A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

Marzban, Hamid Reza

doi:10.1016/j.chaos.2022.112093

Cited by 6 publications

(2 citation statements)

References 61 publications

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“…where 0 < λ ∆ < 1; for ∆ = 1, 2, 3, ..., l. This is one of the well-known delay differential equations (DDEs) that arise in a variety of real-life applications, including electronic systems [22], dynamical systems, population dynamics [23], quantum mechanics, traffic models, biological systems [24], navigational control of aircraft, economy, etc, also see in [25][26][27][28]. The DDEs are a specific kind of differential equation that considers not only the current state of a system but also its past states.…”

Section: Introductionmentioning

confidence: 99%

Fractional order Jacobi wavelet-based numerical analysis of fractal-fractional multi-pantograph delay differential equation with variable coefficients

Singh,

Verma

2024

Preprint

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In this study, the fractal-fractional Caputo and Caputo-Fabrizio derivatives are used to formulate the fractal-fractional model of multi-pantograph delay differential equations with variable coefficients. The wavelet method is constructed to provide a numerical solution by using fractional-order Jacobi wavelets. This methodology relies on the operational matrix for fractal-fractional integration of fractional order Jacobi wavelets and the collocation method. We defined pseudo code and stability analysis of the proposed approach for the given model. The error analysis and comparison of the numerical results are also shown in the tables and graphs for the three illustrative examples. In the proposed methods, the data are obtained on different values of fractal \((\nu)\) and fractional \((\mu,\phi)\) parameters and it is noteworthy to point out that the classical case is recovered for \(\mu=1\) and \(\nu=1\).

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Section: Introductionmentioning

confidence: 99%

Fractional order Jacobi wavelet-based numerical analysis of fractal-fractional multi-pantograph delay differential equation with variable coefficients

Singh,

Verma

2024

Preprint

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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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A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy

Ahmed,

Azar,

Abdel-Aty

et al. 2024

Ain Shams Engineering Journal

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A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

Cited by 6 publications

References 61 publications

Fractional order Jacobi wavelet-based numerical analysis of fractal-fractional multi-pantograph delay differential equation with variable coefficients

Fractional order Jacobi wavelet-based numerical analysis of fractal-fractional multi-pantograph delay differential equation with variable coefficients

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

A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy

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