Solutions Of The Time Fractional Partial Equations And Applications

Fractional calculus is a branch of studying the property of any order integral or derivative.Fractional order differential equation is the equation containing the noninteger order derivative,raising from the standard differential equations by replacing the integer-order derivatives with fractional-order derivatives.Its application is very broad, many researchers find that the fractional differential equations more precisely describe the property of some materials with memory and heredity.Fractional order differential equations are playing an increasingly important role in engineering,physics and other fields,such as the fractal theory and the diffusion in porous media,fractional capacitance theory,electrolysis chemical,fractional biological neurons,condensate physics,vibration control of viscoelastic system,statistical mechanics and so on.In this paper,we mainly consider the time-fractional anomalous diffusion equation, discuss its analytic solution,numerical solution and its application.In Chapter 1,the developmental history of fractional calculus and the existing work about fractional calculus are reviewed.We also recall some definitions and properties of the fractional derivatives used in this paper.In Chapter 2,two time-fractional anomalous diffusion equations are deduced from the random walk and a stable law.These two equations will be investigated numerically in the next two chapters.In Chapter 3,the solution of time fractional anomalous diffusion equation is discussed. Using separation of variable methods and Laplace transform,the analytical solutions of a non-homogeneous anomalous sub-diffusion equation with Dirichlet,Neumann and Robin boundary conditions are derived respectively.The solution is expressed in terms of the Mittag-Leffler function.These techniques can be applied to solve other kinds of anomalous diffusion problems.In Chapter 4,we consider a time fractional anomalous diffusion equation on a finite domain.We propose an efficient finite difference/spectral method to solve the time fractional diffusion equation.Stability and convergence of the method are rigourously established.We prove that the full discretization is unconditionally stable,and the numerical solution converges to the exact one with order O(△t^2-α+N^-m),where△t,N and m are the time step size,polynomial degree,and regularity of the exact solution respectively.Numerical experiments are carried out to support the theoretical claims.In Chapter 5,we generalize the method that we have proposed in the Chapter 4 to the time fractional Cable equation for modeling neuronal dynamics.Numerical results are presented to show the applicability of the method.In Chapter 6,we discuss one class of nonlinear time fractional Fokker-Planck equation with initial-boundary value on a finite domain.The stability and convergence of a finite difference method are discussed by energy methods.A numerical example is presented to compare with the exact analytical solution.