Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Hosseininia, M.; Heydari, M. H.

doi:10.1016/j.chaos.2019.07.017

Cited by 38 publications

(9 citation statements)

References 42 publications

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“…The moving least square method is used in [21] for a category of VO fractional 2D telegraph equations including derivatives with non-singular kernel. A wavelet method utilized in [22] for a category of VO fractional 2D reaction-diffusion equations including derivatives with non-singular kernel. In [23], the author applied a cardinal method for a class of nonlinear VO fractional optimal control problems generated by this kind of derivatives.…”

Section: Introductionmentioning

confidence: 99%

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

2021

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In this work, a nonlinear singular variable-order fractional Emden–Fowler equation involved with derivative with non-singular kernel (in the Atangana–Baleanu–Caputo type) is introduced and a computational method is proposed for its numerical solution. The desired method is established upon the shifted Jacobi polynomials and their operational matrix of variable-order fractional differentiation (which is extracted in the present study) together with the spectral collocation method. The presented method transforms obtaining the solution of the main problem into obtaining the solution of an algebraic system of equations. Several numerical examples are examined to show the validity and the high accuracy of the established method.

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Section: Introductionmentioning

confidence: 99%

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

2021

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“…Note that the main issue about V-O fractional functional equations is difficulty in obtaining their exact solutions. In recent years, many approximate methods have been applied to solve such problems; for instance, see Heydari et al (2019a, 2019b), Bhrawy and Zaky (2017, 2016), Hafez and Zaky (2019), Parsa Moghaddam and Tenreiro Machado (2017), Keshi et al (2018), Hosseininia and Heydari (2019a, 2019b), Hassani et al (2019a, 2019b) and Hassani and Naraghirad (2019).…”

Section: Introductionmentioning

confidence: 99%

Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations

Heydari

Mahmoudi

Avazzadeh

et al. 2020

Journal of Vibration and Control

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The main objectives of this study are to introduce a new class of optimal control problems governed by a dynamical system of weakly singular variable-order fractional integral equations and to establish a computational method by utilizing the Chebyshev cardinal functions for their numerical solutions. In this way, a new operational matrix of variable-order fractional integration is generated for the Chebyshev cardinal functions. In the established method, first the control and state variables are approximated by the introduced basis functions. Then, the interpolation property of these basis functions together with their mentioned operational matrix is applied to derive an algebraic equation instead of the objective function and an algebraic system of equations instead of the dynamical system. Eventually, the constrained extrema technique is applied by adjoining the constraints generated from the dynamical system to the objective function using a set of Lagrange multipliers. The accuracy of the established approach is examined through several test problems. The obtained results confirm the high accuracy of the presented method.

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“…The finite difference method 31 and the meshless method 32 have been implemented for VO fractional telegraph problem. The method of the Legendre wavelets were utilized in Hosseininia and Heydari 33 for finding solution of VO fractional reaction‐diffusion problem. The Legendre and Chebyshev cardinal functions were utilized, respectively, in Heydari and Atangana 34 and Heydari 35 for VO fractional Schrödinger problem and VO fractional optimal control problems.…”

Section: Introductionmentioning

confidence: 99%

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Heydari

Avazzadeh

Cattani

2020

Math Methods in App Sciences

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In this article, nonlinear variable-order (VO) fractional Korteweg-de Vries (KdV) Burgers' equation with nonsingular VO time fractional derivative is introduced and discussed. The approximate solution of the expressed problem is obtained in the form of a series expansion in terms of the shifted discrete Chebyshev polynomials (CPs) with great accuracy. The method is a computational procedure based on the collocation technique and the shifted discrete CPs together with their operational matrices (ordinary and VO fractional derivatives). The main advantage of the designed approach is that it provides a global solution for the problem. In order to examine the efficiency of the designed algorithm, some numerical problems have been provided. The obtained solutions confirm that the present method is computationally effective and sufficiently accurate in solving this equation.

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Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel

Cited by 38 publications

References 42 publications

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

Shifted Jacobi polynomials for nonlinear singular variable-order time fractional Emden–Fowler equation generated by derivative with non-singular kernel

Chebyshev cardinal functions for a new class of nonlinear optimal control problems with dynamical systems of weakly singular variable-order fractional integral equations

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

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