The Research On Singularity-Separation Chebyshev Collocation Method For Fractional Two-Point Boundary Value Problems

With the development of science and technology,many practical engineering calculations show that fractional differential equations can more accurately describe some practical problems.Because the analytical solutions are often difficult to obtain directly,numerical methods are always the important issues in studying the definite problems of fractional differential equations.In this paper,we study the two-point boundary value problems of fractional differential equations.For linear and nonlinear equations,the fractional integral operators are used to obtain equivalent integral equations,but the transformation methods are different.The linear problems are converted into equivalent Volterra-Fredholm integral equations,whereas the nonlinear problems are converted into equivalent hypersingular integral equations.Due to the solution of the fractional differential equation is singular at the left endpoint of the interval,the calculation accuracy is usually deteriorated by standard numerical methods.In this paper we aim to describe the singular behavior of the solution at the left endpoint of the interval.First,we design a Picard iterative algorithm to find the fractional series solution of the equation at the left endpoint with an undetermined parameter,which is used to approximate the solution of the equation on the subinterval containing the singular point.Then,the Chebyshev collocation method is used on the regular interval to obtain the numerical solution and the undetermined parameter in the series expansion simultaneously.This singularity-separation collocation method avoids the influence of singular behavior on the accuracy of the approximate solution.The first chapter introduces the research progress of fractional differential and integral equations by reviewing literature and gives the research objectives and the arrangement of this paper.In the second chapter,we study the singular expansion of the solution at the left endpoint of the interval for the fractional two-point boundary value problem.We use Picard iteration and series decomposition to find the fractional series expansion with an undetermined parameter for the solution of the equivalent integral equation at the left endpoint.In theory,the undetermined parameter can be determined by the boundary condition at the right endpoint,but because the series solution is usually only accurate near the left endpoint,this method is not practical.For the series solution generated by iteration,a special representation about the undetermined parameter is also given,which is used for the discretization and error analysis of the collocation method in the following chapters.In the third chapter,we consider the linear fractional two-point boundary value problem.The interval is split into singular subinterval and regular subinterval by using the fractional series expansion with a parameter for the solution at the left endpoint.On the singular subinterval,the series expansion is used to approximate the solution and on the regular subinterval,an extended Chebyshev collocation method is designed to simultaneously obtain the numerical solution of the equation and the undetermined parameter in the series expansion.It is strictly proved that the collocation solution has optimal convergence with respect to maximum norm.Numerical examples verify the correctness of the series expansion and the high accuracy of the Chebyshev collocation method.In the fourth chapter,the collocation method based on Hadamard finite-part integral for nonlinear fractional differential equation is studied.First,the relationship between the fractional derivative and Hadamard finite-part integral for singular function is studied.Second,the fractional two-point boundary value problem is transformed into a hypersingular integral equation by using the Hadamard finite-part integral.Third,the computational interval is still divided into singular and regular parts by using the fractional series expansion for the solution at the left endpoint derived in Chapter 2.The Chebyshev collocation method is designed on the regular interval to discretize the transformed hypersingular integral equation.Meanwhile,the undetermined parameter in the series expansion is also obtained.In this chapter,we also provide a recursive algorithm to quickly calculate the hypersingular integrals related to Chebyshev polynomials.Finally,numerical examples confirm the effectiveness and high accuracy of the scheme in this chapter.Finally,a brief summary is given to illustrate the shortcomings and deficiencies of this paper and the future research direction is also prospected.