New concept in calculus: Piecewise differential and integral operators

Atangana, Abdon; Araz, Seda İğret

doi:10.1016/j.chaos.2020.110638

Cited by 160 publications

(99 citation statements)

References 13 publications

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“…In this section, we present a numerical scheme based on Newton polynomial for the numerical solution of the considered model with piecewise di¤erential and integral operators [14,15]. We start with the piecewise SEIR model with classical and stochastic Caputo-Fabrizio case which is given by 8 > > < > > :…”

Section: Numerical Solution Of the Model With Exponential Decay Processmentioning

confidence: 99%

“…In this section, we present a numerical scheme based on Newton polynomial for the numerical solution of the considered model with piecewise differential and integral operators [14,15]. We start with the piecewise SEIR model with classical and stochastic Caputo-Fabrizio case which is given by Here Applying the associated integral, we can have We divide [0, T ] in two Interpolating f i ( t, X i ) using the Newton polynomial within [ t n , t n +1 ] yields The parameters and initial conditions are given as and The simulation of the numerical solution of the model is depicted in Figure 2.…”

Section: Numerical Solution Of the Model With Exponential Decay Processmentioning

confidence: 99%

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Advanced analysis in epidemiological modeling: Detection of wave

Atangana

Araz

2021

Preprint

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Some mathematical concepts have been used in the last decades to predict the behavior of spread of infectious diseases. Among them, the reproductive number concept has been used in several published papers for study the stability of the spread. Some conditions were suggested to predict there would be either stability or instability. An analysis was also suggested to determine conditions under which infectious classes will increase or die out. Some authors pointed out limitations of the reproductive number, as they presented its inability to fairly help understand the spread patterns. The concept of strength number and analysis of second derivatives of the mathematical models were suggested as additional tools to help detect waves. In this paper, we aim at applying these additional analyses in a simple model to predict the future. Keywords: Strength number, second derivative analysis, waves, piecewise modeling.

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Section: Numerical Solution Of the Model With Exponential Decay Processmentioning

confidence: 99%

Section: Numerical Solution Of the Model With Exponential Decay Processmentioning

confidence: 99%

Advanced analysis in epidemiological modeling: Detection of wave

Atangana

Araz

2021

Preprint

Self Cite

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“…Many mathematical models on the elements various illnesses in term of non-integer order derivatives were proposed see for occasion [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] and the literature referenced therein. Fractal fractional calculus is the generalization of classical calculus [25] , [26] , [27] , [28] , [29] . To get a better insight into a mathematical model and to deeply understand phenomena, non-integer order operators can be used.…”

Section: Introductionmentioning

confidence: 99%

Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study

et al. 2022

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This research study consists of a newly proposed Atangana–Baleanu derivative for transmission dynamics of the coronavirus (COVID-19) epidemic. Taking the advantage of non-local Atangana–Baleanu fractional-derivative approach, the dynamics of the well-known COVID-19 have been examined and analyzed with the induction of various infection phases and multiple routes of transmissions. For this purpose, an attempt is made to present a novel approach that initially formulates the proposed model using classical integer-order differential equations, followed by application of the fractal fractional derivative for obtaining the fractional COVID-19 model having arbitrary order and the fractal dimension . With this motive, some basic properties of the model that include equilibria and reproduction number are presented as well. Then, the stability of the equilibrium points is examined. Furthermore, a novel numerical method is introduced based on Adams–Bashforth fractal-fractional approach for the derivation of an iterative scheme of the fractal-fractional ABC model. This in turns, has helped us to obtained detailed graphical representation for several values of fractional and fractal orders and , respectively. In the end, graphical results and numerical simulation are presented for comprehending the impacts of the different model parameters and fractional order on the disease dynamics and the control. The outcomes of this research would provide strong theoretical insights for understanding mechanism of the infectious diseases and help the worldwide practitioners in adopting controlling strategies.

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“…The representation of different real phenomena have formulated by fractional order integral or differential equation like mathematical fractional order model for microorganism population, a logistic non-linear model for the human population, tuberculosis model, dingy problem, hepatitis B, C models and the basic Lotka-Volterra models being the basis of all infectious problems [21] , [22] , [23] , [24] , [25] , [26] . The aforesaid problems have been analyzed for qualitative analysis with help of some well-known theorems of fixed point theory [27] , [28] , [29] , [30] . The feasibility and stability analyses have also been done through various theorems.…”

Section: Introductionmentioning

confidence: 99%

Fractal-fractional order mathematical vaccine model of COVID-19 under non-singular kernel

Algehyne

Ibrahim

2021

Chaos, Solitons & Fractals

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In this paper,the severe acute respiratory syndrome coronavirus (SARS-CoV-2) or COVID-19 is researched by employing mathematical analysis under modern calculus. In this context, the dynamical behavior of an arbitrary order p and fractal dimensional q problem of COVID-19 under Atangana Bleanu Capute (ABC) operator for the three cities, namely, Santos, Campinas, and Sao Paulo of Brazil are investigated as a case-study. The considered problem is analyzed for at least one solution and unique solution by the applications of the theorems of fixed point and non-linear functional analysis. The Ulam-Hyres stability condition via nonlinear functional analysis for the given system is derived. In order to perform the numerical simulation, a two-step fractional type, Lagrange plynomial (Adams Bashforth technique) is utilized for the present system. MATLAB simulation tools have been used for testing different fractal fractional orders considering the data of aforementioned three regions. The analysis of the results finally infer that, for all these three regions, the smaller order values provide better constraints than the larger order values.

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New concept in calculus: Piecewise differential and integral operators

Cited by 160 publications

References 13 publications

Advanced analysis in epidemiological modeling: Detection of wave

Advanced analysis in epidemiological modeling: Detection of wave

Modeling the dynamics of COVID-19 using fractal-fractional operator with a case study

Fractal-fractional order mathematical vaccine model of COVID-19 under non-singular kernel

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