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Register a new userRetarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
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Authors
A. V. Chechkin
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We propose diffusion-like equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, can not be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish the relation to the Continuous Time Random Walk theory. We show that the distributed order time fractional diffusion equation describes the sub-diffusion random process which is subordinated to the Wiener process and whose diffusion exponent diminishes in time (retarding sub-diffusion) leading to superslow diffusion, for which the square displacement grows logarithmically in time. We also demonstrate that the distributed order space fractional diffusion equation describes super-diffusion phenomena when the diffusion exponent grows in time (accelerating super-diffusion).
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Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.
Super-diffusion, characterized by a spreading rate $t^{1/alpha}$ of the probability density function $p(x,t) = t^{-1/alpha} p left( t^{-1/alpha} x , 1 right)$, where $t$ is time, may be modeled by space-fractional diffusion equations with order $1 < alpha < 2$. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond than this range ($alpha > 2$), known as high-order diffusion, or hyperdiffusion. Recently, space-time duality, motivated by Zolotarevs duality law for stable densities, established a link between time-fractional and space-fractional diffusion for $1 < alpha leq 2$. This paper extends space-time duality to fractional exponents $1<alpha leq 3$, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order $2<alpha leq 3$ model sub-diffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.
In this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the Fourier-Laplace transform method. We represent solutions in terms of infinite series in Fox $H$-functions. The fractional and second moments are derived by using Mittag-Leffler functions. We observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives. Generalized uniformly distributed order diffusion equation, as a model for strong anomalous behavior, is analyzed by using Tauberian theorem. Some previously obtained results are special cases of those presented in this paper.
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A. V. Chechkin
, V. Yu. Gonchar (Institute for Theoretical Physics
, n Kharkov. Institute for Single Crystals
1999
We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known kinetic equations of the Brownian motion theory and contain symmetric fractional derivatives over velocity and space, respectively. With the help of these equations we study analytically the processes of linear relaxation in a force - free case and for linear oscillator. For a weakly damped oscillator we also get kinetic equation for the distribution in slow variables. Linear relaxation processes are also studied numerically by solving corresponding Langevin equations with the source which is a discrete - time approximation to a white Levy noise. Numerical and analytical results agree quantitatively.
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.
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