Solutions Of The Fractional Anomalous Diffusion Equations And Its Application

Fractional calculus is an extension of the classical integer-order calculus. Just like the classical integer-order calculus, fractional calculus has a long history. How-ever, because of lack of application, fractional calculus developed slowly at its lie-ginning. Fortunately. It is known that the integer-order calculus is a powerful tool to describe the dynamic processes in applied science, but. experiments and reality teach us that there are many complex systems in nature with anomalous dynamics which can not be characterized by classical integer-order derivative model. In re-cent years, the fractional operator is different from the integer-order one. The former one is non-local, and has memory effect, which makes it be suitable to describe the hereditary viscoelastic materials and stochastic models with memory. After that, the fractional calculus developed very quickly. Now, fractional calculus plays an im-portant role in Physics, Mathematics, Mechanical Engineering, Biology. Electrical Engineering, Control theory and Finance, and so on.Since the fractional calculus is non-local, then, by applying fractional calculus to classical diffusion equation and wave equation, a fractional differential equation with time and space fractional derivatives can be obtained, which can be used to describe the anomalous diffusion in Physics. So, the fractional calculus is becoming a useful tools to describe anomalous diffusion phenomenon, Such as constitutive relationships of viscoelastic materials, porous media seepage and random walk in fractal media. After getting the fractional differential equation, how to get its solution attracts many researchers. It also becomes a promising area.This dissertation is divided into six chapters:In Chapter1, we introduce; background of fractional anomalous diffusion equa-tion and the methods employed to deal with relevant problem, then present our questions and main results.In Chapter2, we present the background of the fractional calculus and the relevant knowledge, including some definitions and properties of fractional calculus,such as Riemann-Liouville fractional calculus. Caputo fractional calculus and Grumuald-Letnikov fractional calculus. Some relations between these definitions are also listed. Then, we introduce two important functions for solving the fractional differential equation:Mittag-Lefner function and Fox-H function, as well as some of their properties. At last, we also give the relations between the continuous time random walk model and fractional differential equation.In Chapter3, By using Laplace transform. Fourier transform. Mellin trans-form and Green function method, we get the solution of a generalized fractional differential equation with absorbent term, where the diffusion term, external term and absorbent term have different order of fractional derivatives. We find that its solution has heavier tail and higher peak, in contrast to Normal distribution.In Chapter4, we prove the validity of the fractional Taylor’s Formula pro-posed by Jumarie, which is an useful tool to get the numerical solution of fractional differential equations and its accuracy.In Chapter5, we study the stochastic representation of the fractional differential equation. First, the stochastic representation of a modified advection dispersion equation is proved to be a subordinated process, where the parent process is a classical diffusion process driven by Brownian motion, and the subordinator is the inverse of a Levy motion, whose characteristic function is dependent on the function presented in the convolution. Then we extend the parent process to the one driven by Levy motion. A spacial fractional differential equation is obtained. After that, we also answer the question proposed by Magdziarz in his paper, which means we obtain the stochastic representation of a fractional Fokker-Planck equation with time and space dependent drift and diffusion. At last, taking advantage of stochastic representation, we picture the solution of these fractional differential equation.In Chapter6, Comparing with the Normal distribution, the solution of frac- tional diffusion equation follows a stretched Gaussion distribution. i.e. has heavier tail and higher peak. So.we suppose that the stock price follows the stochastic pro-cess obtained in Chapter4.then, we obtain the option pricing formula. The "smirk" volatility is also obtained. The results got in this chapter extend the Black-Scholes Model, and more close to historical data.