Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations

Jumarie, Guy

doi:10.1016/j.chaos.2004.03.020

Cited by 47 publications

(16 citation statements)

References 22 publications

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“…Fractional calculus has become of increasing use for analyzing not only stochastic processes driven by fractional Brownian processes [1][2][3][4][5][6][7][8][9][10][11], but also nonrandom fractional phenomena in physics [12][13][14][15][16][17], like the study of porous systems, for instance, and quantum mechanics [18][19][20][21][22][23][24]. Whichever the flamework is, we believe that the very reason for introducing and using fractional derivative is to deal with nondifferentiable functions.…”

Section: Introductionmentioning

confidence: 99%

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

Jumarie¹

2006

Computers & Mathematics with Applications

921

516

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The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form fwhere E~ is the Mittag-Leffler function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent tile problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in tile form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

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Section: Introductionmentioning

confidence: 99%

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

Jumarie¹

2006

Computers & Mathematics with Applications

921

516

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“…The diffusion and drift coefficient may also be functions depending on the space variable ξ and the time τ to be in the form of FokkerPlanck equation(FPE) [3]. For some applications of this equation, see [4,5,6].…”

Section: Introductionmentioning

confidence: 99%

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Hikal¹,

Ibrahim²

2015

Int. J. of Pure and Appl. Math.

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In this paper, the Adomian's decomposition method (ADM) is considered to solve a fractional advection-dispersion model. This model can be represented if the first order derivative in time is replaced by the Caputo fractional derivative of order α (0 < α ≤ 1). In addition, the space derivative orders are replaced by the alternative orders 0 < β ≤ 1 and 1 < γ ≤ 2. The obtained solutions are formulated in a convergent infinite series in terms of Mittage-Leffler functions. Finally, two illustrative examples are introduced to ensure the effectiveness of the used method.

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“…This equality is based on Taylor's series expressed in terms of fractional derivatives. Using the notation (Jumarie, 2004) db(t, a) = σw(t) (dt) α ,…”

Section: Discussionmentioning

confidence: 99%

Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations

Cited by 47 publications

References 22 publications

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence

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