discrete non-commutative space-time can be interpreted as a special physical (crystal) lattice with long-range interaction.

scholarly journals Exact Discrete Analogs of Derivatives of Integer Orders: Differences as Infinite Series

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fourier Series ◽  
Power Law ◽  
Infinite Series ◽  
Finite Differences ◽  
Characteristic Property ◽  
Discrete Analogs ◽  
Derivatives Of

New differences of integer orders, which are connected with derivatives of integer orders not approximately, are proposed. These differences are represented by infinite series. A characteristic property of the suggested differences is that its Fourier series transforms have a power-law form. We demonstrate that the proposed differences of integer ordersnare directly connected with the derivatives∂n/∂xn. In contrast to the usual finite differences of integer orders, the suggested differences give the usual derivatives without approximation.

scholarly journals Convergence rates of derivatives of a family of barycentric rational interpolants

2011 ◽  
Vol 61 (9) ◽  
pp. 989-1000 ◽  
Author(s):  
Jean-Paul Berrut ◽  
Michael S. Floater ◽  
Georges Klein
Keyword(s):  
Convergence Rates ◽  
Rational Interpolants ◽  
Derivatives Of
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