Abstract

The general and approximate self-similar solutions for the Green’s function have been obtained for a wide class of integrodifferential equations for the two- and three-dimensional (in spatial coordinates) nonstationary superdiffusive transport of a perturbation of a homogeneous medium for a finite fixed velocity of carriers. This problem concerns the transport of resonance radiation in astrophysical gases and plasma, migration of animals, and the transfer of energy of electromagnetic waves in the plasma. The case of a model free path distribution function decreasing by a power law with increasing distance has been considered. Numerical calculations have been performed for two particular types of free path distribution functions, including the case with a Lorentzian shape of wings of the spectral line profile for the emission of photons by atoms or ions. The method developed in [A.B. Kukushkin and P.A. Sdvizhenskii, J. Phys. A: Math. Theor. 49, 255002 (2016)] for an infinite velocity of carriers has been used to construct the approximate self-similar solution. The inclusion of a finite velocity corresponds to the generalization of transport based on Lévy flights to transport based on “Lévy walks with stops.” The self-similar solution for arbitrary one-dimensional superdiffusive transport obtained in [A.B. Kukushkin and A.A. Kulichenko, Phys. Scripta 94, 115009 (2019)] has been applied to the case of the two- and three-dimensional transports. The accuracy of the self-similar solution has been tested by comparing it to the numerically calculated general solution.

中文翻译：

---

均质介质中基于LévyWalk的超扩散运输：一般和近似自相似解

摘要

对于一类有限的固定速度的均质介质扰动的二维和三维（在空间坐标中）非平稳超扩散输运，我们获得了格林函数的一般和近似自相似解。的载体。这个问题涉及天体气体和等离子体中共振辐射的传输，动物的迁移以及等离子体中电磁波能量的传输。考虑了模型自由路径分布函数随距离增加而幂函数减少的情况。已经针对两种特殊类型的自由路径分布函数执行了数值计算，包括具有谱线轮廓的机翼的洛伦兹形状的情况，用于通过原子或离子发射光子。[AB Kukushkin和PA Sdvizhenskii，J. Phys。答：数学。理论。49，255002（2016）]为载体的无限速度已被用于构建近似自相似的解决方案。包含有限速度对应于基于Lévy航班的运输到基于“Lévy有站步行”的运输的一般化。在[AB Kukushkin and AA Kulichenko，Phys。SCRIPTA 94，115009（2019）]已被施加到所述二维和三维运输的情况下。通过将自相似解与数值计算的一般解进行比较，测试了其准确性。