Tempered fractional calculus

Sabzikar, Farzad; Meerschaert, Mark M.; Chen, Jinghua

doi:10.1016/j.jcp.2014.04.024

Cited by 287 publications

(184 citation statements)

References 68 publications

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“…In order to overcome this modeling barrier, there are different techniques such as discarding the very large jumps and employing truncated Lévy flights [17], or, adding a high-order power-law factor [27]. However, the most popular, and perhaps most rigorous, approach to get finite moments is exponentially tempering the probability of large jumps of Lévy flights, which results in tempered-stable Lévy processes with finite moments [8,5,18,26]. The corresponding fluid (continuous) limit for such models yields the tempered fractional diffusion equation, which complements the previously known models in fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Tempered Fractional Sturm--Liouville EigenProblems

Zayernouri

Ainsworth

Karniadakis

2015

SIAM J. Sci. Comput.

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Continuum-time random walk is a general model for particle kinetics, which allows for incorporating waiting times and/or non-Gaussian jump distributions with divergent second moments to account for Lévy flights. Exponentially tempering the probability distribution of the waiting times and the anomalously large displacements results in tempered-stable Lévy processes with finite moments, where the fluid (continuous) limit leads to the tempered fractional diffusion equation. The development of fast and accurate numerical schemes for such nonlocal problems requires a new spectral theory and suitable choice of basis functions. In this study, we introduce two classes of regular and singular tempered fractional Sturm-Liouville problems of two kinds (TFSLP-I and TFSLP-II) of order ν ∈ (0, 2). In the regular case, the corresponding tempered differential operators are associated with tempering functions p I (x) = exp(2τ ) and p II (x) = exp(−2τ ), τ ≥ 0, respectively, in the regular TFSLP-I and TFSLP-II, which do not vanish in [−1, 1]. In contrast, the corresponding differential operators in the singular setting are associated with different forms of p I (x) = exp(2τ )(1 − x) 1+α (1 + x) 1+β and p II (x) = exp(−2τ )(1 − x) 1+α (1 + x) 1+β , vanishing at x = ±1 in the singular TFSLP-I and TFSLP-II, respectively. The aforementioned tempered fractional differential operators are both of tempered Riemann-Liouville and tempered Caputo type of fractional order μ = ν/2 ∈ (0, 1). We prove the well-posedness of the boundary-value problems and that the eigenvalues of the regular tempered problems are real-valued and the corresponding eigenfunctions are orthogonal. Next, we obtain the explicit eigensolutions to TFSLP-I and -II as nonpolynomial functions, which we define as tempered Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with TFSLP-I and -II. Finally, we introduce these eigenfunctions as new basis (and test) functions for spectrally accurate approximation of functions and tempered-fractional differential operators. To this end, we further develop a Petrov-Galerkin spectral method for solving tempered fractional ODEs, followed by the corresponding stability and convergence analysis, which validates the achieved spectral convergence in our simulations.

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Section: Introductionmentioning

confidence: 99%

Tempered Fractional Sturm--Liouville EigenProblems

Zayernouri

Ainsworth

Karniadakis

2015

SIAM J. Sci. Comput.

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“…We begin with the definitions of α-th order left and right Riemann-Liouville (RL) normalized tempered fractional derivatives [5,7,21]. …”

Section: Derivation Of the Quasi-compact Approximations For The Tempementioning

confidence: 99%

“…Replacing the way of truncation in [3] with exponentially truncating the Lévy flight, some analytic results to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process are presented [4]. Exponentially tempering the power-law PDF of waiting times or jump lengths seems to become popular nowadays, since it can bring many technical conveniences [5], e.g., making the tempered stochastic process still be Lévy process. For capturing the slow convergence of sub-diffusion to a diffusion limit for passive tracers in heterogeneous media, the model with exponentially tempered power-law waiting time distribution is introduced in [6].…”

Section: Introductionmentioning

confidence: 99%

Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

Deng

2017

Applied Numerical Mathematics

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Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

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“…The nonlinear dynamics model based on fractional calculus [4][5][6][7][8] usually can be formalized as d α f dt α = g(k, f ). The function g is constructed by the scientists according to the real system corresponding to the model.…”

Section: Introductionmentioning

confidence: 99%

Optimal nonlinear dynamics modeling method for big data based on fractional calculus and simulated annealing

Zhu

2015

Nonlinear Dyn

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No nonlinear dynamics modeling method for big data has been proposed, and no nonlinear dynamics modeling method based on fractional calculus and simulated annealing has been proposed until now. This paper provided the optimal nonlinear dynamics modeling method for big data based on fractional calculus and simulated annealing, which can find the optimal nonlinear dynamics model for big data based on fractional-order calculus. This paper also provided another alternative method that the nonlinear dynamics modeling method for big data is based on integer-order calculus. This paper took price, supply-demand ratio and selling rate big data as an application to illustrate the proposed methods. And this paper compared the optimal nonlinear dynamics modeling method for big data based on fractional calculus and simulated annealing with the alternative method based on integer-order calculus, and concluded that the nonlinear dynamics model found by the method based on fractional calculus and simulated annealing is closer to the real system than the nonlinear dynamics model found by the method based on integer-order calculus.

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Tempered fractional calculus

Cited by 287 publications

References 68 publications

Tempered Fractional Sturm--Liouville EigenProblems

Tempered Fractional Sturm--Liouville EigenProblems

Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

Optimal nonlinear dynamics modeling method for big data based on fractional calculus and simulated annealing

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