Numerical Methods Of The Fractional Kinetic Equations And Theoretic Analysis

Fractional kinetic equations have been of great interest recently.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics,chemistry,biology, environmental sciences,engineering and finance.Fractional kinetic equations provide a powerful instrument for the description of memory and hereditary properties of different substances.However,many analytical solutions for the fractional kinetic equations are complicated, which include the complicated series or especial function.Moreover,analytic solutions of most fractional kinetic equations cannot be obtained explicitly.At present numerical methods and analysis of stability and convergence for fractional partial differential equations are quite limited and difficult to derive.This motivates us to develop effective numerical methods for the fractional differential equations.In this thesis,we consider two kind of fractional kinetic equations.The first kind of the fractional kinetic equations is the fractional kinetic equations of the diffusion,diffusion-advection, and Fokker-Planck type.Numerical methods and theoretical analysis for the fractional kinetic equations are discussed in Chapters 2,3 and 4,respectively.The second kind of the fractional kinetic equations is the fractional kinetic equations of anomalous subdiffusion type,such as the anomalous subdiffusion equation,a nonlinear fractional reaction-subdiffusion process and the fractional cable equation.Numerical methods and theoretical analysis for the fractional kinetic equations are discussed in Chapters 5,6 and 7,respectively.These fractional kinetic equations above-mentioned have been presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.These fractional equations can be derived asymptotically from basic random walk models,and from a generalised master equation.In the first chapter,we summarize the history of the theory of fractional calculus, the background and significance of this dissertation,and the previous works about the fractional kinetic equations.Our research group and the framework of this thesis are given.In Chapter 2,we consider a space-time fractional diffusion equation on a finite domain.The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann-Liouville fractional derivative of orderβ∈(1,2],and the first-order time derivative by a Caputo fractional derivative of orderα∈(0,1].An implicit and an explicit difference approximations for the spacetime fractional diffusion equation with initial and boundary values are investigated. Stability and convergency results for the methods are discussed.Using mathematical induction,we prove that the implicit difference method is unconditionally stable and convergent,but the explicit difference method is conditionally stable and convergent. Some numerical results show the system exhibits anomalous diffusive behaviour.In this chapter,we also consider a two-dimensional fractional diffusion equation on a finite domain.We examine an implicit difference approximation to solve the space-time fractional diffusion equation.Stability and convergency of the method are discussed. Some numerical examples are presented to show the application of the present technique.In Chapter 3,we consider a space-time fractional advection dispersion equation on a finite domain.This equation is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of orderα∈(0,1],and the first-order and second-order space derivatives by the Riemman-Liouville fractional derivatives of orderβ∈(0,1]and of orderγ∈(1,2], respectively,n implicit and an explicit difference approximations is proposed.Using mathematical induction,we prove that the implicit difference method is unconditionally stable and convergent,but the explicit difference method is conditionally stable and convergent.Numerical results are in good agreement with theoretical analysis.In Chapter 4,we consider a space-time fractional Fokker-Planck equation on a finite domain.This equation is obtained from the standard Fokker-Planck equation by replacing the first-order time derivative by the Caputo fractional derivative,the second- order space derivative by the left and right Riemann-Liouville fractional derivatives. We propose a computationaUy effective implicit numerical method to solve this equation. Stability and convergence of the methods are discussed.Numerical example is given,which is in good agreement with the exact solution.In Chapter 5,we consider anomalous subdiffusion equation.A new implicit numerical method and two solution techniques for improving the order of convergence of the implicit numerical method for solving the anomalous subdiffusion equation are proposed.The stability and convergence of the new implicit numerical method are investigated by the energy method.Some numerical examples are given.The numerical results demonstrate the effectiveness of theoretical analysis.These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.In Chapter 6,a nonlinear fractional reaction-subdiffusion process is considered. We propose a new computationally efficient numerical method to simulate the process. Firstly,the nonlinear fractional reaction-subdiffusion equation is decoupled,which is equivalent to solving a nonlinear fractional reaction-subdiffusion equation.Secondly, we propose an implicit numerical method to approximate this equation.Thirdly,the stability and convergence of the method are discussed using a new energy method.Finally, some numerical examples are presented to show the application of the present technique.This method and supporting theoretical results can also be applied to fractional integro-differential equations.In Chapter 7,a fractional cable equation is discussed.An implicit difference method is proposed.The stability and convergence of the method are discussed using an energy method.Moreover,we also propose the finite element approximation of the fractional cable equation.The stability and error estimates are established.We derive the convergent order of the method.Numerical examples are presented which demonstrate the effectiveness of the methods and confirm the theoretical analysis.