The Hp Type Continuous Galerkin And Spectrum Collocation Method For Second-order Nonlinear Delay Differential Equations

In this paper,we consider the numerical methods with high accuracy for the second order nonlinear delay differential equations.On the one hand,for the nonlinear second-order Delay differential equations,we propose an h-p version of the continuous Petrov-Galerkin method,analyze the existence and uniqueness of the numerical solution,and deduce a priori error estimates in the L2-and L?-norm that are completely explicit in the local time steps,in the local polynomial degrees,and in the local regularity of the exact solutions.Moreover,it is proved that,for analytic solutions with start-up singularities exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders.On the other hand,we introduce an h-p version of the Chebyshev-Gauss-Lobatto spectral collocation method for the nonlinear second-order delay differential equations,and design a fast algorithm with high accuracy by using properties of the chebyshev polynomials.Some numerical examples are given to verify the proposed algorithm.