A new kind of double Chebyshev polynomial approximation on unbounded domains

Koc, Ayse Betul; Kurnaz, Aydın

doi:10.1186/1687-2770-2013-10

Cited by 12 publications

(17 citation statements)

References 18 publications

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“…Furthermore, there are also novel methods developed in recent years [11,50]. In this paper, we will use the Taylor series approach to obtain the polynomial approximation system which is expressed in the following lemma.…”

Section: Algorithm For Polynomial Approximationmentioning

confidence: 99%

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Polynomial-Approximation-Based Control for Nonlinear Systems

Zhou

Chen

Lam

2016

Circuits Syst Signal Process

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This paper is concerned with the stabilization problem for nonlinear systems. A new polynomial-approximation-based approach for modeling nonlinear systems is first proposed. The nonlinearity is approximated by polynomials and the approximation errors are treated as modeling uncertainties. The original nonlinear systems are converted into polynomial systems with modeling uncertainties. In order to highlight the approximation accuracy, the piecewise polynomial approximation functions are utilized. A novel polynomial state-feedback controller is designed to solve the stabilization problem. Furthermore, switched polynomial state-feedback controllers are designed to improve the performance. The stabilization conditions are presented in terms of sum of squares, which can be numerically solved via SOSTOOLS. Finally, simulation examples are provided to demonstrate the feasibility of the proposed method and show its advantage over the polynomial-fuzzy-model-based approach.

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Section: Algorithm For Polynomial Approximationmentioning

confidence: 99%

“…where 11 are the elements at the first row and first column of N , ∆A p (x (t)) and Q 1 , respectively. .…”

Section: Remarkmentioning

confidence: 99%

Polynomial-Approximation-Based Control for Nonlinear Systems

Zhou

Chen

Lam

2016

Circuits Syst Signal Process

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“…Several research groups have further developed the definition proposed by Boyd for solving differential equations [11,12,13,14,15,16,17,18,19,20]. Furthermore, Koc and Kurnaz [21] have proposed a modified type of Chebyshev polynomials as an alternative to the solutions of the partial differential equations defined in real domain. In their study, the basis functions called exponential Chebyshev (EC) functions E n (x) which are orthogonal in (−∞, ∞).…”

Section: Introductionmentioning

confidence: 99%

“…In our previous work [22,23] we have developed the operational matrix of the derivatives (ODEs) by processing the truncation made by Koc and Kurnaz [21] and applied it to ordinary and systems of differential equations defined in whole range. Recently, we reported novel operational matrix of EC functions for solving ODEs in unbounded domains [24].…”

Section: Introductionmentioning

confidence: 99%

Exponential second kind Chebyshev approximation for differential equations in unbounded domains, with applications

Salam¹,

Ramadan²

2019

Communications in Numerical Analysis

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In this paper, exponential Chebyshev polynomial of the second kind (ESC) as a basis functions with the collocation method is investigated. All Principles and the derived properties of the ESC functions are introduced as a new basis defined in the whole domain. ESC functions are employed to deal with high-order linear ordinary differential equations with variable coefficients defined in unbounded domain. Numerical test examples are given to demonstrate the applicability of the method. Physical applications of the differential equations naturally defined in the infinite domains are demonstrated.

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“…Over the last three decades, the scientists have paid much attention to spectral methods due to their high accuracy (see, for instance, [1][2][3][4][5][6][7], and the references therein). On the other hand, spectral methods typically give rise to full matrices, partially negating the gain in efficiency due to the fewer number of grid points.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

Bhrawy

Doha

Băleanu

et al. 2015

Math Methods in App Sciences

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In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

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A new kind of double Chebyshev polynomial approximation on unbounded domains

Cited by 12 publications

References 18 publications

Polynomial-Approximation-Based Control for Nonlinear Systems

Polynomial-Approximation-Based Control for Nonlinear Systems

Exponential second kind Chebyshev approximation for differential equations in unbounded domains, with applications

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

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