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17 pages, 360 KiB

Open AccessArticle

Novel Results on Legendre Polynomials in the Sense of a Generalized Fractional Derivative

by Francisco Martínez, Mohammed K. A. Kaabar and Inmaculada Martínez

Math. Comput. Appl. 2024, 29(4), 54; https://doi.org/10.3390/mca29040054 - 12 Jul 2024

Cited by 2 | Viewed by 1780

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function

f \in L_{α}^{2} ([- 1, 1])

in a series of generalized Legendre polynomials is addressed. Full article

19 pages, 1541 KiB

Open AccessArticle

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

by Akanksha Verma, Wojciech Sumelka and Pramod Kumar Yadav

Symmetry 2023, 15(9), 1753; https://doi.org/10.3390/sym15091753 - 13 Sep 2023

Cited by 5 | Viewed by 1421

Abstract

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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13 pages, 512 KiB

Open AccessArticle

Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

by Hari M. Srivastava, Abedel-Karrem N. Alomari, Khaled M. Saad and Waleed M. Hamanah

Fractal Fract. 2021, 5(3), 131; https://doi.org/10.3390/fractalfract5030131 - 20 Sep 2021

Cited by 25 | Viewed by 2570

Abstract

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters. Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)

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23 pages, 369 KiB

Open AccessArticle

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

by Mohra Zayed, Mahmoud Abul-Ez, Mohamed Abdalla and Nasser Saad

Mathematics 2020, 8(1), 136; https://doi.org/10.3390/math8010136 - 18 Jan 2020

Cited by 14 | Viewed by 3772

Abstract

The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work. Full article

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