The Research On The Asymptotic Solutions And Collocation Methods For Initial Value Problems Of Two-term Caputo Fractional Ordinary Differential Equations

Fractional differential equations are suitable for describing objective phenomena with the nature of historical memory.In general,it is difficult to obtain analytical solutions for these equations and so numerical methods are needed to solve them.The typical feature of these equations is that the solutions are not sufficiently smooth at the origin,which makes the standard numerical methods have very low accuracy.In this paper,we focus on describing the singular behavior of Caputo fractional ordinary differential equations and designing high precision numerical methods,involving linear and nonlinear,constant and variable coefficient problem models.In the first chapter,we briefly discuss the development of fractional differential equations,and outline the research objectives and main content of this paper.In the second chapter,we consider the asymptotic properties of solutions at singularities for the initial value problems of two-term linear Caputo fractional ordinary differential equations with constant coefficients.The asymptotic solutions about origin and infinity are formulated via Laplace transform.The asymptotic expansion about the origin reflects the singular behavior of the solution,including singularity type and degree,which is also a good approximation when the variable is near the origin.The expansion at infinity exhibits the structure of the solution,as well as the stable or unstable property of the solution,which becomes more accurate as the variable tends to large.Numerical examples show the easy calculation and high accuracy of the proposed expansions and their Pade approximations when they are approximate to the exact solution for small or large variable,respectively.At the end of this chapter,the method is applied to solve the initial value problem of the Bagley-Torvik equation.In the third chapter,we continue to study the piecewise collocation method for two-term linear fractional differential equations based on the results of chapter 2.First,the singular equation is transformed into a regular equation by using the asymptotic expansion obtained in chapter 2 to approximate the equation in a small subinterval including the origin.Second,a collocation method with the piecewise Lagrange interpolation is designed on the remaining interval to obtain numerical solution,which is called the piecewise collocation method via singularity separation.Third,the error analysis of the collocation scheme is conducted,and the optimal convergence order is obtained.Finally,two numerical examples are provided to show that the proposed method has optimal approximate accuracy on the computational interval.In the fourth chapter,we study the Chebyshev collocation method for nonlinear fractional ordinary differential equations with variable coefficients.In order to accurately identify the singularity of the solution,we design a modified Picard iteration algorithm to solve the second kind Volterra integral equation transformed from the differential equation,and the psi series expansion of the solution at the origin is obtained.Due to the asymptotic solution about the origin has high accuracy near the origin,we introduce an interval splitting point to separate the singular part of the equation,and then transform the singular problem into a regular one,which can be solved by the high accuracy Chebyshev collocation method on the remaining interval.At this stage,the Caputo fractional derivative in the equation is replaced by its equivalent Hadamard finite part integral form to avoid dealing with the derivative directly.When the transformed hypersingular integral equation is discretized by Chebyshev collocation method,we design an efficient recursive formula to calculate the hypersingular weight integrals related to the Chebyshev polynomials.Further,Newton iteration method is used to solve the derived system of nonlinear algebraic equations.Two numerical examples are given to confirm the effectiveness of the proposed method.We call the method as the Chebyshev collocation method via singularity separation,which provides a new idea for the algorithm exploration of such fractional singular problems.Finally,a brief summary of the paper is given,and the further research ideas in the future for this singular problem are also considered.