Fast Multistep Collocation Methods And Block Boundary Value Methods For Volterra Integral Equations

The numerical approximation of differential equations and integral equations hasbeen one of the principal concerns of numerical analysis. It is still a very active field ofresearch due to the proposition of new methods and the appearance of new problems.Here, we mainly concern with two kinds of numerical methods, namely multistepcollocation (MC) methods and block boundary value methods (B2VMs). The classicalnumerical methods are extended to solve some new problems.Multistep collocation methods are high convergence and easy to be constructed. Itis known that the numerical treatment of Volterra integral equations (VIEs) has a veryhigh computational cost, since, for each time step, we have to compute the history term(lag term). It is important to seek for numerical methods which can pull down thecomputational cost. Firstly, fast multistep collocation methods are proposed for VIEswith convolution kernel by making use of discretized inverse Laplace transformation. Itis proved that new methods own both high order of convergence and efficientimplementation, which needs less cost in time and space. Moreover, the fast methodscan preserve the convergence and stability of the original methods.B2VMs are relatively new and efficient numerical methods. Differs from multistepmethods, which are step-by-step approach, B2VMs are called global methods. In thesecond part of this paper, we applied B2VMs and reducible quadrature to neutralVolterra delay integro-differential equations. It is proved that every A-stable B2VMs canpreserve the delay-independent asymptotic stability of the analytic solution under somecertain conditions.The new methods in the first part give an efficient approach for solving VIEs,which avoid the very high computational cost both in time and space. It makes themultistep collocation methods more practical for VIEs. The problem in the second partcontains several kinds of numerical problems, such as delay differential equations, delayalgebraic differential equations and neutral Volterra delay integro-differential equations.This extension brings B2VMs much wider applications. What’s more, the relationbetween asymptotic stability and A-stability of ordinary differential equations makes theanalysis of asymptotic stability much easier since only A-stability is needed to beconsidered.