On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel

Băleanu, Dumitru; Jajarmi, Amin; Hajipour, Mojtaba

doi:10.1007/s11071-018-4367-y

Cited by 142 publications

(95 citation statements)

References 31 publications

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“…3. for m = 3, if the discriminant of H (Λ), D(H ) is positive, then Routh-Hurwitz conditions are the necessary and sufficient conditions for (11) (11) is satisfied for all σ ∈ [0, 1). 6.…”

Section: Theoremmentioning

confidence: 99%

“…Although [2][3][4][5][6] has completed a great deal of work on dynamic modeling of influenza, it is limited to ordinary differential equations. However, currently, it has been found that the use of fractional differential equations to model many different fields of phenomena has been very successful [7][8][9][10][11][12][13][14][15][16][17][18]. For instance, in mathematical epidemiology, Ebola virus epidemic has been modeled with fractional-order differential equations by [19].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model

Yaro

Apeanti

Akuamoah

et al. 2019

Int. J. Appl. Comput. Math

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The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number R 0 . We discuss the stability of the disease-free equilibrium when R 0 < 1, and the necessary conditions for the stability of the endemic equilibrium when R 0 > 1. A sensitivity analysis shows that R 0 is most sensitive to the probability of the disease transmission rate. The results from the numerical simulations of optimal control strategies disclose that the utmost way of controlling or probably eradicating the transmission of respiratory diseases should be quarantining the exposed individuals, monitoring and treating infected people for a substantial period.

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Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model

Yaro

Apeanti

Akuamoah

et al. 2019

Int. J. Appl. Comput. Math

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“…These quantities are essentially defined by convolutions integrals with kernel of type power law functions that define the well-know Caputo, Riemann-Liouville and Riesz-Feller fractional operators, which are focus of recent mathematical applications [19][20][21][22][23][24][25][26]. Nowadays, a new family of kernels has been used to describe generalised diffusive models [27][28][29][30][31][32][33][34]. In this context, it is possible to define the generalised diffusion equation…”

Section: Introductionmentioning

confidence: 99%

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Santos

2018

Fractal Fract

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Abstract:The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context.

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“…Example of Equation(10) and error Equation(19), and in this is table, y is constant and t, x, n are variable with = 0.5, 3 = 0.6 3.44413849 × 10 −4 4.13886868 × 10 −6 3.83621593 × 10 −7 1 = 0.1, 2 = 0.5, 3 = 0.9 8.22822045 × 10 −4 4.20641342 × 10 −6 3.83627607 × 10 −7 1 = 0.3, 2 = 0.4, 3 = 0.8 4.69460091 × 10 −4 4.16779793 × 10 −6 3.83664363 × 10 −7 1 = 0.1, 2 = 0.2, 3 = 0.3 3.57606141 × 10 −4 1.26366819 × 10 −6 9.50441948 × 10 −7…”

mentioning

confidence: 99%

“…Example of Equation(10) and error Equation(19), and in this table, y is constant and t, x, n are variable with 1 , 2 = 0.2 7.33866827 × 10 −6 1.99765798 × 10 −6 3.76046720 × 10 −7 1 = 0.3, 2 = 0.6 4.24195754 × 10 −6 1.14902327 × 10 −6 2.18345928 × 10 −7 1 = 0.1, 2 = 0.6 5.06981634 × 10 −5 1.37448038 × 10 −6 2.59171862 × 10 −7 1 = 0.3, 2 = 0.4 5.06981634 × 10 −5 1.36667106 × 10 −6 2.55903570 × 10 −7 = 0.5, 3 = 0.6 2.53215157 × 10 −6 6.69354326 × 10 −7 1.279053657 × 10 −7 1 = 0.1, 2 = 0.5, 3 = 0.9 2.53215157 × 10 −6 5.69364388 × 10 −7 1.1932046303 × 10 −7 1 = 0.3, 2 = 0.4, 3 = 0.8 2.53215157 × 10 −6 5.01438508 × 10 −7 1.0589717284 × 10 −7 1 = 0.2, 2 = 0.6, 3 = 0.7 5.55083179 × 10 −6 1.03736449 × 10 −7 2.2208255269 × 10 −7…”

mentioning

confidence: 99%

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

Ramezani

2019

Math Methods in App Sciences

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In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for 0<αi<1,i=1,2,⋯, double-struckN. The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation.

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On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel

Cited by 142 publications

References 31 publications

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model

Analysis and Optimal Control of Fractional-Order Transmission of a Respiratory Epidemic Model

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

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