A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

Jafari, Hossein; Ganji, R.M.; Sayevand, Khosro; Băleanu, Dumitru

doi:10.1177/10775463211016967

Cited by 37 publications

(4 citation statements)

References 33 publications

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“…(2.12) Lemma 2.4 (Jafari et al [28]). Let 𝜇 ∈ (0, 1), then 𝜇-order ABC derivative of q(s) can be rewritten by…”

Section: Preliminariesmentioning

confidence: 99%

“…Baleanu et al [26] give necessary optimality conditions and propose a numerical method by using the finite difference scheme. Jajarmi and Baleanu [27] present a new iterative method to get approximate solutions of obtained necessary optimality conditions, other direct numerical approaches in terms of Legendre orthogonal polynomials [28], and B‐spline functions [29] are also proposed. However, to the best of our knowledge, there is few work on FOCPs involving both integer‐order and ABC derivatives; we will add some new results on this topic.…”

Section: Introductionmentioning

confidence: 99%

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Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Liu

et al. 2023

Asian Journal of Control

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This article considers fractional optimal control problems (FOCPs) including both integer‐order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method.

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“…(2.12) Lemma 2.4 (Jafari et al [28]). Let 𝜇 ∈ (0, 1), then 𝜇-order ABC derivative of q(s) can be rewritten by…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Liu

et al. 2023

Asian Journal of Control

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show abstract

“…It is crucial to find efficient numerical methods to solve these VO‐FOCPs. Some numerical approaches to solve these problems are transcendental Bernstein series [17, 18], Chebyshev cardinal functions [19], and Legendre polynomials [20].…”

Section: Introductionmentioning

confidence: 99%

An approximate solution for variable‐order fractional optimal control problem via Müntz‐Legendre wavelets with an application in epidemiology

Razzaghi

Iordache

2023

Math Methods in App Sciences

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A class of variable‐order fractional optimal control problems (VO‐FOCPs) is solved by applying Müntz‐Legendre wavelets. Different from classical wavelets (such as Legendre and Chebyshev), the Müntz‐Legendre wavelets have an extra parameter representing the fractional order; therefore, they provide more reliable results for certain fractional calculus problems. Using the regularized beta function, the Riemann‐Liouville fractional integral operator (RLFIO) of these wavelets is precisely determined. We then transform the given VO‐FOCPs into parameter optimization problems that can be easily solved. The convergence analysis and error estimation of the proposed method are provided. Four examples are solved to illustrate the high accuracy of the approach. The method is also applied to a cancer‐obesity interaction model to analyze the interactions between the tumor, immune, normal, and fat cells when the chemotherapeutic drugs are injected into a body.

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“…However, the same complex modelling feature renders finding the analytical solutions to those FDEs very difficult, and therefore it is crucial to have accurate numerical solutions. In recent years, certain numerical methods have been used for FDEs, such as spectral method [15], Galerkin method [16], Homotopy analysis method [17], Jacobi Tau method [18], waveform relaxation method [19], fractional finite difference method [20], Adomian decomposition method [21,22], Fourier transform [23], power series method [24], B-spline collocation method [25], block-by-block method [26], variational iteration method [27], polynomial methods [28][29][30][31][32] and wavelet methods [33][34][35][36][37][38][39]. Wavelets are mathematical functions that can be used to decompose data into various time-frequency components.…”

Section: Introductionmentioning

confidence: 99%

Wavelet methods for fractional electrical circuit equations

Tural-Polat,

Dincel

2023

Phys. Scr.

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Classical electric circuits consists of resistors, inductors and capacitors which have irreversible and lossy properties that are not taken into account in classical analysis. FDEs can be interpreted as basic memory operators and are generally used to model the lossy properties or defects. Therefore, employing fractional differential terms in electric circuit equations provides accurate modelling of those circuit elements. In this paper, the numerical solutions of fractional LC, RC and RLC circuit equations are considered to better model those imperfections. To this end, the operational matrices for Bernoulli and Chebyshev wavelets are used to obtain the numerical solutions of those fractional circuit equations. Chebyshev wavelets are orthogonal, and under some circumstances, Bernoulli wavelets can be orthogonal. The wavelet methods' quick convergence and minimal processing load depend on the orthogonality principle. In the proposed method, those FDEs are transformed into algebraic equation systems using operational matrices employing the discrete Wavelets. The performance of those two wavelet methods are compared and contrasted for computational load, speed, and absolute error values. The paper exploits discrete Bernoulli and Chebyshev wavelets for the numerical solution of fractional LC, RC and RLC circuit equations. The fast convergence, low processing burden, and compactness of the Bernoulli and Chebyshev wavelet methods for fractional circuit equation solutions represent the novel contributions of this paper. Numerical solutions and comparisons are also presented to validate the method.

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A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

Cited by 37 publications

References 33 publications

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

Fractional optimal control problems with both integer‐order and Atangana–Baleanu Caputo derivatives

An approximate solution for variable‐order fractional optimal control problem via Müntz‐Legendre wavelets with an application in epidemiology

Wavelet methods for fractional electrical circuit equations

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