A new numerical approximation of the fractal ordinary differential equation

Atangana, Abdon; Jain, Sonal

doi:10.1140/epjp/i2018-11895-1

Cited by 49 publications

(22 citation statements)

References 16 publications

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“…In this section, we develop an efficient approximation scheme, namely the predictorcorrector method, for the numerical solution of fractional differential equation (14). Note that this technique has been previously investigated in [23][24][25] for different types of fractional differential equations. To develop this method for Eq.…”

Section: Methodsmentioning

confidence: 99%

“…There are several fractional operators, such as Caputo and Caputo-Fabrizio, which have been applied to study the behavior of biological systems. Regarding these operators, some efficient numerical techniques have also been developed to solve different types of fractional differential equations [22][23][24][25]. However, a few attempts have been made to compare these operators in order to determine the most efficient one.…”

Section: Introductionmentioning

confidence: 99%

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New aspects of poor nutrition in the life cycle within the fractional calculus

et al. 2018

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The nutrition of pregnant women is crucial for giving birth to a healthy baby and even for the health status of a nursing mother. In this paper, the poor nutrition in the life cycle of humans is explored in the fractional sense. The proposed model is examined via the Caputo fractional operator and a new one with Mittag-Leffler (ML) nonsingular kernel. The stability analysis as well as the existence and uniqueness of the solution are investigated, and an efficient numerical scheme is also designed for the approximate solution. Comparative numerical analysis of these two operators reveals that the model based on the new fractional derivative with ML kernel has a different asymptotic behavior to the classic Caputo. Thus, the new aspects of fractional calculus provide more flexible models which help us to adjust the dynamical behaviors of the real-world phenomena better.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New aspects of poor nutrition in the life cycle within the fractional calculus

et al. 2018

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“…Introduction. In recent decade, the movement of fluid within a non-conventional media has attracted attention of many scholars and this has also emerged a separate field of investigation [1,4,2]. This study of more general behavior of porous media including deformation of solid frame is called poro-mechanics.…”

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Models of fluid flowing in non-conventional media: New numerical analysis

Atangana¹,

Jain²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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confidence: 99%

“…Of the total submitted papers, only 22 paper were accepted, providing a well-balanced representation of all continents. Sonal suggested a new numerical scheme for solving ordinary differential equations with power law setting differential operator [1]. A new numerical method for partial differential equations based on the two-step Laplace transform Adams-Bashforth approximation was applied by Jain and Alkahtani [2,3].…”

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confidence: 99%

Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

et al. 2018

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The complexities of nature have led researchers to develop more sophisticated and complex mathematical operators to capture, or accurately replicate observed realities [1][2][3][4][5][6][7]. In the recent decades, it has been widely accepted that the differential operators, based on the concept of rate of change, are not always suitable candidates for modelling complex real-world problems, as they cannot capture heterogeneous behaviours [8][9][10][11]. Specific examples are provided by long-range, random walk, anomalous diffusion, non-Markovian processes, fractal behaviours, and groundwater flowing in non-conventional media. To accurately replicate these natural processes, the concept of non-local differential operators was suggested as well as that of local differential operators along with power law setting. Non-local operators are grouped into three classes, including fractional differential operators with singular and non-local kernel, fractional differential operators with non-singular and local kernel, and fractional differential operators with non-singular and non-local kernel. Those with singular kernel were recently introduced and have been applied in several fields of science, technology and engineering, with great success [12][13][14][15][16][17][18]. One of their key properties is the crossover behaviour, described for given-range random walk, deterministic and stochastic behaviour. For instance, those with Mittag-Leffler kernel have crossover behaviour from waiting time distribution, mean square displacement to density probability distribution. These operators do not impose singularities to any model, especially those with no singularities, as in the case of singular kernel operators. The Focus Point was devoted to creating a discussion and to collecting the latest innovative results, where non-singular kernel differential operators and local operators, based on power law setting, are applied to more complex problems arising in physics and other related topics. We invited top researchers in the field and received many submissions from all the continents. Of the total submitted papers, only 22 paper were accepted, providing a well-balanced representation of all continents. Sonal suggested a new numerical scheme for solving ordinary differential equations with power law setting differential operator [1]. A new numerical method for partial differential equations based on the two-step Laplace transform Adams-Bashforth approximation was applied by Jain and Alkahtani [2,3]. Owolabi presented the model of a dynamical system using the Atangana-Baleanu differential operator [4], and Ahmed et al.suggested a modified exponential rational function method for non-linear fractional differential equations [5]. Allwright applied a power-law setting differential operator to the advection dispersion equation and suggested a new numerical scheme [6]. Karaagac presented an analysis of the cable equation using the Atangana-Baleanu differential operator [7]. Atangana and Alqahtani studied a tumour model with intrusive mo...

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A new numerical approximation of the fractal ordinary differential equation

Cited by 49 publications

References 16 publications

New aspects of poor nutrition in the life cycle within the fractional calculus

New aspects of poor nutrition in the life cycle within the fractional calculus

Models of fluid flowing in non-conventional media: New numerical analysis

Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation

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