This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay.Using a Halanay inequality generalized by Liz and Trofimchuk,we give two sufficient conditions for the stability of the true solution to this class of equations.Runge-Kutta methods with compound quadrature rule are considered.Nonlinear stability conditions for the proposed methods are derived.As an illustration of the application of these investigations,the asymptotic stability of the presented methods for Volterra delay-integro-differential equations are proved under some weaker conditions than those in the literature.An extension of the stability results to such equations with weakly singular kernel is also discussed.
A two-grid method for solving the Cahn-Hilliard equation is proposed in this paper.This two-grid method consists of two steps.First,solve the Cahn-Hilliard equation with an implicit mixed finite element method on a coarse grid.Second,solve two Poisson equations using multigrid methods on a fine grid.This two-grid method can also be combined with local mesh refinement to further improve the efficiency.Numerical results including two and three dimensional cases with linear or quadratic elements show that this two-grid method can speed up the existing mixed finite method while keeping the same convergence rate.
A series of eontractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations （NFDEs） in Banach spaces are obtained, which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations （FDEs）, neutral delay differential equations （NDDEs） and NFDEs of other types which appear in practice.
This article investigates the fractional derivative order identification, the coefficient identification, and the source identification in the fractional diffusion problems. If 1 〈 α〈 2, we prove the unique determination of the fractional derivative order and the dif- fusion coefficient p（x） by fo u（0, s）ds, 0 〈 t 〈 T for one-dimensional fractional diffusion-wave equations. Besides, if 0 〈 α 〈 1, we show the unique determination of the source term f（x, y） by U（0, 0, t）, 0 〈 t 〈 T for two-dimensional fractional diffusion equations. Here, a denotes the fractional derivative order over t.
