Generalized shifted Chebyshev polynomials for fractional optimal control problems

Hassani, Hossein; Machado, J. A. Tenreiro; Naraghirad, Eskandar

doi:10.1016/j.cnsns.2019.03.013

Cited by 57 publications

(24 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To check the stability of the proposed new algorithms to reconstruct an image, we have used Mean Square Error (MSE) to measure the performance of the proposed FLeMs constructed from fractional Legendre orthogonal functions, FLaMs from fractional order orthogonal Laguerre functions, and compare our results with FCMs recently introduced in Benouini et al 2019and Hassani et al (2019). The MSE between the original image ( , ) and reconstructed image ̂( , ) is computed from the following formula:…”

Section: Image Representationmentioning

confidence: 99%

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Farouk¹,

Awad²

2021

Int. j. adv. appl. sci.

View full text Add to dashboard Cite

In this paper, we have introduced new sets of fractional order orthogonal basis moments based on Fractional order Legendre orthogonal Functions (FLeFs) and Fractional order Laguerre orthogonal Functions (FLaFs) for image representation. We have generated a novel set of Fractional order Legendre orthogonal Moments (FLeMs) from fractional order Legendre orthogonal functions and a new set of Fractional order Laguerre orthogonal Moments (FLaMs) from the fractional order Laguerre orthogonal functions. The new presented sets of (FLeMs) and (FLaMs) are tested with the recently introduced Fractional order Chebyshev orthogonal Moments (FCMs). This edge detection filter can be used successfully in the gray level image and color images. The new sets of fractional moments are used to reconstruct the gray level image. The numerical results show FLeMs and FLaMs are promised techniques for image representation. The computational time of the proposed techniques is compared with the computational time of Chebyshev orthogonal Moments techniques and gives better results. Also, the fractional parameters give the flexibility of studying global features of the image at different positions of moments.

show abstract

Section: Image Representationmentioning

confidence: 99%

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Farouk¹,

Awad²

2021

Int. j. adv. appl. sci.

View full text Add to dashboard Cite

show abstract

“…Heydari 22 proposed a direct scheme based on the Chebyshev cardinal functions to solve a class of variable-order FOCP. Hassani et al 23 derived an optimization method based on the generalized shifted Chebyshev polynomials. Hosseinpour et al 24 applied the Müntz-Legendre spectral collocation technique for solving delay FOCP.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of nonlinear fractional optimal control problems using generalized Bernoulli polynomials

Hassani

Machado

Asl

et al. 2021

Optim Control Appl Methods

Self Cite

View full text Add to dashboard Cite

This article introduces a new class of basis functions, namely, the generalized Bernoulli polynomials (GBP). The GBP are adopted for solving nonlinear fractional optimal control problems (NFOCP) generated by nonlinear fractional dynamical systems (NFDS) and boundary conditions (BC). The corresponding operational matrices (OM) of fractional derivatives (FD) expand the solution of the problem in terms of the GBP. The method transforms the NFOCP into systems of nonlinear algebraic equations. First, the state and control variables are approximated by the GBP with unknown coefficients and parameters and substituted in the objective function, NFDS and BC. Then, the Gaussian quadrature rule and the OM of FD allow the formulation of a constrained problem, which is solved using Lagrange multipliers. The accuracy of the method is tested by means of several examples and the results confirm its good performance.

show abstract

“…During last years, various numerical methods have been used for numerically solving nonlinear fractional OCPs (for instance, see previous studies [2][3][4][5][6][7] ). The interested reader can also consult on previous studies [8][9][10][11][12][13][14] to see more details about the latest published papers about the fractional OCPs.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Heydari

Atangana

Avazzadeh

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

This article is devoted to developing an accurate operational matrix (OM) method for the solution of a new category of nonlinear optimal control problems (OCPs) explained by fractal‐fractional dynamical systems. The fractal‐fractional differentiation is considered in the sense of Atangana‐Riemann‐Liouville. The method is based on the shifted Legendre polynomials (LPs). Through the way, a new OM of fractal‐fractional differentiation is extracted for the shifted LPs. The optimality conditions are converted to an algebraic system of equations by using the shifted LP expansions of the state and control variables and applying the method of constrained extrema. More precisely, these variables are approximated by the shifted LPs with undetermined coefficients. Then, these expansions are replaced in the performance index, and the Gauss‐Legendre integration method is used to approximate the performance index for extracting a nonlinear algebraic equation. In the sequel, the mentioned OM and the collocation method are used to generate some algebraic equations from the dynamical system. Finally, the method of the constrained extrema is used by coupling the algebraic constraints generated from the dynamical system and the initial condition with the algebraic equation elicited from the performance index by a set of unknown Lagrange multipliers. The accuracy of the method is studied by solving some numerical examples.

show abstract

Generalized shifted Chebyshev polynomials for fractional optimal control problems

Cited by 57 publications

References 32 publications

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Image representation based on fractional order Legendre and Laguerre orthogonal moments

Numerical solution of nonlinear fractional optimal control problems using generalized Bernoulli polynomials

Numerical solution of nonlinear fractal‐fractional optimal control problems by Legendre polynomials

Contact Info

Product

Resources

About