Numerical Methods For Two Kinds Of Linear Time-Space Variable-Order Fractional Advection-Diffusion Equations

Fractional operator is a non-local operator, has the properties such as inheritance and memory, using fractional order differential equation model can more accurately simulate the actual process of this nature. Variable-order fractional differential equation is a generalization of the fractional order differential equation,namely the order of the equation can be expressed as a function of time and space variables. For many non-locality problems in physics, mechanics, materials, civil engineering, biology, finance and other fields, a variable-order fractional model will be able to get sharper and more detailed description of the problem.For some simple variable-order fractional differential equations, we can use Fourier transform, Laplace transform and Mellin transform method to get the analytic solutions, but the obtained analytical solution commonly include some complex functions such as Green function, Fox-function, Mittag-Leffler function.For many complex variable-order fractional differential equations, we cannot get the analytic solution. Therefore, there is a strong need to study the numerical methods for variable-order fractional differential equations.In this paper, we study difference methods for one-dimensional and twodimensional time-space variable-order fractional advection-diffusion equations,where the order of time fractional derivative is only associated with space variable, and the order of space fractional derivative is associated with time and space variables. We can present a new implicit difference schemes for the equations by discretizing the time and space fractional operators respectively, and then prove that the schemes is unconditionally stable and convergent. Finally, some numerical experiments are given, and the numerical results show that the effectiveness of the schemes and the correctness of theoretical analysis.In the first chapter, we introduce some previous research results and some basic definitions and lemmas needed in this paper.In the second chapter, we study the one-dimensional time-space variableorder fractional advection-diffusion equation, and present an implicit difference method. The stability and convergence of the method are discussed, and some numerical experiments are given.In the third chapter, a new implicit difference method is presented for the two-dimensional time-space variable-order fractional advection-diffusion equation. The stability and convergence of the method are also discussed, and some numerical experiments are given.In the fourth chapter, the discussed problems in the present paper are summarized and prospected.