An Introduction to Fractional Calculus

Butzer, Paul L.; Westphal, U.

doi:10.1142/9789812817747_0001

Cited by 349 publications

(210 citation statements)

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“…It is here briefly recalled that fractional calculus is a valuable tool to observe the influence of memory effects on the dynamics of systems [41][42][43][44][45][46], and has been recently used in epidemiological models [47][48][49][50]. Typically, the evolution of epidemiological models is described with differential equations, the derivatives being of integer order.…”

Section: Introductionmentioning

confidence: 99%

Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model

et al. 2017

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Memory has a great impact on the evolution of every process related to human societies. Among them, the evolution of an epidemic is directly related to the individuals' experiences. Indeed, any real epidemic process is clearly sustained by a non-Markovian dynamics: memory effects play an essential role in the spreading of diseases. Including memory effects in the susceptible-infected-recovered (SIR) epidemic model seems very appropriate for such an investigation. Thus, the memory prone SIR model dynamics is investigated using fractional derivatives. The decay of long-range memory, taken as a power-law function, is directly controlled by the order of the fractional derivatives in the corresponding nonlinear fractional differential evolution equations. Here we assume "fully mixed" approximation and show that the epidemic threshold is shifted to higher values than those for the memoryless system, depending on this memory "length" decay exponent. We also consider the SIR model on structured networks and study the effect of topology on threshold points in a non-Markovian dynamics. Furthermore, the lack of access to the precise information about the initial conditions or the past events plays a very relevant role in the correct estimation or prediction of the epidemic evolution. Such a "constraint" is analyzed and discussed.

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Section: Introductionmentioning

confidence: 99%

Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model

et al. 2017

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“…(a) α = 2 Taking α i = 1, 1 ≤ i ≤ n − 1, β = 1, applying the generalized n-dimensional transform to both sides of (33) - (34), and using theorem (17), we get…”

Section: Example 51 Consider the Following N-dimensional Heat-like mentioning

confidence: 99%

Analytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional Differential Transform Method

Baranwal

Pandey

Tripathi

et al. 2011

Zeitschrift Für Naturforschung A

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In this paper, we have introduced a generalized n-dimensional differential transform method to propose a user friendly algorithm to obtain the closed form analytic solution for n-dimensional fractional heat-and wave-like equations. Three examples are given to establish the simplicity of the algorithm. In Example 5.3, we show that ten terms of the series representing the solution, even in fractional order, give a very accurate solution.

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“…A theory of the generalized fractional calculus, as well as the corresponding applications, can be found in a monograph written by Kiryakova [23]. We also mention a few the most cited papers on this subject [17,26,27,5].…”

Section: Introductionmentioning

confidence: 99%