Lévy Processes in the Physical Sciences

Woyczyński, Wojbor A.

doi:10.1007/978-1-4612-0197-7_11

Cited by 127 publications

(106 citation statements)

References 35 publications

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“…(28) differs from the fractional Burgers equation [18,19] in the structure of the nonlinear term. Nevertheless, an analysis similar to that in Refs.…”

Section: Ginzburg-landau Equation With Fractional Derivativesmentioning

confidence: 99%

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Fractional Ginzburg–Landau equation for fractal media

Tarasov

Zaslavsky

2005

Physica A: Statistical Mechanics and its Applications

185

129

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“…(28) differs from the fractional Burgers equation [18,19] in the structure of the nonlinear term. Nevertheless, an analysis similar to that in Refs.…”

Section: Ginzburg-landau Equation With Fractional Derivativesmentioning

confidence: 99%

“…Nevertheless, an analysis similar to that in Refs. [18,19] may be performed to obtain some estimates on the solution.…”

Section: Ginzburg-landau Equation With Fractional Derivativesmentioning

confidence: 99%

Fractional Ginzburg–Landau equation for fractal media

Tarasov

Zaslavsky

2005

Physica A: Statistical Mechanics and its Applications

185

129

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“…The study of PDEs involving g λ is motivated by a number of physical problems, such as overdriven detonations in gases [9] or anomalous diffusion in semiconductor growth [27], and by mathematical models in finance (see below for references). We consider perturbations by g λ of Hamilton-Jacobi equations or scalar conservation laws, that is to say    ∂ t u(t, x) + g λ [u(t, ·)](x) = F (t, x, u(t, x), ∇u(t, x)) t > 0 , x ∈ R N , u(0, x) = u 0 (x)…”

Section: Introductionmentioning

confidence: 99%

Fractal First-Order Partial Differential Equations

Droniou

Imbert

2006

Arch Rational Mech Anal

150

222

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The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton-Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one side, the key a priori estimates for the scalar conservation law and the Hamilton-Jacobi equation and, on the other side, the smoothing effect of the operator. As far as Hamilton-Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law. Mathematical subject classifications: 35B45, 35B65, 35A35, 35S30Key words. Lévy operator -fractal conservation laws -maximum principle -non-local Hamilton-Jacobi equations -smoothing effect -a priori estimates -global-in-time existence -rate of convergence.

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“…While the linear FE has attracted fairly broad research activity, the study of nonlinear FE is at its very beginning [11]- [14]. This paper consists of two parts on the FE properties.…”

Section: Introductionmentioning

confidence: 99%

Some applications of fractional equations

Weitzner

Zaslavsky

2003

Communications in Nonlinear Science and Numerical Simulation

161

114

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We present two observations related to the application of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power tails in the long time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear media with fractal properties is considered. A corresponding fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations is proposed.

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Lévy Processes in the Physical Sciences

Cited by 127 publications

References 35 publications

Fractional Ginzburg–Landau equation for fractal media

Fractional Ginzburg–Landau equation for fractal media

Fractal First-Order Partial Differential Equations

Some applications of fractional equations

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