Numerical solutions for distributed-order fractional optimal control problems by using generalized fractional-order Chebyshev wavelets

Ghanbari, Ghodsieh; Razzaghi, Mohsen

doi:10.1007/s11071-021-07195-4

Cited by 18 publications

(6 citation statements)

References 21 publications

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“…Mehandiratta [16] has derived the necessary optimality conditions for a FOCP on a star graph and solved it by using the finite difference method. A direct method based on the fractional‐order Chebyshev wavelets for distributed‐order FOCP has been proposed by Ghanbari and Razzaghi [17]. Hosseinpour et al [18] have proposed a Müntz–Legendre spectral collocation method for solving delay FOCP.…”

Section: Introductionmentioning

confidence: 99%

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

Kumar

Mehra

2023

Math Methods in App Sciences

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This paper develops a Müntz-Legendre wavelet method for solving a fractional optimal control problem with dynamic constraint as a fractional Sturm-Liouville problem. We first derive the necessary optimality conditions as a two-point boundary value problem by using the calculus of variation and integration by part formula. The operational matrices for Müntz-Legendre scaling functions have been obtained, and by utilizing it, the two-point boundary value problem has been converted into a system of algebraic equations. The L 2 -error estimates in the approximation of operations matrices and in the approximation of unknown variable by the Müntz-Legendre wavelet has been derived. In the last, illustrative examples have been taken to show the applicability of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

Kumar

Mehra

2023

Math Methods in App Sciences

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“…Alipanah et al [13] solved the classic brachistochrone problem using the non classical pseudo‐spectral method. The generalized fractional‐order Chebyshev wavelets method is used to solve distributed‐order fractional optimal control problems [14]. The Ritz‐generalized Pell wavelets method was suggested for two classes of fractional pantograph problems [15].…”

Section: Introductionmentioning

confidence: 99%

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Rahimkhani¹,

Ordokhani

Sabermahani

2023

Math Methods in App Sciences

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In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

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“…If the problem solution includes fractional and piecewise expressions at the same time, fractional piecewise functions are a suitable choice for the stated method. Some of the basic functions with this aspect that have been used in recent years are fractional‐order Chebyshev wavelets [19], Müntz–Legendre wavelets [20], fractional‐order Bessel wavelets [21], fractional‐order Boubaker wavelets [22], and the combination of Müntz–Legendre polynomials and block‐pulse functions [23].…”

Section: Introductionmentioning

confidence: 99%

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non‐smooth solutions

Zhagharian,

Heydari,

Razzaghi

2023

Asian Journal of Control

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In this study, to solve fractional problems with non‐smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non‐smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.

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Numerical solutions for distributed-order fractional optimal control problems by using generalized fractional-order Chebyshev wavelets

Cited by 18 publications

References 21 publications

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

Müntz–Legendre wavelet method for solving Sturm–Liouville fractional optimal control problem with error estimates

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non‐smooth solutions

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