Fractal fractional optimal control for a novel malaria mathematical model; a numerical approach

Sweilam, N. H.; AL–Mekhlafi, S.M.; Almutairi, Alanoud

doi:10.1016/j.rinp.2020.103446

Cited by 7 publications

(9 citation statements)

References 20 publications

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“…We also note an inconsistency in Sweilam et al (2020a, 2020c), Sweilam et al (2021a), and Sweilam et al (2020b). Based on Theorem 1 and as a consequence of the formulae of integration by parts (16), (20), and (24), we deduce that when the control system is defined with a Caputo derivative, then the adjoint system should be in the Riemann–Liouville sense.…”

Section: Fractional Optimal Control Problemsmentioning

confidence: 75%

“…Regarding constant-, distributed-, and variable-order Caputo fractional optimal control problems Agrawal (2006, 2008a, 2008b), Sweilam et.al. (2020a, 2020c), Sweilam et al (2021a), and Sweilam et.al.…”

Section: Fractional Optimal Control Problemsmentioning

confidence: 99%

“…Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions proposed in the literature Agrawal (2006, 2008a, 2008b), Sweilam et al (2020a, 2020c), Sweilam et al (2021a), and Sweilam et al (2020b). The transversality conditions were introduced similarly to those of the integer-order case, with a vanishing multiplier at the terminal of the interval.…”

Section: Introductionmentioning

confidence: 99%

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On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

Hendy

Zaky

Machado

2021

Journal of Vibration and Control

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The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

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Section: Fractional Optimal Control Problemsmentioning

confidence: 75%

Section: Fractional Optimal Control Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

Hendy

Zaky

Machado

2021

Journal of Vibration and Control

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“…Some scientific studies on fractional order optimal control problems have been proposed in previous studies. [21][22][23][24][25] For further learning, the interested researchers can refer to earlier research. [26][27][28][29][30] Motivated from the above facts, the optimal reachability problem for the fractional dynamical systems in terms of 𝜓-Hilfer derivative is not yet studied.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, Rakhshan and Effati 20 solve FOCP in the Caputo sense using Laplace transforms and the Chebyshev–Gauss collocation method. Some scientific studies on fractional order optimal control problems have been proposed in previous studies 21–25 . For further learning, the interested researchers can refer to earlier research 26–30 …”

Section: Introductionmentioning

confidence: 99%

Fractional optimal reachability problems withψ‐Hilfer fractional derivative

Vellappandi¹,

Govindaraj²,

Sousa

2022

Math Methods in App Sciences

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The aim of this article is to investigate the optimal reachability problem in terms of the 𝜓-Hilfer fractional derivative for a fractional dynamical system.Euler-Lagrange equations are used to obtain the necessary optimality constraints for the fractional optimal reachability problem (FORP). In terms of the 𝜓-Hilfer fractional derivative, we developed the predictor-corrector algorithm to provide numerical solutions to the FORP. Three numerical examples are provided to show the correctness of theoretical outputs.

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Numerical solution of fractal‐fractional differential equations system via Vieta‐Fibonacci polynomials fractal‐fractional integral operators

Rahimkhani,

Ordokhani,

Sabermahani

2024

Int J Numerical Modelling

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The main idea of this work is to present a numerical method based on Vieta‐Fibonacci polynomials (VFPs) for finding approximate solutions of fractal‐fractional (FF) pantograph differential equations and a system of differential equations. Although the presented scheme can be applied to any fractional integral, we focus on the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio integrals with due to their privileges. To carry out the method, first, we introduce FF integral operators in the Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio senses. Then, by applying the Vieta‐Fibonacci polynomials and their FF integral operators together with the collocation method, the problem becomes reduced to a system of algebraic equations that can be solved by Mathematical software. In the presented scheme, acceptable approximate solutions are achieved by employing only a few number of the basic functions. Moreover, the error analysis of the presented method is investigated. Finally, the accuracy of the presented method is examined through the numerical examples. The proposed scheme is implemented for some famous systems of FF differential equations, such as memristor, which is a fundamental circuit element so called universal charge‐controlled mem‐element, convective fluid motion in rotating cavity, and Lorenz chaotic system.

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Fractal fractional optimal control for a novel malaria mathematical model; a numerical approach

Cited by 7 publications

References 20 publications

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

Fractional optimal reachability problems withψ‐Hilfer fractional derivative

Numerical solution of fractal‐fractional differential equations system via Vieta‐Fibonacci polynomials fractal‐fractional integral operators

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