Abstract
The Continuous Time Random Walk and the Finite Difference Method for the Space Time Fractional Diffusion Equations
Highlights
In 1827 an English botanist, Robert Brown, noticed that small particles suspended in fluids perform peculiarly erratic movements. This phenomenon, which can be observed in gases, is referred to as Brownian motion [1]
Einstein considered the case of the free particle, that is, a particle in which no forces other than those due to the molecules of the surrounding medium are acting. He was able to show that the probability density u ( x, t ) must satisfy the partial differential equation where a > 0 is a certain physical constant
The extension to Lévy stable motion is a straight forward generalization due to the common properties of Lévy stable motion and Brownian motion, but the Lévy flights differ from the regular Brownian motion by the occurrence of extremely long jumps whose length is distributed according to the Lévy long tail x −1−α, 0 < α < 2 Many physical, biological, medical, and chemical models exhibits a power law with a non integer frequency of order t−β, where 0 < β < 1. for diffusion processes and 0 < β < 2 for wave propagation
Summary
The space fractional diffusion equations are a linear partial pseudo-differential equation with spatial fractional derivatives in space This equation arises when the motion of the particle is not Brownian and there are extremely long jumps whose length is distributed according to the Lévy long tail x −1−α , 0 < α < 2 Some physical, biological and chemical models exhibit a power law with a non integer frequency of order t−β , 0 < β < 1. In this case, one needs to replace the first order time differential operator by the Caputo time fractional operator.
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