In a recent paper, Saxena et al. [1] developed the solutions of three generalized fractional kinetic equations in terms of Mittag-Leffler functions. The object of the present paper is to further derive the solution of further generalized fractional kinetic equations. The results are obtained in a compact form in terms of generalized Mittag-Leffler functions. Their relation to fundamental laws of physics is briefly discussed.
An Alternative Method for Solving a Certain Class of Fractional Kinetic Equations
Saxena and Kalla（2007）最近给出了一种求解Haubold and Mathai（2000）和Saxena et al.（2002，2004 a，2004 b）早期求解的分数阶动力学方程的替代方法。这种方法也可用于求解比上述作者所解的分数阶动力学方程更一般的分数阶动力学方程。考虑到动力学方程在某些控制复杂系统中反应扩散和反常扩散的物理问题中的有用性和重要性，作者提出了另一种简单的方法来导出上述作者和Nonnenmacher和Metzler（1995）所解的分数阶动力学方程的广义形式的解.该方法依赖于Riemann-Liouville分数阶微积分算子的使用。应用Riemann-Liouville分数阶积分算子及其有趣的性质，可以直接求出分数阶动力学方程的解.这种方法不使用拉普拉斯变换。
We investigate the fractional diffusion approximation of a kinetic equation set in a bounded interval with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we show that the asymptotic density function is the {\it unique solution} of a fractional diffusion equation with Neumann boundary condition. This analysis completes a previous work by the same authors in which a limiting fractional diffusion equation was identified on the half-space, but the uniqueness of the solution (which is necessary to pro…
We investigate the fractional diffusion approximation of a kinetic equation in the upper-half plane with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time asymptotic, we derive a fractional diffusion equation with a nonlocal Neumann boundary condition for the density of particles. Interestingly, this asymptotic equation is different from the one derived by L. Cesbron in [7] in the case of specular reflection conditions at the boundary and does not seem to have receive a lot of attention previously.
We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian …
In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem.
We consider the limit of a linear kinetic equation, with reflection-transmission-absorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, re…
Computable Solutions of Fractional Reaction-Diffusion Equations Associated with Generalized Riemann-Liouville Fractional Derivatives of Fractional Order
Confined kinetics and heterogeneous diffusion driven by fractional Gaussian noise: A path integral approach
Institute for Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）；School of Natural and Computing Sciences, University of Aberdeen（阿伯丁大学自然与计算科学学院）；Department of Physics, Universidad Nacional de Colombia（哥伦比亚国立大学物理系）；Utrecht University（乌得勒支大学）；University of Aberdeen（阿伯丁大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）
In this paper, we propose the invariant subspace approach to find exact solutions of time-fractional partial differential equations (PDEs) with time delay. An algorithmic approach of finding invariant subspaces for the generalized non-linear time-fractional reaction-diffusion equations with time delay is presented. We show that the fractional reaction-diffusion equations with time delay admit several invariant subspaces which further yields several distinct analytical solutions. We also demonstrate how to derive exact solutions for time-fractional PDEs with multiple time delays. Finally, we ex…
In this paper we study generalized time-fractional diffusion equations on the Poincarè half plane $\mathbb{H}_2^+$. The time-fractional operators here considered are fractional derivatives of a function with respect to another function, that can be obtained by starting from the classical Caputo-derivatives essentially by means of a deterministic change of variable. We obtain an explicit representation of the fundamental solution of the generalized-diffusion equation on $\mathbb{H}_2^+$ and provide a probabilistic interpretation related to the time-changed hyperbolic Brownian motion. We finally…
