Efficient Difference Method For Time-Space Fractional Diffusion Equation With Fractional Derivative Boundary Condition

Fractional differential equations play a very important role in physics,chemistry,biology and so on.In recent years,many scholars have considered numerical methods for spatial fractional diffusion equations with fractional derivative boundary conditions.Based on this,under the low regularity problem of the solution,the initial-boundary value problems of a class of time-space fractional diffusion equations with fractional derivative boundary conditions is studied by using finite difference method.In the first chapter,the research background of the fractional-order differential equations with fractional derivative boundary condition,and the main research work of this paper are given.In the second chapter,a numerical scheme for these fractional equations is constructed under the non-uniform grid.Then the stability and convergence analysis of the numerical scheme are given.It is proved that the numerical scheme is unconditionally stable,and the convergence rate satisfies O(N^{-min{2-?₁,r?₁}+?} in the sense of maximum norm.In the third chapter,a numerical scheme for fast solution?henceforth referred to as fast scheme?is established under the non-uniform mesh based on the exponential summation approximation to kernel function t^-?.And the stability and convergence analysis are discussed.It is proved that its convergence rate satisfies convergent precision O(N^{-min{2-?₁,r?₁}+?} in the sense of maximum norm.Lastly,the extrapolation method is applied to the spatial direction of the fast scheme to reach the second-order convergence precision.At this time,the convergence rate of numerical scheme satisfies O(N^{-min{2-?₁,r?₁}+ ?2+ ?}in the sense of maximum norm.In the fourth chapter,numerical experiments are given to verify the correctness and effectiveness of the algorithm.