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REVIEW ON THE USE OF CHEBYSHEV POLYNOMIALS FOR
NUMERICAL MODELING OF APPLIED PROBLEMS
Djurayeva Nasiba Turakhanovna
Doctoral student of the Department of Applied
Mathematics of Termez State University
Termez State University, Uzbekistan
https://doi.org/10.5281/zenodo.13842246
Chebyshev polynomials are widely used for numerical modeling of various
applied problems. The main advantage of methods using Chebyshev polynomials
is the high speed of convergence of calculation results. If
N
- is the number of
degrees of freedom (the number of Chebyshev polynomials or the number of
nodes in a difference scheme), then when using difference methods the
maximum error decreases as
2
N
or
4
N
(in difference methods of the second or
fourth order of accuracy, respectively). However, in methods using Chebyshev
polynomials, this error decreases at large
N
faster than
N
to any negative
power. In difference methods, difference approximations are used to satisfy
boundary conditions; in methods with Chebyshev polynomials, these conditions
can be satisfied exactly. The use of Chebyshev polynomials in modeling the
equation of diffusion and mass transfer arising in the food industry is described
in [1]. In this work, polynomials are used to model the two-dimensional mass
transfer equation occurring during convective air drying of food products. This
modeling is used to improve the quality of dried food products and to predict
moisture distribution.
To solve the problem of high-frequency stock price forecasting, paper [2]
proposed a forecasting model based on Chebyshev stacking and weighted neural
network. The proposed method extracts information about the characteristics of
a function from a high-frequency series of stock prices through Chebyshev
polynomial expansion.
Work [3] presents the development and research of a method for
calculating transient processes in electrical circuits based on Chebyshev
polynomials. Long-term electromagnetic transient processes occur in electrical
systems due to switching and impulse effects, as a result of which the simulation
time for such processes can be large, which is undesirable. The simulation time
increases significantly if the circuit in the study is complex, and also if this circuit
is described by a rigid system of equations of state. In this work, methods for
calculating transient processes in electrical circuits based on the approximation
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of solution functions by series using Chebyshev polynomials have been
developed and studied.
A method for predicting stability during milling based on Chebyshev
polynomials is given in [4]. The research carried out in this work relates to the
field of modern production, in particular to the method of predicting stability
during milling.
The general origin of the description of biological forms and special
functions is the subject of study in [5]. Gilis transforms, originating from botany,
are used to define square waves and Chebyshev polynomials.
The work [6] presents extremely reliable algorithms for solving density for
multiparameter mixed models from the search for roots of the Chebyshev
expansion. Calculation of mixture density for a given temperature, pressure and
composition based on multi-parameter mixture models with Helmholtz energy
sometimes fails; failures are caused by insufficiently accurate estimates of the
density root, as well as insufficiently reliable numerical methods that may not
converge to the desired density solution. Expansions in Chebyshev polynomials
have the property that all roots of the expansion can be reliably obtained.
A modified multi-output feed-forward Chebyshev polynomial network for
classifying wine regions by patterns is described in [7]. Based on existing results,
this paper presents, analyzes and applies a modified multi-output feedforward
Chebyshev polynomial neural network to classify vipodel region patterns.
Due to the intensive use of discrete transformations in image coding, the
search for fast and energy-efficient approaches to their hardware
implementation has become relevant. In [8], a new approximation for the
integer discrete Chebyshev transform with better quality and energy efficiency
due to the study of truncation and quality and energy efficiency due to the study
of truncation and trimming is proposed. The basic idea is that reducing the
coefficient values to fractions allows truncation by shifts in the internal
transformation calculations and leads to smaller values of the non-diogonal
remainders, which reduces non-orthogonality.
The work [9] presents a numerical simulation of a singularly perturbed
fourth-order equation using the preliminary integration method. In this article,
Chebyshev polynomials are used as basis functions in the pre-integration
method. Tabular and graphical results show the high accuracy and efficiency of
the proposed method.
The movement of an incompressible viscous fluid is described by the
system of Navier-Stokes equations. From this system, the Orr-Sommerfeld
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equation is obtained by the method of small perturbations, which is a fourth-
order nonlinear ordinary differential equation with a small parameter at the
highest derivative [10]. This article outlines methods for solving the eigenvalue
problem for the Orr-Sommerfeld equation.
Literature:
[1] Ishtiaq Ali., Maliha Tehseen Saleem. Applications of Orthogonal Polynomials
in Simulations of Mass Transfer Diffusion Equation Arising in Food Engineering
// Submission received: 28 January 2023 / Revised: 9 February 2023 /
Accepted: 11 February 2023 / Published: 16 February 2023
[2] Yiwen Wang., Yida Zhang ., Zixuan Pan. Research on High-frequency stock
price prediction based on Chebyshev-Stacking and Weighted LSTM neural
network [1] // Highlights in Science, Engineering and Technology MMML 2022
Volume 22 (2022).
[3] Zhenghu Yan., Xibin Wang., Zhibing Liu., Dongqian Wang.,Yongjian Ji.
Orthogonal polynomial approximation method for stability prediction in milling
// The International Journal of Advanced Manufacturing Technology, //
Published: 13 February 2017, Volume 91, pages 4313–4330, (2017).
[4] Johan Gielis., Diego Caratelli., Carlos Moreno de Jong van Coevorden., Paolo
Emilio Ricci. The Common Descent of Biological Shape Description and Special
Functions // DOI:10.1007/978-3-319-75647-9_10.
[5] Gielis J., Caratelli D., Moreno de Jong van Coevorden C., Ricci P.E. The
common descent of biological shape description and special functions //
Springer Proceedings in Mathematics & Statistics; Conference: Differential and
Difference Equations with Applications: ICDDEA, Amadora, Portugal, June 2017,
in the press.
[6] Ian H. Bell., Bradley K. Alpert. Exceptionally robust density solution
algorithms for multiparameter mixture models from the Chebyshev expansion.
[7] On Modified Multi-Output Chebyshev-Polynomial Feed-Forward Neural
Network for Pattern Classification of Wine Regions // date of current version
January 7, 2019. Digital Object Identifier 10.1109/ACCESS.2018.2885527.
[8] Guilherme Pai., Leandro M. G. Rocha., Gustavo M. Santana., Leonardo B.
Soares., Eduardo A. C. da Costa and Sergio Bampi. Using Pruning and Truncation
for Power-Efficient 2-D Approximate Tchebichef Transform Hardware
Architecture // Journal of Integrated Circuits and Systems, vol. 13, n. 1, 2018.
[9] Abdurakhimov B.F., Djuraeva N.T. Numerical modeling of a singularly
perturbed fourth-order equation by the preliminary integration method //
Problems of Computational and Applied Mathematics. – 2024. – No. 4(58). – P. 8-
17.2024. – №4(58). – С. 8-17.
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[10] Normurodov Ch.B., Juraeva N.T. Review on methods for solving the problem
of hydrodynamic stability // Problems of computational and applied
mathematics, Tashkent, 2022. - № 1(38). C 77-90
