| In the past few decades,the spectral method has developed rapidly in numerical sim-ulations in many fields,such as fluid dynamics,quantum mechanics,numerical weather-prediction and so on.Nowadays,it has become one of the most important tools for numer-ical solutions of differential equations,and has been successfully used for computations in science and engineering.By employing global functions as trial functions for discretization of differential equations,the spectral method possesses the high accuracy,i.e.,the so called convergence of "infinite order".It means that the convergence rates of discrete solutions increase as the regularity of the exact solutions increases.Volterra integro-differential equations(VIDEs)and fractional differential equations(FDEs)are models of evolutionary problems with memory arising in many applications,such as physical and biological phenomena,lasers and population growth,etc.In recent years,the numerical analysis in this field has become increasingly popular.However,the non-locality and weakly singularities of such problems usually lead to great challenges to general approaches,which are based on local operations.To overcome these challenges,the spec-tral method is often a good candidate,which appears to be a global approach and very suitable for non-local problems.However,the existing work for the second kind of Volterra-type equations mainly fo-cus on linear equations or smooth solutions.And most of the spectral schemes proposed recently are based on single step methods,which is inefficient to deal with singular prob-lems or long time simulations.Factually,plenty of the models rising in the real world are nonlinear and having weakly singular solutions,which is scarcely studied in the lit-erature.Therefore,we present a multi-step Legendre-Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels.We derive the error bounds for smooth solutions and weakly singular solutions.Numerical experiments illustrate the validity and efficiency of the proposed method.We next consider the spectral method for boundary value problems of nonlinear FDEs with a Caputo derivative.To suit the non-locality and nonlinearity of the problem,we employ two kinds of interpolations based on Legendre Gauss and Jacobi Gauss points,and construct single-step Legendre-Jacobi spectral collocation method.We present the convergence analysis for the method.Numerical experiments are included to verify the validity.This dissertation consists of the following several parts.Firstly,we briefly review the basic ideas and development on spectral method.the background of VIDEs and FDEs,as well as the research progress on numerical methods of them.Secondly,we recall some basic concepts and theories of spectral method,which is fundamental and important for the theoretical analysis in our work.Thirdly,we introduce a multi-step Legendre-Jacobi spectral collocation method for the nonlinear VIDEs of the second kind with singular kernels.we derive the hp-version error bounds of the Legendre-Jacobi collocation methods for smooth solutions on an arbitrary mesh and the hp-version error bounds for singular solutions on a quasi-uniform mesh.The theoretical results are verified by the numerical experiments.Then,for the fractional boundary value problems with a Caputo derivative,we refor-mulate the original equation as an equivalent Volterra-Fredholm equation and propose the Legendre-Jacobi spectral collocation method for the equivalent nonlinear integral equation.We derive the error bolnds of the proposed method in the function spaces L2(-1,1)and L?(-1,1).Our theoretical results are verified by the numerical experiments.Finally,we make a conclusion about the important results and main achievements in this thesis.We also point out the deficiency of the work and the possible remedial measurements in the future study. |
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