Automodel solutions for superdiffusive transport by the L\'evy walks

TL;DR

This paper extends the automodel solution method for superdiffusive transport equations to include finite carrier velocity, modeling more realistic scenarios like photon transport and predator migration with retardation effects.

Contribution

It introduces an automodel solution for superdiffusive transport with finite velocity, covering Le9vy walk + rests, applicable to various physical and biological systems.

Findings

Derived a generalized automodel solution for superdiffusive transport.

Verified the solution against exact results for specific power-law PDFs.

Applicable to photon transport, heat transfer, and ecological migration models.

Abstract

The method of approximate automodel solution for the Green's function of the time-dependent superdiffusive (nonlocal) transport equations (J. Phys. A: Math. Theor. 49 (2016) 255002) is extended to the case of a finite velocity of carriers. This corresponds to extension from the L\'evy flights-based transport to the transport of the type, which belongs to the class of "L\'evy walk + rests", to allow for the retardation effects in the L\'evy flights. This problem covers the cases of the transport by the resonant photons in astrophysical gases and plasmas, heat transport by electromagnetic waves in plasmas, migration of predators, and other applications. We treat a model case of one-dimensional transport on a uniform background with a simple power-law step-length probability distribution function (PDF). A solution for arbitrary superdiffusive PDF is suggested, and the verification of solution for a particular power law PDF, which corresponds, e.g., to the Lorentzian wings of atomic spectral line shape for emission of photons, is carried out using the computation of the exact solution.