A Chebyshev-Gauss-Lobatto Spectral Collocation Method For Nonlinear Volterra Integral Equations With Vanishing Variable Delays

As an important numerical method, spectral method often provides exceedingly accurate numerical results with relatively less degree of freedoms, and has been widely used for scientific computations. On the other hand, Volterra integral equations with delays(DVIEs) are the models of evolutionary problems with memory arising in many applications, such as physical and biological phenomena, lasers and population growth, etc.. In recent years, some numerical approaches for such problems have become increasingly popular.Among the existing works, numerical methods based on the single-step spectral schemes have been frequently used for various linear Volterra-type equations. However, this kind of algorithms is not suitable for long time simulations. Therefore, it is also very important for us to study the multistep spectral methods of the nonlinear Volterra-type equations.The main purpose of this dissertation is to investigate the multistep Chebyshev-Gauss-Lobatto spectral collocation methods for nonlinear Volterra integral equations(VIEs) with vanishing variable delays. We introduce a multistep Chebyshev-Gauss-Lobatto spectral collocation scheme and design an ecient algorithm. Numerical experiments demonstrate that the suggested methods possess the spectral accuracy. In particular, they are very appropriate for various problems with highly oscillating solutions and steep gradient solutions, as well as numerical simulations of long time behaviors.This dissertation consists of the following three parts.Firstly, we briefly review the research progress on numerical methods of DVIEs.Secondly, we propose a multistep Chebyshev-Gauss-Lobatto spectral collocation method for the nonlinear second-kind VIEs with vanishing variable delays.Thirdly, numerical experiments confirm the theoretical expectations.