The Nature Of The Fractional Order Singular Diffusion Equations

In this paper, we discuss several fractional anomalous diffusion equations and con-sider the properties of each diffusion equation, concluding the diffusion character, gen-eralized Einstein Relation and the asymptotic behavior.In recent years, many researchers begin to study the problems in physics and fi-nance using the theories of fractional Brownian motion. But there have no paper to conclude the properties of the diffusion model driven by the composite-subdiffusive fractional Brownian motion. There are also many researchers study the effect of dif-ferent subordinator to the composite process, but there have no thorough analysis to the properties of the composite processes either. We are able to solve the two kinds of problems above in our paper.This dissertation is divided into five chapters:In Chapter1, we introduce the background of fractional anomalous diffusion equa-tion and expound the related research and applications about the stochastic representa-tion, the diffusion character, generalized Einstein Relation and the asymptotic behavior. Also, we summarize the results of the paper briefly.In Chapter2, we get the composite process X(Sa(t)) which combines the inverse of the α—stable Levy process Sα(t) and the main stochastic process X(r) driven by the fractional Brownian motion, basing on the research of the stochastic process driven by Brownian motion by M.Magdziarrz and A.Weron [20] in2007. Further more, we ob-tain the fractional Fokker-Planck equation which is satisfied by the probability density function of X(Sa(t)) and prove the properties of the equation.In Chapter3, we introduce the stochastic representation of a group transport equa-tion, as well as some properties of the transport equation.In Chapter4, we get the generalized fractional Fokker-Planck equation and diffu-sion character which is satisfied the probability density function of the composite pro-cess X(Sα,λ(t)) which is the process combined the inverse of the tempered α—stable Leuy process and the main stochastic process X(r) driven by the Brownian motion. Further more, we prove the equation satisfies the generalized Einstein Relation, also give the asymptotic behavior of the solution of the equation.In Chapter5, we definite a subordinator process and give the anomalous diffusion equation and properties of the composite process,We calculate the second moment in the absence of the force and the relation be-tween it with the first moment in the presence of a uniform force field to prove the dif-fusion character and generalized Einstein Relation. By using the second-order Bessel equation and the method of steepest descent, we obtain the asymptotic behavior of the diffusion equation.