scholarly journals Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-4
Author(s):  
M. L. Mittal ◽  
Mradul Veer Singh
Keyword(s):  
Error Estimation ◽  
Laguerre Polynomials ◽  
Degree Of Approximation ◽  
Continuity Condition ◽  
Frontier Point ◽  
Laguerre Series ◽  
Summability Methods ◽  
The Individual ◽  
Euler Operators ◽  
Cesàro Matrix

Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order α: Cα, Euler Eq, and Nörlund Np) means of its Fourier-Laguerre series at the point x=0 after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by (T·Eq)-transform of its Fourier-Laguerre series at frontier point x=0, where T is a general lower triangular regular matrix. A particular case, when T is a Cesáro matrix of order 1, that is, C1, has also been discussed as a corollary of main result.

scholarly journals A Study on Degree of Approximation by Summability Means of the Fourier-Laguerre Expansion

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
H. K. Nigam ◽  
Ajay Sharma
Keyword(s):  
Generating Function ◽  
Degree Of Approximation ◽  
Laguerre Expansion ◽  
Frontier Point ◽  
Laguerre Series

A very new theorem on the degree of approximation of the generating function by means of its Fourier-Laguerre series at the frontier point is obtained.

scholarly journals Error Analysis and Error Estimates for Co-Simulation in FMI for Model Exchange and Co-Simulation V2.0

2013 ◽  
Vol 60 (1) ◽  
pp. 75-94 ◽  
Author(s):  
Martin Arnold ◽  
Christoph Clauss ◽  
Tom Schierz
Keyword(s):  
System Dynamics ◽  
Error Estimation ◽  
Size Control ◽  
Industrial Applications ◽  
Physical Models ◽  
Modular Structure ◽  
Step Size ◽  
Simulation Tools ◽  
Communication Step ◽  
The Individual

Complex multi-disciplinary models in system dynamics are typically composed of subsystems. This modular structure of the model reflects the modular structure of complex engineering systems. In industrial applications, the individual subsystems are often modelled separately in different mono-disciplinary simulation tools. The Functional Mock-Up Interface (FMI) provides an interface standard for coupling physical models from different domains and addresses problems like export and import of model components in industrial simulation tools (FMI for Model Exchange) and the standardization of co-simulation interfaces in nonlinear system dynamics (FMI for Co-Simulation), see [10]. The renewed interest in algorithmic and numerical aspects of co-simulation inspired some new investigations on error estimation and stabilization techniques in FMI for Model Exchange and Co-Simulation v2.0 compatible co-simulation environments. In the present paper, we focus on reliable error estimation for communication step size control in this framework.

scholarly journals On the Degree of Approximation of a Function by Nörlund Means of its Fourier Laguerre Series

2020 ◽  
Vol 1 ◽  
pp. 65-70
Author(s):  
Suresh Kumar Sahani ◽  
Vishnu Narayan Mishra ◽  
Narayan Prasad Pahari
Keyword(s):  
Degree Of Approximation ◽  
End Point ◽  
Laguerre Series ◽  
Nörlund Means ◽  
The Given ◽  
Approximation Of A Function ◽  
Approximation Of Function

In this paper, we have proved the degree of approximation of function belonging to L[0, ∞) by Nörlund Summability of Fourier-Laguerre series at the end point x = 0. The purpose of this paper is to concentrate on the approximation relations of the function in L[0, ∞) by Nörlund Summability of Fourier- Laguerre series associate with the given function motivated by the works [3], [9] and [13].  

scholarly journals Counting Words with Laguerre Series

10.37236/3500 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Generating Function ◽  
Laguerre Polynomials ◽  
Equal Length ◽  
Weighted Sums ◽  
Combinatorial Interpretation ◽  
Laguerre Series

We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter $\alpha = -1$.  We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of $k$-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length.

scholarly journals A numerical scheme for the solution of neutral integro-differential equations including variable delay

2021 ◽  
Author(s):  
Burcu Gürbüz
Keyword(s):  
Differential Equations ◽  
Numerical Technique ◽  
Laguerre Polynomials ◽  
Algebraic Equations ◽  
Initial Value ◽  
First Order ◽  
Variable Delays ◽  
Laguerre Series ◽  
Collocation Points ◽  
Fundamental Matrices

AbstractIn this study, an effective numerical technique has been introduced for finding the solutions of the first-order integro-differential equations including neutral terms with variable delays. The problem has been defined by using the neutral integro-differential equations with initial value. Then, an alternative numerical method has been introduced for solving these type of problems. The method is expressed by fundamental matrices, Laguerre polynomials with their matrix forms. Besides, the solution has been obtained by using the collocation points with regard to the reduced system of algebraic equations and Laguerre series.

scholarly journals On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its Fourier series

2003 ◽  
Vol 34 (3) ◽  
pp. 245-247 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fourier Series ◽  
Series Representation ◽  
Triangular Matrix ◽  
Degree Of Approximation ◽  
Lipschitz Class ◽  
The Matrix ◽  
Hausdorff Matrices ◽  
Cesàro Matrix ◽  
Hausdorff Means ◽  
Euler Matrix

In a recent paper Lal and Yadav [1] obtained a theorem on the degree of approximation for a function belonging to a Lipschitz class using a triangular matrix transform of the Fourier series representation of the function. The matrix involved was the product of $ (C, 1) $, the Cesaro matrix of order one, with $ (E, 1) $, the Euler matrix of order one. In this paper we extend this result to a much wider class of Hausdorff matrices.

scholarly journals Mathematical Model for Metal Transfer Study in Additive Manufacturing with Electron Beam Oscillation

Crystals ◽  
2021 ◽  
Vol 11 (12) ◽  
pp. 1441
Author(s):  
Alexey Shcherbakov ◽  
Daria Gaponova ◽  
Andrey Sliva ◽  
Alexey Goncharov ◽  
Alexander Gudenko ◽  
...  
Keyword(s):  
Electron Beam ◽  
High Speed ◽  
Metal Transfer ◽  
Continuity Condition ◽  
Finite Difference Approximation ◽  
Navier Stokes ◽  
Difference Approximation ◽  
Deposition Processes ◽  
Spatial Oscillation ◽  
The Individual

A computer model has been developed to investigate the processes of heat and mass transfer under the influence of concentrated energy sources on materials with specified thermophysical characteristics, including temperature-dependent ones. The model is based on the application of the volume of fluid (VOF) method and finite-difference approximation of the Navier–Stokes differential equations formulated for a viscous incompressible medium. The “predictor-corrector” method has been used for the coordinated determination of the pressure field which corresponds to the continuity condition and the velocity field. The modeling technique of the free liquid surface and boundary conditions has been described. The method of calculating surface tension forces and vapor recoil pressure has been presented. The algorithm structure is given, the individual modules of which are currently implemented in the Microsoft Visual Studio environment. The model can be applied for studying the metal transfer during the deposition processes, including the processes with electron beam spatial oscillation. The model was validated by comparing the results of computational experiments and images obtained by a high-speed camera.

scholarly journals Groups defined on images in fluid diffusion

1988 ◽  
Vol 30 (1) ◽  
pp. 101-119 ◽  
Author(s):  
A. J. Bracken ◽  
H. S. Green ◽  
L. Bass
Keyword(s):  
Recurrence Relation ◽  
Differential Operators ◽  
Laguerre Polynomials ◽  
Chemical Reactor ◽  
Degree Of Approximation ◽  
Simple Relationship ◽  
Invariance Group ◽  
Third Order ◽  
Point Of Entry ◽  
Method Of Solution

AbstractA method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

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