Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods

Saad, Khaled M.; Khader, M. M.; Gómez-Aguilar, J. F.; Băleanu, Dumitru

doi:10.1063/1.5086771

Cited by 119 publications

(48 citation statements)

References 14 publications

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“…In 2015, Caputo and Fabrizio discovered a new operator of arbitrary order, namely, Caputo‐Fabrizio (CF) operator with arbitrary order and enforced to the several linear and nonlinear physical problems . In 2016, Atangana and Baleanu introduced another nonsingular derivative based on Miitag‐Leffler kernel and applied to the many problems . The parabolic heat equation was first developed and introduced Joseph Fourier in 1822.…”

Section: Introductionmentioning

confidence: 99%

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

179

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The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multi dimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator. KEYWORDS HPTM and RPSM, multidimensional heat equations, new fractional derivative, new YAC operator, Prabhakar or generalized Mittag-Leffler function MSC CLASSIFICATION 26A33; 34A08; 34A34; 60G22 Math Meth Appl Sci. 2020;43:6062-6080. wileyonlinelibrary.com/journal/mma

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Section: Introductionmentioning

confidence: 99%

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

179

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“…Very recently, Agbavon et al studied the Fisher's equation numerically by taking the diffusion term to be smaller than the reaction term, Mickens and Oyedeji presented travelling wave solutions to the modified Burgers and diffusionless Fisher's equations, and so on. See References .…”

Section: Introductionmentioning

confidence: 99%

Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

Loyinmi

Akinfe

2020

Engineering Reports

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In this research work, we propose an unprecedented hybrid algorithm which involves the coupling of a new integral transform, namely, the Elzaki transform and the well‐known homotopy perturbation method called the Elzaki homotopy transformation perturbation method (EHTPM) to solve for the exact solution of three distinct types of Fisher's equation, namely, the Fisher's equation of two cases, the sixth‐order Fisher's equation and the nonlinear diffusion equation of the Fisher type. These equations are prominent in mathematical biology and highly applicable in genetic propagation, population dynamics, stochastic processes, combustion theory, as well as a prototype model for a spreading flame, and so on. The efficacy and authenticity of this method was established via convergence and error analysis, and shows that the solutions obtained from EHTPM are unique and convergent. The results of EHTPM when compared with the exact results, homotopy results, and results from the other existing literature via table of comparison, 3D plots, and the convergence plots validates that EHTPM is an efficient and reliable tool of providing exact solutions to a wider class of nonlinear partial differential equations in a simple and straightforward manner, with no discretization, linearization, computation of Adomian polynomials, and devoid of errors.

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“…27 Application of the new derivatives to the partial differential equations is considered in Yusuf et al 28 A two-strain epidemic model with fractional derivatives has been proposed in Yusuf et al 29 The application of fractional operator known as Atangana-Baleanu to Korteweg-de Vries (KDV) equation is analyzed in Inc et al 30 The dynamics of chaotic attractors with fractional conformable derivative is proposed in Pe´rez et al 31 A fractional-time wave equation with regular kernel is studied by Cuahutenango-Barro et al 32 The authors studied a fractional Hunter-Saxton equation using Riemann-Liouville and Liouville-Caputo derivatives. 33 Numerical solution of Fisher's type equations of fractional nature with Atangana-Baleanu derivative is considered in Saad et al 34 The application of Feng's first integral technique to the fractional modified Korteweg-de Vries (MKDV) equation and their analysis and solutions is presented in Ye´pez-Martnez et al 35 Motivated from the recent literature on the chaotic models, we consider a new chaotic model in two fractional operators, that is, the Caputo-Fabrizio derivative and the Atangana-Baleanu derivative, and present comparison results. Then, we present numerical approaches for both the derivatives and give various graphical results for the fractional-order parameter a.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of a new chaotic system through the Caputo–Fabrizio and Atanagan–Baleanu fractional operators

Khan

2019

Advances in Mechanical Engineering

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The aim of this article is to analyze the dynamics of the new chaotic system in the sense of two fractional operators, that is, the Caputo–Fabrizio and the Atangana–Baleanu derivatives. Initially, we consider a new chaotic model and present some of the fundamental properties of the model. Then, we apply the Caputo–Fabrizio derivative and implement a numerical procedure to obtain their graphical results. Further, we consider the same model, apply the Atangana–Baleanu operator, and present their analysis. The Atangana–Baleanu model is used further to present a numerical approach for their solutions. We obtain and discuss the graphical results to each operator in details. Furthermore, we give a comparison of both the operators applied on the new chaotic model in the form of various graphical results by considering many values of the fractional-order parameter [Formula: see text]. We show that at the integer case, both the models (in Caputo–Fabrizio sense and the Atangana–Baleanu sense) give the same results.

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Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods

Cited by 119 publications

References 14 publications

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

The dynamics of a new chaotic system through the Caputo–Fabrizio and Atanagan–Baleanu fractional operators

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