2019
DOI: 10.1063/1.5086771
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Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods

Abstract: The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher’s equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. Th… Show more

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Cited by 120 publications
(48 citation statements)
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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%
“…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%
“…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%
“…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%