| The fractional differential equations,which have been widely applied,have become one of the research hotspots in many fields.In this thesis,we focus on two important properties and difficulties of fractional differential equations: nonlocality and singularity.The nonlocality brings about the problems of large storage and long computational time,which seriously affects the practicability and feasibility of numerical algorithms in the engineering field,so it is imperative to construct efficient fast algorithms.The singularity will affect the convergence orders of the numerical algorithms for the nonsmooth solution problems,and adopting the nonuniform mesh is one of the best methods to deal with the singularity.The tempered fractional advection-dispersion equations,distributed-order time-space reaction-diffusion equations and the time fractional diffusion equations are three important classes of fractional diffusion equations with definite application background.In this thesis,graded mesh,general nonuniform mesh and adaptive mesh are used to study the singularities of these three kinds of equations step by step.This thesis has some enlightenment for dealing with the singularity of fractional differential equations,especially the posterior error estimation and adaptive algorithm for solving time fractional differential equations is one of the few works in this field.The main work of this paper includes:Firstly,a fast L1 algorithm based on the graded mesh is developed for a class of tempered fractional advection-dispersion equations.Based on the nonsmooth regularity assumption and the sum-of-exponentials technique,a semi-discrete L1 time scheme on the graded mesh is proposed.Then the finite element method is used to obtain the results on the stability and convergence of the semi-discrete scheme and the convergence of the fully discrete scheme.Numerical experiments based on different regions verify the effectiveness of the algorithm and the correctness of the theoretical results.Secondly,a fast Alikhanov algorithm based on the general nonuniform mesh is developed for a class of time-space distributed-order fractional reaction-diffusion equations.Using the composite mid-point quadrature,we change the distributed-order time fractional derivative into multi-term time fractional derivative.With the help of the sum-of-exponentials technique,a fast algorithm is constructed to greatly reduce the influence of the nonlocality of fractional derivative.For the initial singularity of time fractional derivative,the general nonuniform mesh is selected.The Alikhanov scheme is used in the temporal direction,and is extended to the multi-term fractional derivative on the nonuniform mesh for the first time.Then the space direction is discretized by the finite element method,and the results on the stability and convergence of the semi-discrete scheme and the convergence of the fully discrete scheme are obtained.Numerical experiments show that the theory is correct and the numerical algorithm is efficient.Finally,a posteriori error estimation and an adaptive algorithm are developed on the L1 method for the time fractional parabolic equations.The time discretization is carried out by using the L1 scheme,and the continuous approximation solution is obtained by linear interpolation of the numerical solution.The residual posterior error estimation is given based on the continuous approximation solution,but numerical experiments show that the posterior error estimator can only reach the first order,and cannot match the optimal convergence order of L1 scheme.Then,we continue to reconstruct the numerical solution,obtain a new posterior error estimator,and give error upper bound and error lower bound under different norms.The subsequent numerical experiments verify that the error estimator can reach order 2-α in the case of smooth solutions,but only α order in the case of nonsmooth solutions.To achieve the optimal convergence order in the case of nonsmooth solutions,we construct an adaptive algorithm based on the barrier function,and the subsequent numerical experiments verify that the optimal convergence order can be achieved on the adaptive mesh. |
