scholarly journals Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 53
Author(s):  
Cristina B. Corcino ◽  
Baby Ann A. Damgo ◽  
Joy Ann A. Cañete ◽  
Roberto B. Corcino
Keyword(s):  
Fourier Series ◽  
Asymptotic Approximation ◽  
Fourier Expansion ◽  
Asymptotic Formulas ◽  
Asymptotic Approximations

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter λ. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases λ=1 and λ=−1 were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials.

scholarly journals Asymptotic Approximations of Apostol-Genocchi Numbers and Polynomials

2021 ◽  
Vol 14 (3) ◽  
pp. 666-684
Author(s):  
Cristina Bordaje Corcino
Keyword(s):  
Fourier Series ◽  
Generating Function ◽  
Asymptotic Formulas ◽  
Asymptotic Approximations ◽  
Genocchi Numbers ◽  
Special Cases

Asymptotic approximations of the Apostol-Genocchi numbers andpolynomials are derived using Fourier series and ordering of poles ofthe generating function. Asymptotic formulas for the Apostol-Eulernumbers and polynomials are obtained as consequence. Asymptoticformulas for special cases which include the Genocchi numbers andpolynomials are also explicitly stated.

Asymptotic stepwise solutions of the Korteweg--de Vries equation with a singular perturbation and their accuracy

2021 ◽  
Vol 8 (3) ◽  
pp. 410-421
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
I. V. Gapyak ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Vries Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Main Term ◽  
Asymptotic Approximations ◽  
De Vries ◽  
The Given

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

scholarly journals Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alejandro Urieles ◽  
William Ramírez ◽  
María José Ortega ◽  
Daniel Bedoya
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Zeta Function ◽  
Fourier Expansion ◽  
Series Representation ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Lerch Zeta Function ◽  
Rational Arguments

Abstract The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (02) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
Simple Equations ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

Errors in the numerical solution of the torsion Schr¨odinger equation with Mathieu functions basis set

2019 ◽  
Author(s):  
А.Н. Белов ◽  
В.В. Туровцев ◽  
Ю.Д. Орлов
Keyword(s):  
Fourier Series ◽  
Fourier Expansion ◽  
Continued Fraction ◽  
Basis Set ◽  
Hamiltonian Matrix ◽  
Mathieu Function ◽  
Matrix Elements ◽  
Mathieu Functions ◽  
Expansion Coefficients ◽  
Relative Errors

Рассмотрена погрешность алгоритма аппроксимации функций Матье рядами Фурье, когда коэффициенты ряда Фурье представлены сходящимися цепными дробями. На основании проведенного анализа получены рекуррентные соотношения для абсолютной и относительной погрешностей удерживаемых звеньев цепной дроби и коэффициентов фурье-разложения. Предложен метод оценки точности расчета элементов матрицы гамильтониана торсионного уравнения Шрёдингера в базисе функций Матье. Эффективность предложенного алгоритма подтверждена численными примерами The dependence for the Hamiltonian matrix elements of the Schrodinger torsion equation on the calculation errors of the Mathieu basis set is considered. The Mathieu functions are represented with continued fractions in this study. The analysis of the Mathieu function approximation algorithm using Fourier series expansion is carried out when the coefficients of the Fourier series are represented by convergent continued fractions. It is shown that the major contribution to the errors at the Fourier coefficient calculation is made by the error accumulating in the corresponding elements of the continued fraction. Recurrence relations for the absolute and relative errors of the kept elements of the continued fraction and the Fourier expansion coefficients are obtained. It is shown and illustrated by a numerical example that the absolute and relative errors of the Fourier expansion coefficients in the proposed algorithm are negligible. It is noted that the maximum relative errors of continued fraction are in the highest elements of the kept part. The results of our work are used to estimate the calculation error in the integrals containing Mathieu functions. These integrals constitute the Hamiltonian matrix elements of the Schr¨odinger torsion equation. We developed an algorithm to estimate of the calculation accuracy of the Hamiltonian matrix elements of the Schr¨odinger torsion equation in the basis set of Mathieu functions. We provide the example of this algorithm. The results of the work indicate the adequacy and effectiveness at the application of the Mathieu function basis set to the solution of the Schrdinger torsion equation.¨

Fourier Series Expansion of Irregular Curves

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 105-119 ◽  
Author(s):  
M. Eugenia Montiel ◽  
Alberto S. Aguado ◽  
Ed Zaluska
Keyword(s):  
Fourier Series ◽  
Fourier Expansion ◽  
Spectral Synthesis ◽  
Fourier Series Expansion ◽  
Irregular Shapes ◽  
Analysis And Synthesis ◽  
Important Approach ◽  
Fourier Theory ◽  
Fractal Functions ◽  
Shape Analyses

Fourier theory provides an important approach to shape analyses; many methods for the analysis and synthesis of shapes use a description based on the expansion of a curve in Fourier series. Most of these methods have centered on modeling regular shapes, although irregular shapes defined by fractal functions have also been considered by using spectral synthesis. In this paper we propose a novel representation of irregular shapes based on Fourier analysis. We formulate a parametric description of irregular curves by using a geometric composition defined via Fourier expansion. This description allows us to model a wide variety of fractals which include not only fractal functions, but also fractals belonging to other families. The coefficients of the Fourier expansion can be parametrized in time in order to produce sequences of fractals useful for modeling chaotic dynamics. The aim of the novel characterization is to extend the potential of shape analyses based on Fourier theory by including a definition of irregular curves. The major advantage of this new approach is that it provides a way of studying geometric aspects useful for shape identification and extraction, such as symmetry and similarity as well as invariant features.

Stochastic approximation for the general epidemic

1973 ◽  
Vol 10 (2) ◽  
pp. 263-276 ◽  
Author(s):  
Donald Ludwig
Keyword(s):  
Exact Solution ◽  
Population Size ◽  
Stochastic Approximation ◽  
Asymptotic Approximation ◽  
Asymptotic Approximations ◽  
System Of Equations ◽  
Numerical Comparisons ◽  
Stochastic Epidemic ◽  
General Stochastic

A system of equations is introduced whose solutions are remarkably close to corresponding solutions of the “general stochastic epidemic”. If N is the population size, then there are approximately equations to be solved for the general stochastic epidemic while the number of equations in the approximating system is proportional to N. An asymptotic approximation to the general stochastic epidemic is also introduced. Numerical comparisons of the stochastic and asymptotic approximations with the exact solution are presented.

Stochastic Expansions and Asymptotic Approximations

1992 ◽  
Vol 8 (3) ◽  
pp. 343-367 ◽  
Author(s):  
Michael A. Magdalinos
Keyword(s):  
Distribution Function ◽  
Asymptotic Approximation ◽  
Asymptotic Methods ◽  
Asymptotic Approximations ◽  
Test Statistic ◽  
Chi Square ◽  
Stochastic Expansion ◽  
Stochastic Expansions ◽  
General Conditions

Under general conditions the distribution function of the first few terms in a stochastic expansion of an econometric estimator or test statistic provides an asymptotic approximation to the distribution function of the original estimator or test statistic with an error of order less than that of the limiting normal or chi-square approximation. This can be used to establish the validity of several refined asymptotic methods, including the comparison of Nagar-type moments and the use of formal Edgeworth or Edgeworth-type approximations.

scholarly journals Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements

2021 ◽  
Vol 25 (1) ◽  
pp. 67-85
Author(s):  
Edward K. Ngailo ◽  
Dietrich Von Rosen ◽  
Martin Singull
Keyword(s):  
Discriminant Analysis ◽  
Linear Discriminant Analysis ◽  
Asymptotic Approximation ◽  
Repeated Measurements ◽  
Asymptotic Approximations ◽  
Multivariate Normal ◽  
Linear Discriminant ◽  
Normal Populations ◽  
Misclassification Errors ◽  
Observation Vector

We propose asymptotic approximations for the probabilities of misclassification in linear discriminant analysis when the group means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of several multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain asymptotic approximation of misclassification errors for the two group case. Finally, we perform Monte Carlo simulations to evaluate the reliability of the proposed results.

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