Chebyshev Approximation by a Rational Expression for Functions of Many Variables

Malachivskyy, P. S.; Pizyur, Ya. V.; Malachivsky, R. P.

doi:10.1007/s10559-020-00302-0

Cited by 13 publications

(11 citation statements)

References 7 publications

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“…В працях [10,11] описано алгоритми обчислення параметрів чебишовського наближення функцій однієї змінної на основі схеми Ремеза з використанням диференціальної корекції. Метод побудови чебишовського наближення раціональним виразом на основі обчислення середньостепеневих наближень функцій багатьох змінних описано в працях [12,13].…”

Section: лев мельничокunclassified

“…Для побудови чебишовського наближення функцій двох змінних раціональним виразом з інтерполюванням, застосовуємо метод описаний в працях [12,13], який полягає в послідовній побудові середньостепеневих наближень раціональним виразом з врахуванням інтерполяційної умови (3). Для обчислень застосовуємо ітераційну схема на основі методу найменших квадратів з використанням двох змінних вагових функцій, значення яких уточнюються з врахуванням всіх попередніх наближень.…”

Section: лев мельничокunclassified

“…Параметри раціонального наближення за методом найменших квадратів визначаємо з використанням лінеаризації [14]. [12,13]. Нехай для функції   y x f , , таблично-заданої на множині точок  , існує чебишовське наближення раціональним виразом  …”

Section: лев мельничокunclassified

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Chebyshev approximation of functions of two variables by a rational expression with interpolation

Melnychok

2021

Fìz.-mat. model. ìnf. tehnol.

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A method for constructing a Chebyshev approximation by a rational expression with interpolation for functions of two variables is proposed The idea of the method is based on the construction of the ultimate mean-power approximation in the norm of space Lp at p¬∞ . An iterative scheme based on the least squares method with two variable weight functions was used to construct such a Chebyshev approximation. The results of test examples confirm the effectiveness of the proposed method for constructing the Chebyshev approximation by a rational expression with interpolation.

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Section: лев мельничокunclassified

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Chebyshev approximation of functions of two variables by a rational expression with interpolation

Melnychok

2021

Fìz.-mat. model. ìnf. tehnol.

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“…The rational Chebyshev (RC) functions are used in the domain [0, L], where 0 ≤ L < ∞, which provides great success in dealing with differential equations defined in the open domain [0, L]. Many authors studied RC functions for solving different problems of differential, integro-differential equations, and some other physical problems with variable coefficients [23][24][25]. Saeid Abbasbandy et al [26] investigated the use of the RC collocation method to get an approximate solution of magneto hydro dynamic flow of an incompressible viscous fluid over a stretching sheet.…”

Section: Introductionmentioning

confidence: 99%

Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations

et al. 2021

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In this work, a numerical technique for solving general nonlinear ordinary differential equations (ODEs) with variable coefficients and given conditions is introduced. The collocation method is used with rational Chebyshev (RC) functions as a matrix discretization to treat the nonlinear ODEs. Rational Chebyshev collocation (RCC) method is used to transform the problem to a system of nonlinear algebraic equations. The discussion of the order of convergence for RC functions is introduced. The proposed base is specified by its ability to deal with boundary conditions with independent variable that may tend to infinity with easy manner without divergence. The technique is tested and verified by two examples, then applied to four real life and applications models. Also, the comparison of our results with other methods is introduced to study the applicability and accuracy.

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“…Many researchers considered CPs to solve differential equations in the finite domain ½−1, 1 (see [1][2][3][4][5][6][7][8] and [9]), but they often fail in the larger domain, also if the exact solution of the problem was in a rational form. For this reason, the rational Chebyshev (RC) functions are applied in the large domain ½0, l where l ⟶ ∞, which provide a major success in dealing with differential equations (DEs) defined in the open domain ½0, l. Many researchers studied RC functions for treating man different problems of differential, integrodifferential equations (IDEs), partial, and some other physical-engineering prob-lems as in [10,11] and [12]. Abbasbandy et al [13] applied the RC collocation method to get numerical solution of the magnetohydrodynamic flow (MHF) of an incompressible viscous fluid (VF) over a stretching sheet problem.…”

Section: Introductionmentioning

confidence: 99%

A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

Ramadan

Radwan

Nassar

et al. 2021

Journal of Function Spaces

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A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

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Chebyshev Approximation by a Rational Expression for Functions of Many Variables

Cited by 13 publications

References 7 publications

Chebyshev approximation of functions of two variables by a rational expression with interpolation

Chebyshev approximation of functions of two variables by a rational expression with interpolation

Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations

A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

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