Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann‐Liouville derivative

Owolabi, Kolade M.

doi:10.1002/num.22197

Cited by 43 publications

(18 citation statements)

References 37 publications

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“…The scheme can be easily extended to employ and solve the system of nonlinear differential equations involving Caputo‐Fabrizio operator. In the future, we hope that this can also extend the scheme to solve some other problems related to newly defined differential operators or other types of fractional calculus problems() such as in the study of Owolabi, KM (2018a), Owolabi, KM (2018b), Owolabi, KM (2018c), Owolabi, KM (2018d), Owolabi KM and Atangana A (2017a), Owolabi KM and Atangana (2017b), and Owolabi and Atangana (2017c).…”

Section: Resultsmentioning

confidence: 99%

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

Toh

Phang

Loh

2018

Math Methods in App Sciences

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In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo-Fabrizio operator. To derive this new predictor-corrector scheme, which suits on Caputo-Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo-Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved. KEYWORDSAdams-Bashforth-Moulton method, Caputo fractional derivative, Caputo-Fabrizio operator, nonlinear differential equation, predictor-corrector scheme INTRODUCTIONIn 2015, Michele Caputo and Mauro Fabrizio introduced a new definition of Caputo derivative, 1 which was called Caputo-Fabrizio fractional derivative. The main characteristics of this new definition are including regular kernel, having two different representations for temporal and spatial variables and as memory operator. Although it was verified that the operator does not fit the usual concepts neither for fractional nor for integer derivative/integral, 2 the new operator was successfully applied in describing the control movement of waves on the area of shallow water, 3 unsteady flows of an incompressible Maxwell fluid, 4 anomalous diffusion phenomena, 5 diffusion and the diffusion-advection equation, 6 epidemiological model for computer viruses, 7 model of circadian rhythms, 8 and many more phenomena. In this research direction, some existing numerical methods have been extended to the problem defined in the sense of this new Caputo-Fabrizio operator. Among them is the combination of the Laplace transform and homotopy analysis method to solve the so-called fractional partial differential equation problem with Caputo-Fabrizio operator. 9 Other methods include multiqaudric (MQ)-RBF collocation method, 10 discretization scheme, 11 fractional Adams-Bashforth method, 12 and three-step fractional Adams-Bashforth method. 13 However, since Caputo-Fabrizio operator is relatively new, there are still relatively limited works that have been done to obtain the simple, reliable, and accurate solution for the problem defined in Caputo-Fabrizio operator.On the other hand, predictor-corrector scheme had been extended to solve fractional differential equations. 14 Since that, some modification was done to obtain a better accurate scheme or more suitable to different problems, such as Abbreviation used: Fractional ordinary differential equation (FODE).

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Section: Resultsmentioning

confidence: 99%

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

Toh

Phang

Loh

2018

Math Methods in App Sciences

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“…With the introduction of some mathematical concepts in fractional analysis, it is inevitable to reconsider chaos with these mathematical tools. [1][2][3][4][5][6][7][8][9][10][11] In this paper, we consider new model where are used new differential and integral operators and solve our model with the new numerical scheme introduced by Atangana and Seda. They gave detailed information about the effectiveness and usefulness of the suggested method in Refs.…”

Section: Introductionmentioning

confidence: 99%

Atangana–seda Numerical Scheme for Labyrinth Attractor With New Differential and Integral Operators

Atangana

Araz

2020

Fractals

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In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based on Newton polynomial. The accuracy and efficiency of the method can be easily seen with numerical simulations.

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“…Various chaotic processes have been modelled by a number of fractional derivatives. Chaotic differential equations are largely encountered in various fields of engineering, physics, chemistry, economics, and other related applied science subjects [ 14 , 15 , 16 ]. The main advantage of spectral methods relies mainly in their accuracy for a given number of unknowns.…”

Section: Introductionmentioning

confidence: 99%

Modelling of Chaotic Processes with Caputo Fractional Order Derivative

Owolabi

Gómez‐Aguilar

Fernández‐Anaya

et al. 2020

Entropy

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Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

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Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann‐Liouville derivative

Cited by 43 publications

References 37 publications

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

Atangana–seda Numerical Scheme for Labyrinth Attractor With New Differential and Integral Operators

Modelling of Chaotic Processes with Caputo Fractional Order Derivative

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