scholarly journals Dual Taylor Series, Spline Based Function and Integral Approximation and Applications

2019 ◽  
Vol 24 (2) ◽  
pp. 35 ◽  
Author(s):  
Roy M. Howard
Keyword(s):  
Exponential Function ◽  
Taylor Series ◽  
Natural Extension ◽  
Fractional Power ◽  
Integral Function ◽  
Integral Approximation ◽  
Series Approximation ◽  
Starting Point ◽  
Catalan’S Constant ◽  
Logarithm Function

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series approximation. It is shown that a spline based series approximation to an integral yields, in general, a higher accuracy for a set order of approximation than a dual Taylor series, a Taylor series and an antiderivative series. A spline based series for an integral has many applications and indicative examples are detailed. These include a series for the exponential function, which coincides with a Padé series, new series for the logarithm function as well as new series for integral defined functions such as the Fresnel Sine integral function. It is shown that these series are more accurate and have larger regions of convergence than corresponding Taylor series. The spline based series for an integral can be used to define algorithms for highly accurate approximations for the logarithm function, the exponential function, rational numbers to a fractional power and the inverse sine, inverse cosine and inverse tangent functions. These algorithms are used to establish highly accurate approximations for π and Catalan’s constant. The use of sub-intervals allows the region of convergence for an integral approximation to be extended.

Conservation Law of Energy Using Fractional Taylor Series

2016 ◽  
Vol 841 ◽  
pp. 105-109
Author(s):  
Ali Soroush ◽  
Farzam Farahmand
Keyword(s):  
Taylor Series ◽  
Conservation Law ◽  
Control Volume ◽  
Fractional Differentiation ◽  
Linear Flow ◽  
First Order ◽  
Taylor Series Approximation ◽  
Series Approximation ◽  
Non Linear ◽  
Flow Power

Customary conservation law of energy is commonly derived using first-order Taylor series, which is only valid for situation of linear changes in the flow of energy in control volume. It is shown that using high-order Taylor series will approximate non-linear changes in the flow of energy but in fact some error remains. We used fractional Taylor series which exactly represent non-linear flow of energy in control volume. By replacing the customary integer-order Taylor series approximation with the fractional-order Taylor series approximation, limitation of the linear flow of energy in the control volume and the restriction that the control volume must be infinitesimal is omitted. The innovation of this paper is we show that as long as the order of fractional differentiation is equal with flow power-law, the fractional conservation law of energy will be exact and it can be used for fluid in a porous medium.

scholarly journals Characterizing the continuous functionals

1983 ◽  
Vol 48 (4) ◽  
pp. 965-969 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Rational Numbers ◽  
The Real ◽  
Finite Systems ◽  
Several Variables ◽  
Finite Structures ◽  
Starting Point ◽  
Topological Completion ◽  
Continuous Functionals

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

scholarly journals The asymptotic expansion of integral functions defined by Taylor series

1940 ◽  
Vol 238 (795) ◽  
pp. 423-451 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Complex Plane ◽  
Taylor Series ◽  
Asymptotic Expansions ◽  
Integral Function ◽  
The Other ◽  
General Integral ◽  
Other Hand

1.1. Considerable attention has been devoted to the behaviour of the general integral function for large values of the variable, and many important theorems have been proved in this field. On the other hand, the behaviour of a large number of particular integral functions has been studied in detail and their asymptotic expansions for certain regions of the plane obtained. There is, however, a substantial gap between the two theories. For example, much of the most interesting work on the general integral function deals with the distribution of its zeroes and other values; but many of the asymptotic expansions obtained for particular functions are not valid in the regions in which these functions have zeroes. In this paper and its sequels I propose to study several fairly wide classes of functions defined by Taylor series; from the properties of the coefficients I deduce asymptotic expansions of the function defined by the series. For the sort of functions I consider we can usually divide the whole complex plane, with the exception of certain “ barrier regions” , into a number of regions R , in each of which the function is given asymptotically by an equation of the form

scholarly journals Squaring Technique using Vedic Mathematics

2021 ◽  
Author(s):  
Jasmine Bajaj ◽  
Babita Jajodia
Keyword(s):  
Power Series ◽  
Taylor Series ◽  
Number System ◽  
Approximation Technique ◽  
Approximation Techniques ◽  
Vedic Mathematics ◽  
Decimal Number ◽  
Taylor Series Approximation ◽  
Series Approximation ◽  
Interesting Approach

Vedic Mathematics provides an interesting approach to modern computing applications by offering an edge of time and space complexities over conventional techniques. Vedic Mathematics consists of sixteen sutras and thirteen sub-sutras, to calculate problems revolving around arithmetic, algebra, geometry, calculus and conics. These sutras are specific to the decimal number system, but this can be easily applied to binary computations. This paper presented an optimised squaring technique using Karatsuba-Ofman Algorithm, and without the use of Duplex property for reduced algorithmic complexity. This work also attempts Taylor Series approximation of basic trigonometric and inverse trigonometric series. The advantage of this proposed power series approximation technique is that it provides a lower absolute mean error difference in comparison to previously existing approximation techniques.

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