scholarly journals Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

2021 ◽  
Vol 29 (3) ◽  
pp. 251-259
Author(s):  
Aleksandr A. Belov ◽  
Valentin S. Khokhlachev
Keyword(s):  
Error Estimates ◽  
Complex Plane ◽  
Special Functions ◽  
Integral Form ◽  
Rounding Errors ◽  
Efficient Calculation ◽  
Full Period ◽  
Laplace Transformations ◽  
Coarse Grids ◽  
Integrand Function

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.

A numerical method for the determination of the Titchmarsh-Weyl m -coefficient

1991 ◽  
Vol 435 (1895) ◽  
pp. 535-549 ◽  
Keyword(s):  
Numerical Method ◽  
Error Estimates ◽  
Complex Plane ◽  
Real Line ◽  
Integral Inequality ◽  
Limit Point ◽  
Computational Techniques ◽  
Limit Point Case ◽  
Null Set

A numerical method for determining the Titchmarsh-Weyl m ( λ ) function for the singular eigenvalue equation – ( py' )' + qy = λwy on [ a ,∞), where a is finite, is presented. The algorithm, based on Weyl’s theory, utilizes a result first used by Atkinson to map a point on the real line onto the Weyl circle in the complex plane. In the limit-point case these circles ‘nest’ and tend to the limit-point m ( λ ). Using Weyl’s result for the diameter of the circles, error estimates for m ( λ ) are obtained. In 1971, W. N. Everitt obtained an extension of an integral inequality of Hardy-Littlewood, namely the help inequality. He showed that the existence of that inequality is determined by the properties of the null set of Im[ λ 2 m ( λ )]. In view of the major difficulties in analysing m ( λ ) even in the rare cases when it is given explicitly, very few examples of the help inequality are known. The computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities.

scholarly journals Fractional order of Legendre-type matrix polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
M. Zayed ◽  
M. Hidan ◽  
M. Abdalla ◽  
M. Abul-Ez
Keyword(s):  
Fractional Order ◽  
Recurrence Relations ◽  
Special Functions ◽  
Optimization Theory ◽  
Integral Form ◽  
First Integral ◽  
Point Of View ◽  
Rodrigues Formula ◽  
Matrix Polynomials ◽  
Fractional Orders

Abstract Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

scholarly journals Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

2021 ◽  
Vol 64 (2) ◽  
pp. 152-163
Author(s):  
P. G. Lasy ◽  
I. N. Meleshko
Keyword(s):  
Approximate Solution ◽  
Unit Circle ◽  
Mixed Problem ◽  
Gordon Equation ◽  
Special Functions ◽  
Integral Form ◽  
Third Order ◽  
Complex Power ◽  
Klein Gordon ◽  
Klein Gordon Equation

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

scholarly journals On the Lauwerier formulation of the temperature field problem in oil strata

2005 ◽  
Vol 2005 (10) ◽  
pp. 1577-1588 ◽  
Author(s):  
Lyubomir Boyadjiev ◽  
Ognian Kamenov ◽  
Shyam Kalla
Keyword(s):  
Porous Medium ◽  
Temperature Field ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Special Functions ◽  
Integral Form ◽  
Field Problem ◽  
Fractional Heat Equation ◽  
The Laplace Transform

The paper is concerned with the fractional extension of the Lauwerier formulation of the problem related to the temperature field description in a porous medium (sandstone) saturated with oil (strata). The boundary value problem for the fractional heat equation is solved by means of the Caputo differintegration operatorD∗(α)of order0<α≤1and the Laplace transform. The solution is obtained in an integral form, where the integrand is expressed in terms of a convolution of two special functions of Wright type.

scholarly journals On Generalisation of Polynomials in Complex Plane

2010 ◽  
Vol 2010 ◽  
pp. 1-9
Author(s):  
Maslina Darus ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Complex Plane ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Second Order ◽  
Second Order Differential Equations

The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

scholarly journals Monotonicity of the error term in Gauss-Turán quadratures for analytic function

2007 ◽  
Vol 48 (4) ◽  
pp. 567-581 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Miodrag M. Spalević
Keyword(s):  
Analytic Function ◽  
Weight Function ◽  
Error Estimates ◽  
Complex Plane ◽  
Error Term ◽  
Quadrature Formulae

AbstractFor Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

scholarly journals New integral forms of generalized Mathieu series and related applications

2013 ◽  
Vol 7 (1) ◽  
pp. 180-192 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Tibor Pogány
Keyword(s):  
Integral Equation ◽  
Systematic Study ◽  
Fredholm Integral Equation ◽  
Special Functions ◽  
Integral Form ◽  
Integral Representations ◽  
Integration Path ◽  
Integral Expression ◽  
Integral Forms ◽  
Mathieu Series

The main object of this article is to present a systematic study of integral representations for generalized Mathieu series and its alternating variant, and to derive a new integral expression for these special functions by contour integration using rectangular integration path. Also, by virtue of newly established integral form of generalized Mathieu series, we obtain a new integral expression for the Bessel function of the first kind of half integer order, solving a related Fredholm integral equation of the first kind with nondegenerate kernel.

scholarly journals Dynamic Bending of an Infinite Electromagnetoelastic Rod

2020 ◽  
Vol 20 (4) ◽  
pp. 493-501
Author(s):  
Duc Thong Pham ◽  
◽  
Dmitry V. Tarlakovskii ◽  
Keyword(s):  
Electromagnetic Field ◽  
Initial Conditions ◽  
Quantitative Difference ◽  
Rational Functions ◽  
Integral Form ◽  
Elastic Solution ◽  
Influence Functions ◽  
Longitudinal Coordinate ◽  
The Laplace Transform ◽  
Laplace Transformations

The problem of non-stationary bending of an infinite electromagnetoelastic rod is considered. It is assumed that the material of the rod is a homogeneous isotropic conductor. The closed-form system of process equations is constructed under the assumption that the desired functions depend only on the longitudinal coordinate and time using the corresponding relations for shells which take into account the initial electromagnetic field, the Lorentz force, Maxwell’s equations, and the generalized Ohm’s law. The desired functions are assumed to be bounded, and the initial conditions are assumed to be null. The solution of the problem is constructed in an integral form with kernels in the form of influence functions. Images of kernel are found in the space of Laplace transformations in time and Fourier transformations in spatial coordinates. It is noted that the images are rational functions of the Laplace transform parameter, which makes it quite easy to find the originals. However, for a general model that takes into account shear deformations, the subsequent inversion of the Fourier transform can be carried out only numerically, which leads to computational problems associated with the presence of rapidly oscillating integrals. Therefore, the transition to simplified equations corresponding to the Bernoulli – Euler rod and the quasistationary electromagnetic field is carried out. The method of a small parameter is used for which a coefficient is selected that relates the mechanical and electromagnetic fields. In the linear approximation, influence functions are found for which images and originals are constructed. In this case, the zeroth approximation corresponds to a purely elastic solution. Originals are found explicitly using transform properties and tables. Examples of calculations are given for an aluminum rod with a square cross section. It is shown that for the selected material the quantitative difference from the elastic solution is insignificant. At the same time, taking into account the connectedness of the process leads to additional significant qualitative effects.

scholarly journals Evaluation of peak functions on ultra-coarse grids

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120665 ◽  
Author(s):  
Natalia Petrovskaya ◽  
Nina Embleton
Keyword(s):  
Random Variable ◽  
Spatial Density ◽  
Pest Species ◽  
Sampled Data ◽  
Monitoring And Control ◽  
Density Distributions ◽  
Integration Error ◽  
Peak Functions ◽  
Coarse Grids ◽  
Integrand Function

Integration of sampled data arises in many practical applications, where the integrand function is available from experimental measurements only. One extensive field of research is the problem of pest monitoring and control where an accurate evaluation of the population size from the spatial density distribution is required for a given pest species. High aggregation population density distributions (peak functions) are an important class of data that often appear in this problem. The main difficulty associated with the integration of such functions is that the function values are usually only available at a few locations; therefore, new techniques are required to evaluate the accuracy of integration as the standard approach based on convergence analysis does not work when the data are sparse. Thus, in this paper, we introduce the new concept of ultra-coarse grids for high aggregation density distributions. Integration of the density function on ultra-coarse grids cannot provide the prescribed accuracy because of insufficient information (uncertainty) about the integrand function. Instead, the results of the integration should be treated probabilistically by considering the integration error as a random variable, and we show how the corresponding probabilities can be calculated. Handling the integration error as a random variable allows us to evaluate the accuracy of integration on very coarse grids where asymptotic error estimates cannot be applied.

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