The Spectral Collocation Method For Several Classes Of Fractional Differential Equations With Variable-order Derivatives

Variable-order fractional calculus is the extension and further development of con-stant order fractional calculus,which allow the order of the derivatives and integrals to be non-constant,depending on temporal or spatial variables.Variable-order fractional calcu-lus has a big difference with constant-order fractional calculus.An increasing number of problems in physics,control and signal processing have been modeled via fractional dif-ferential equations with variable-order derivatives.Due to the existence of variable-order fractional derivatives,it is usually impossible to obtain the analytical solutions of such equations.Hence,it is important to develop efficient numerical methods.This thesis con-structs and studies several single-step spectral collocation methods and hp-version spec-tral collocation methods for fractional differential equations with variable-order deriva-tives.The first chapter introduces research background,history of development and present situation of fractional calculus and variable-order fractional calculus,and reviews re-search advances about the spectral collocation methods for fractional differential equa-tions.This chapter also lists some basic definitions and lemmas of fractional calculus,and some basic theories of the spectral collocation methods.Finally,the main contents of this paper are summarized.The second chapter investigates the Legendre spectral collocation method for quasi-linear fractional initial value problems with a variable-order derivative.By using Banach contraction principle,Schaefer's fixed point theorem and the Gronwall-Bellman lemma,some sufficient conditions of the existence and uniqueness of exact solution are obtained.By using the Legendre-Gauss interpolations,the Legendre spectral collocation method is constructed.The priori error estimates for the proposed scheme under the L~2and L~?norms are established.Finally,numerical experiments are given to illustrate the spectral accuracy of the scheme.In chapter three,a Legendre spectral collocation method is presented for nonlin-ear fractional initial value problems with variable-order derivatives.Some sufficien-t conditions for the existence and uniqueness of exact solution are established by us-ing Weissinger's fixed point theorem and the Gronwall-Bellman lemma.By using the Legendre-Gauss interpolations,the spectral collocation scheme with the smoothing pa-rameter is constructed for weakly singular solutions.The bound of the smoothing param-eter is given,then the rigorous error estimates under the L~2and L~?norms are derived.Finally,numerical results are given to support the theoretical conclusions.In chapter four,an hp-version spectral collocation method is established for solving Caputo fractional initial value problems with variable-order derivatives.By using Ba-nach contraction principle and Schaefer's fixed point theorem,some sufficient conditions for the existence and uniqueness of exact solution are obtained.The fractional differ-ential equation is recast into an integral equation based on the properties of fractional calculus operators.By using the shifted Jacobi and Legendre polynomial,the hp-version Legendre-Jacobi spectral collocation scheme is established.The rigorous error estimates under the H~1norm are derived for smooth solutions on arbitrary meshes and weakly sin-gular solutions on quasi-uniform meshes.The theoretical conclusions are verified by the numerical experiments.In chapter five,an hp-version spectral collocation method is established for solving Riemann-Liouville fractional initial value problems with a variable-order derivative.By using Weissinger's fixed point theorem and the Gronwall-Bellman lemma,a existence result of exact solutions is obtained.Based on the shifted Legendre polynomial,the hp-version Legendre spectral collocation scheme is established.The hp-version error bounds of the method for smooth solutions and weakly singular solutions are derived.Finally,the numerical results show the method may refine the mesh and/or increase the degree of the polynomial to achieve higher accuracy.The sixth chapter summarizes the main results and the innovations in this paper.Finally,this chapter prospects some future research work based on this paper.