High Accurate Numerical Methods For Fractional Differential Equations Of Caputo Type

Fractional differential equations of Caputo type have been of great interest recently,owing to the extensive applications to many physical and engineering fields.However,the theoretical solutions of fractional differential equations of Caputo type are usually difficult to obtain.It is meaningful to establish the approximate solutions of this kind of equations by applying various numerical methods.Besides,since fractional differential equations of Caputo type can be converted to a class of second kind weakly singular Volterra integral equations,numerical methods for second kind weakly singular Volterra integral equations lead to those for fractional differential equations of Caputo type.For fractional differential equations of Caputo type and second kind weakly singular Volterra integral equations,this dissertation studies the numerical methods with high accuracy.This dissertation is composed of four parts: high order fractional backward differentiation formulae and fractional generalized backward differentiation formulae for fractional differential equations of Caputo type,multistep collocation methods and super implicit multistep collocation methods for second kind weakly singular Volterra integral equations.In the first part,a class of high order fractional backward differentiation formulae for fractional differential equations of Caputo type is studied.Firstly,the coefficients of high order fractional backward differentiation formulae are given.After that,the stability properties of fractional backward differentiation formulae are discussed.Compared with the explicit fractional linear multistep methods of the same order,it is finally concluded that fractional backward differentiation formulae have two distinct features: high accuracy and good stability in numerical computing.In the second part,fractional boundary value methods for fractional differential equations of Caputo type are introduced firstly.The consistence of fractional boundary value methods is analyzed.As a specific kind of fractional boundary value methods,fractional generalized backward differentiation formulae are then constructed.The coefficients of fractional generalized backward differentiation formulae are given based on the order conditions and the expansions of the generating functions of fractional linear multistep methods.Finally,the convergence properties of fractional generalized backward differentiation formulae are analyzed.In the third part,the regularity of the solutions of second kind weakly singular Volterra integral equations is firstly studied.The properties and effects of the smoothing transformations are discussed.Then,smoothing transformations are applied to the second kind weakly singular Volterra integral equations and the uniform meshes are taken.By multistep techniques,a new family of multistep collocation methods is constructed.Finally,the convergence and stability of multistep collocation methods are analyzed.In the fourth part,after the second kind weakly singular Volterra integral equations being transformed by suitable smoothing transformations,uniform partitions of the intervals are chosen.Based on mutistep collocation methods,super implicit multistep collocation methods for second kind weakly singular Volterra integral equations are then constructed by introducing the same number of collocation points in next subintervals.Finally,the convergence order of super implicit multistep collocation methods is given and the stability properties of super implicit multistep collocation methods are analyzed.