2018
DOI: 10.1137/17m1116222
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Abstract: For characterizing the Brownian motion in a bounded domain: Ω, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on the contrary, for the Lévy flights or tempered Lévy flights in a bounded domain, it involves the information of a solution in the complementary set of Ω, i.e., R n \Ω, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probabilit… Show more

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Cited by 88 publications
(72 citation statements)
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References 30 publications
(49 reference statements)
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“…The first formalism presented in this review was related to the fractional walker, this method has a great focus of investigation in the present day, because daily new techniques, theorems and theories continue to be developed by the mathematicians who investigate the fractional calculus [112,113,114,115,116,117,118,119,120,121,122,123,124]. With all this progress of the fractional calculus, physics advances together, since it allows the modelling of a series of interesting problems in physics, such as diffusion equation with tempered derivatives [125,126,127,128,129,130], memory systems [131,132,133,134,135], non-homogeneous systems [60,136,137,138], etc [139,140,141]. One of the most important roles of fractional calculus in physics has been to introduce ever more sophisticated memory kernels, which capture more precisely the processes observed in the real world [142].…”
Section: Brief Discussion and Some Considerationsmentioning
confidence: 99%
“…The first formalism presented in this review was related to the fractional walker, this method has a great focus of investigation in the present day, because daily new techniques, theorems and theories continue to be developed by the mathematicians who investigate the fractional calculus [112,113,114,115,116,117,118,119,120,121,122,123,124]. With all this progress of the fractional calculus, physics advances together, since it allows the modelling of a series of interesting problems in physics, such as diffusion equation with tempered derivatives [125,126,127,128,129,130], memory systems [131,132,133,134,135], non-homogeneous systems [60,136,137,138], etc [139,140,141]. One of the most important roles of fractional calculus in physics has been to introduce ever more sophisticated memory kernels, which capture more precisely the processes observed in the real world [142].…”
Section: Brief Discussion and Some Considerationsmentioning
confidence: 99%
“…where C α,λ i (i = 1, 2) is a positive constant, α ∈ (0, 1) (1, 2) is called the stable index, and λ i is the positive tempering parameter. Here we consider 'symmetric' tempered Lévy process, i.e., [4]). The mean exit time for the solution orbit X t in Eq.…”
Section: Met For One-dimensional Casementioning
confidence: 99%
“…When the components of the tempered Lévy process L t are independent, the particles (or solutions) spread in either horizontal or vertical direction [4]. The finite measure Γ in (3.12) concentrates on the points of intersection of unit circle S 2 and axes.…”
Section: Met For the Horizontal-vertical Casementioning
confidence: 99%
“…random variables and their length obeys Gaussian distribution. The characteristic function of X(t) has a specific form as [9,Eq. 9] E(e ik·X ) = R n e ik·X p(X, t)dX = e ζt(Φ0(k)−1) ,…”
Section: Nonlocal Normal Diffusionmentioning
confidence: 99%
“…then its probability density function (PDF) of the position of the particles solves ∂p(X, t) ∂t = ∆ β/2 p(X, t) or ∂p(X, t) ∂t = ∆ β/2,λ p(X, t), (1.4) where the operators ∆ β/2 and ∆ β/2,λ are defined in [9,Eq. 34] by Fourier transform g(k) := F [g(X)](k) = R n e ik·X g(X)dX with F [∆ β/2 g(X)] = −|k| βĝ (k) and…”
mentioning
confidence: 99%