The Research On The Asymptotic Expansions Of Solutions To Two Kinds Of Variable-order Fractional Differential Equations And Related Numerical Algorithms

Variable-order fractional differential equation(V-FDE)is the extension and further devel-opment of fractional differential equation(FDE),which has the basic feature that the order of derivative is a function of temporal or spatial variable.In recent ten years,the models of V-FDEs have been successfully used in many fields,whereas academic researches about mathematical theory are relatively insufficient,especially the descriptions of the solutions behaviors for these V-FDEs are always difficult problems.In this paper,we consider two kinds of variable-order fractional differential equations,and aim to obtain the Puiseux expansions for the solutions about the initial point,from which we can accurately describe the singular behaviors of the variable-order differen-tial equations.Based on the Puiseux expansions,we design a backward difference scheme to obtain relatively high accuracy solutions for these two kinds of model equations.The thesis includes four chapters.In chapter one,the development of V-FDEs in history is briefly introduced.The research advances about the numerical algorithms for V-FDEs are reviewed.The content and the outline of the paper are also presented.In chapter two,some preliminaries are provided,including the definitions and properties of variable-order differential operator,the Puiseux series of a function about its singular point.Espe-cially,the Puiseux expansion of a special function is derived.Gamma function and its high-order derivatives,the expansions of Gamma function and its reciprocal are also introduced.Lastly,a numerical backward difference scheme for Caputo variable-order fractional derivative is given.In chapter three,a generalized time-dependent model for viscoelastic deformation is con-sidered.First,the Puiseux expansion is derived when the function involving an algebraic and logarithmic singularity is conducted by the Caputo variable-order differential operator.Then an algorithm is designed to find the finite-term truncation of the Puiseux series for the solution about the initial point,which accurately describes the singular behavior of the solution.Finally,based on the series solution,a hybrid numerical algorithm is presented by discretizing the derivative with backward quotient when the variable is far away from the initial point.In chapter four,a generalized form of variable-order relaxation-type equation is considered.First,we briefly introduce the development of the equation,and design an iteration algorithm to obtain the series solution about the initial point.Then,a numerical algorithm is designed based on the series solution.Finally,we write Mathematica codes to test the algorithm.The computation shows that the hybrid algorithm in this paper possesses the advantages of series expansion and backward difference method,and it can be effectively used to solve the variable-order fractional differential equationsFinally,the conclusions are provided to summarize the thesis.The directions for further research in the future are included.