High--Order Fractional Adams-Type Methods For Fractional Differential Equations

Fractional differential equations can effectively model many practical problems in the fields of science and engineering.Many fractional differential equations are difficult to obtain their analytical solutions.How to effectively numerically solve these equations is an important topic.In this thesis,we construct high-order efficient numerical methods for initial boundary value problems of some nonlinear fractional differential equations,and provide corresponding numerical analysis.The first chapter introduces the research background and purpose of this thesis.In Chapter 2,we propose a fractional Adams-Simpson-type method for the initial value problems of nonlinear fractional ordinary differential equations with fractional order α ∈(0,1).In our method,a nonuniform mesh is used so that the optimal conver-gence order can be recovered for weakly singular solutions.By developing a modified fractional Gronwall lemma,we prove that the method is unconditionally convergent un-der the local Lipschitz condition of the nonlinear function,and show that with a proper mesh parameter,the method can achieve the optimal convergence order 3+α for weakly singular solutions.Under very mild conditions,the nonlinear stability of the method is analyzed by using a perturbation technique.The extensions of the method to the initial value problems of multi-term nonlinear fractional ordinary differential equations and multi-order nonlinear fractional ordinary differential systems are also discussed.Nu-merical results confirm the theoretical analysis results and demonstrate the effectiveness of the method for weakly singular solutions.In Chapter 3,based on piecewise cubic interpolation polynomials,we develop a high-order numerical method with nonuniform meshes for solving the initial value prob-lems of nonlinear fractional ordinary differential equations with weakly singular solu-tions.This method is a fractional variant of the classical Adams-type implicit-explicit method widely used for the initial value problems of integer order ordinary differen-tial equations.By using the modified fractional Gronwall lemma,we rigorously prove that the method is unconditionally convergent under the local Lipschitz condition of the nonlinear function,and when the mesh parameter is properly selected,it can achieve optimal fourth-order convergence for weakly singular solutions.We also prove the sta-bility of the method,and discuss the applicability of the method to the initial value problems of multi-term nonlinear fractional ordinary differential equations and multi-order nonlinear fractional ordinary differential systems with weakly singular solutions.Numerical results are given to confirm the theoretical convergence results.In Chapter 4,using the fractional Adams-Simpson-type method proposed in Chap-ter 2 for time-fractional integration on the graded temporal mesh and the fourth-order compact discretization of spatial derivatives on the uniform spatial mesh,a high-order linearized compact difference method is constructed to solve the initial value problems of nonlinear time-fractional Benjamin-Bona-Mahony-Burgers(BBMB)equation with fractional order α ∈(0,1).The detailed construction is based on a transformation method and a linearization technique.The graded temporal mesh ensures the optimal convergence order in time for weakly singular solutions with weak singularity at the initial time t = 0.Using a modified fractional Gronwall lemma,the unconditional con-vergence of the proposed method is proved.It is shown that the method has the optimal fourth-order convergence in space,and can achieve the optimal convergence order 3+αin time for weakly singular solutions when selecting the proper mesh grading parame-ter.In comparison with existing methods in the literature,this new method significantly improves the numerical accuracy.The stability of the method is also proved using a per-turbation technique.Since only one linear system needs to be solved at each time level,the proposed method is very effective in practical calculations.Numerical results con-firm the theoretical convergence result and show the superiority of the proposed method over the existing methods.The applicability and effectiveness of our method for the ini-tial value problems of integer order BBMB equations are also demonstrated through a numerical example.The fifth chapter summarizes the work of this thesis and makes a statement of possible future research.