Fractional Differential Equations

Milici, Constantin; Drăgănescu, Gheorghe; Machado, J.A.T.

doi:10.1007/978-3-030-00895-6_4

Cited by 6 publications

(4 citation statements)

References 20 publications

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“…Here, we illustrate the basic solution procedure of 4th-order RK (RK4) scheme with fractional order [29,30]. Now, consider equation with u(t) ä C p+1 ([t 0 , t 0 + T])…”

Section: Solution Procedures Of the Projected Methodsmentioning

confidence: 99%

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Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

İlhan¹

2022

Phys. Scr.

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The coupled system exemplifying the damped and driven oscillators (namely, Duffing equations) is examined with a familiar and robust numerical method. In the framework, we hired a reliable and most cited Caputo fractional operator to capture essential and stimulating behaviours of the hired physical model. The existence of the solution for the considered model is presented, and we captured the nature of the strange attractor for the Duffing equations with a period of the driving force. The effect of chaotic nature associated parameter is illustrated. The essence of generalizing the classical models is presented with plots, and associated consequences are demonstrated for the substantial time period.

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“…Here, we illustrate the basic solution procedure of 4th-order RK (RK4) scheme with fractional order [29,30]. Now, consider equation with u(t) ä C p+1 ([t 0 , t 0 + T])…”

Section: Solution Procedures Of the Projected Methodsmentioning

confidence: 99%

“…These German mathematicians proposed many numerical schemes to solve differential equations exemplifying atomic spectra and airfoil lift in aerodynamics [26][27][28]. In this paper, we used the results derived by researchers in [29,30] with the projected scheme.…”

Section: Introductionmentioning

confidence: 99%

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

İlhan¹

2022

Phys. Scr.

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“…Fractional differential operators in mathematical modeling are well-known and it has been successful in the fields like science, engineering, and epidemiology. We give a few fundamental definitions along with some proprieties that be will used in the rest of the study 30 , 31 .…”

Section: Basic Definitionsmentioning

confidence: 99%

Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach

Liu,

Farid,

Ullah

et al. 2023

Sci Rep

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This present paper aims to examine various epidemiological aspects of the monkeypox viral infection using a fractional-order mathematical model. Initially, the model is formulated using integer-order nonlinear differential equations. The imperfect vaccination is considered for human population in the model formulation. The proposed model is then reformulated using a fractional order derivative with power law to gain a deeper understanding of disease dynamics. The values of the model parameters are determined from the cumulative reported monkeypox cases in the United States during the period from May 10th to October 10th, 2022. Besides this, some of the demographic parameters are evaluated from the population of the literature. We establish sufficient conditions to ensure the existence and uniqueness of the model’s solution in the fractional case. Furthermore, the stability of the endemic equilibrium of the fractional monkeypox model is presented. The Lyapunov function approach is used to demonstrate the global stability of the model equilibria. Moreover, the fractional order model is numerically solved using an efficient numerical technique known as the fractional Adams-Bashforth-Moulton method. The numerical simulations are conducted using estimated parameters, considering various values of the fractional order of the Caputo derivative. The finding of this study reveals the impact of various model parameters and fractional order values on the dynamics and control of monkeypox.

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“…(2006), Kiryakova (1994), Li et al. (2011), Magin (2006), Milici et al. (2018), Oldham and Spanier (1974), Rudolf (2000), Sabatier et al.…”

Section: Introductionmentioning

confidence: 99%

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

Ghosh

2024

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PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

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Fractional Differential Equations

Cited by 6 publications

References 20 publications

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

Mathematical assessment of monkeypox disease with the impact of vaccination using a fractional epidemiological modeling approach

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

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