On a connection between the discrete fractional Laplacian and superdiffusion

Ciaurri, Óscar; Lizama, Carlos; Roncal, Luz; Malumbres, Juan

doi:10.1016/j.aml.2015.05.007

Cited by 24 publications

(31 citation statements)

References 11 publications

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“…We observe that the discrete fractional Laplace operator (−Δ d ) α is bounded on

ℓ^{p} false(double-struckZ, X false)

for any Banach space X and 1 ≤ p ≤ ∞ . Indeed, by Ciaurri et al,, p.121 we have that

K^{α} false(n false) \sim \frac{C}{n^{2 α + 1}}

, and hence, the result follows from a simple application of the Young inequality. That is,

‖ {(- {normalΔ}_{d})}^{α} f {false‖}_{p} = ‖ K^{α} * f {false‖}_{p} \leq ‖ K^{α} {false‖}_{1} ‖ f {false‖}_{p} .

For more information on the discrete fractional Laplacian, we refer to previous studies() and the references therein.…”

Section: Preliminariesmentioning

confidence: 75%

“…Let (−Δ) α be the fractional Laplacian of order 0< α <1. We recall from Ciaurri et al, section 3; that its discrete counterpart is defined as the discrete convolution operator in the following way (see, eg, Bogdan et al and references therein):

{false(- Δ_{d} false)}^{α} f false(n false) = \sum_{k \in double-struckZ} K^{α} false(n - k false) f false(k false), n \in double-struckZ, f \in l^{2} false(double-struckZ false),

where the coefficients K α are given by (see Ciaurri et al):

K^{α} false(n false) = \frac{1}{2 π} \int_{- π}^{π} {false(4 \sin^{2} false(θ false/ 2 false) false)}^{α} e^{- i n θ} d θ = \frac{{false(- 1 false)}^{n} normalΓ false(2 α + 1 false)}{normalΓ false(1 + α + n false) normalΓ false(1 + α - n false)}, n \in double-struckZ .

This sequence encapsulates all the information about the discrete fractional Laplacian and will be very important in what follows. A graphical representation is given in Ciaurri et al Some properties are K α ( n )>0 only when n =0,

\sum_{n \in double-struckZ} K^{σ} false(n false) = 0

,, Proposition 1

false‖ K^{σ} ‖_{1} = 2 \frac{normalΓ false(1 + 2 σ false)}{normalΓ {false(1 + σ false)}^{2}},

, Lemma 3.2 and

false| K^{α} false(n false) false| \sim \frac{normalΓ false(2 α + 1 false)}{π} false| n |^{- 2 α - 1}

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Several authors have studied equations involving the discrete fractional Laplacian, see, for example, the references. 4,[7][8][9][10] As pointed out in Lizama and Roncal, 4 lattice differential equations with a discrete fractional Laplacian arise in the quest to understand the behavior of processes related to small objects where the continuum limit cannot describe events on length scales comparable with nanometers. The lattice approach gives a possible microstructural basis for anomalous diffusion in media that are characterized by the nonlocality of power law type.…”

Section: )mentioning

confidence: 99%

“…This sequence encapsulates all the information about the discrete fractional Laplacian and will be very important in what follows. A graphical representation is given in Ciaurri et al 8 Some properties are K (n) > 0 only when n = 0, ∑ n∈Z K (n) = 0, 4, Proposition 1 ||K || 1 = 2 Γ(1+2 ) Γ(1+ ) 2 , 4, Lemma 3.2 and |K (n)| ∼ Γ(2 +1) |n| −2 −1 . In the borderline case = 1, we have that…”

Section: The Discrete Fractional Laplace Operatormentioning

confidence: 99%

“…Several authors have studied equations involving the discrete fractional Laplacian, see, for example, the references. ()…”

Section: Introductionmentioning

confidence: 99%

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Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

González-Camus

Keyantuo

Lizama

et al. 2019

Math Methods in App Sciences

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We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem double-struckDtβufalse(n,tfalse)=−false(−Δdfalse)αufalse(n,tfalse)+gfalse(n,tfalse), where 0<β ≤ 2, 0<α ≤ 1, n∈double-struckZ, (−Δd)α is the discrete fractional Laplacian, and double-struckDtβ is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.

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“…We observe that the discrete fractional Laplace operator (−Δ d ) α is bounded on

ℓ^{p} false(double-struckZ, X false)

for any Banach space X and 1 ≤ p ≤ ∞ . Indeed, by Ciaurri et al,, p.121 we have that

K^{α} false(n false) \sim \frac{C}{n^{2 α + 1}}

, and hence, the result follows from a simple application of the Young inequality. That is,

‖ {(- {normalΔ}_{d})}^{α} f {false‖}_{p} = ‖ K^{α} * f {false‖}_{p} \leq ‖ K^{α} {false‖}_{1} ‖ f {false‖}_{p} .

For more information on the discrete fractional Laplacian, we refer to previous studies() and the references therein.…”

Section: Preliminariesmentioning

confidence: 75%

{false(- Δ_{d} false)}^{α} f false(n false) = \sum_{k \in double-struckZ} K^{α} false(n - k false) f false(k false), n \in double-struckZ, f \in l^{2} false(double-struckZ false),

where the coefficients K α are given by (see Ciaurri et al):

K^{α} false(n false) = \frac{1}{2 π} \int_{- π}^{π} {false(4 \sin^{2} false(θ false/ 2 false) false)}^{α} e^{- i n θ} d θ = \frac{{false(- 1 false)}^{n} normalΓ false(2 α + 1 false)}{normalΓ false(1 + α + n false) normalΓ false(1 + α - n false)}, n \in double-struckZ .

\sum_{n \in double-struckZ} K^{σ} false(n false) = 0

,, Proposition 1

false‖ K^{σ} ‖_{1} = 2 \frac{normalΓ false(1 + 2 σ false)}{normalΓ {false(1 + σ false)}^{2}},

, Lemma 3.2 and

false| K^{α} false(n false) false| \sim \frac{normalΓ false(2 α + 1 false)}{π} false| n |^{- 2 α - 1}

.…”

Section: Preliminariesmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: The Discrete Fractional Laplace Operatormentioning

confidence: 99%

“…Several authors have studied equations involving the discrete fractional Laplacian, see, for example, the references. ()…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

González-Camus

Keyantuo

Lizama

et al. 2019

Math Methods in App Sciences

Self Cite

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On fractional semidiscrete Dirac operators of Lévy–Leblond type

Faustino

2023

Mathematische Nachrichten

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In this paper, we introduce a wide class of space‐fractional and time‐fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space‐time lattice hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty )$ (h>0$h>0$), resembling to fractional semidiscrete counterparts of the so‐called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup {}exp(−teiθ(−normalΔh)α)t≥0$\left\lbrace \exp (-te^{i\theta }(-\Delta _h)^{\alpha })\right\rbrace _{t\ge 0}$, carrying the parameter constraints 0<α≤1$0<\alpha \le 1$ and |θ|≤απ2$|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on hZn×[0,∞)$h{\mathbb {Z}}^n\times [0,\infty )$.

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Well-Posedness for Fractional Cauchy Problems Involving Discrete Convolution Operators

González-Camus

2023

Mediterr. J. Math.

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On a connection between the discrete fractional Laplacian and superdiffusion

Cited by 24 publications

References 11 publications

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

On fractional semidiscrete Dirac operators of Lévy–Leblond type

Well-Posedness for Fractional Cauchy Problems Involving Discrete Convolution Operators

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