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.

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.

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

scholarly journals Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
N. Mindu ◽  
D. P. Mason
Keyword(s):  
Conservation Laws ◽  
Power Law ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Multiplier Method ◽  
Reduction Theorem ◽  
Double Reduction ◽  
Exponential Law ◽  
Spatial Derivatives ◽  
Derivatives Of

The derivation of conservation laws for the magma equation using the multiplier method for both the power law and exponential law relating the permeability and matrix viscosity to the voidage is considered. It is found that all known conserved vectors for the magma equation and the new conserved vectors for the exponential laws can be derived using multipliers which depend on the voidage and spatial derivatives of the voidage. It is also found that the conserved vectors are associated with the Lie point symmetry of the magma equation which generates travelling wave solutions which may explain by the double reduction theorem for associated Lie point symmetries why many of the known analytical solutions are travelling waves.

CONVERGENCE IN RELAXATION SPECTRUM RECOVERY

2016 ◽  
Vol 95 (1) ◽  
pp. 121-132 ◽  
Author(s):  
R. J. LOY ◽  
F. R. DE HOOG ◽  
R. S. ANDERSSEN
Keyword(s):  
Infinite Series ◽  
Relaxation Spectrum ◽  
Sufficient Conditions ◽  
Convolution Equation ◽  
Series Expansions ◽  
Practical Application ◽  
Rigorous Analysis ◽  
Spectrum Recovery ◽  
Derivatives Of

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$, given $g$. There are several approaches involving the use of series expansions of derivatives of $g$, which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$) for the convergence of the series concerned.

Using Maple to Study the Partial Differential Problems

2013 ◽  
Vol 479-480 ◽  
pp. 800-804 ◽  
Author(s):  
Chii Huei Yu
Keyword(s):  
Infinite Series ◽  
Partial Derivative ◽  
Higher Order ◽  
The Other ◽  
Mathematical Software ◽  
Partial Differential ◽  
Differential Problem ◽  
Partial Derivatives ◽  
Order Partial Derivative ◽  
Derivatives Of

This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples of multivariable functions to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.

scholarly journals Generalization on some theorems of $ L^1 $-convergence of certain trigonometric series

2008 ◽  
Vol 39 (1) ◽  
pp. 63-74
Author(s):  
Zivorad Tomovski
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Sigma 1 ◽  
Complex Valued ◽  
Derivatives Of

In this paper we study $ L^1 $-convergence of the $ r $-th derivatives of Fourier series with complex-valued coefficients. Namely new necessary-sufficient conditions for $L^1$-convergence of the $ r $-th derivatives of Fourier series are given. These results generalize corresponding theorems proved by several authors (see [7], [10], [13], [19]). Applying the Wang-Telyakovskii class $ ({\bf B}{\bf V})_r^\sigma $, $ \>\sigma>0 $, $ \>r=0,1,2,\ldots\, $ we generalize also the theorem proved by Garrett, Rees and Stanojevi\'{c} in [5]. Finally, for $ \sigma=1 $ some corollaries of this theorem are given.

Sign in / Sign up

Export Citation Format

Share Document