Convergence Of One-leg Methods For Nonlinear Neutral Delay-integro-differential Equations

Let Rd be a d dimensional real Euclidian space with the inner product<·,·> and the corresponding norm ?·?. Consider the following initial value problems (IVPs) of nonlinear neutral delay integro-differential equations(NDIDEs)of Hale type where ?>0 and T>0 are given constants,N?Rd×d stands for a constant matrix with ?N?<1,?:[-?,0]?Rd is a continuous mapping,and f:[0,T]×Rd×Rd×Rd?Rd and g:[0,T]×[-?,T]×Rd?Rd are continuous mappings and satisfy the following conditions,respectively, where ?,?,?,L and ? are real constants and ?,?,L,? are nonnegative,Here we denote this problems class as R(?,?,?,L,?).In this paper we study the convergence of one-leg methods for solving above problem and we obtain the following results.If the one-leg method is A-stable and consisistent of order p?2 in the classical sense for ODEs,the accuracy of quadrature rule is q+1.The sequence {yn} is produced by the method applied to a problem belong to R(?,?,?,L,?)with initial values y0,y1,…,yk-1.Then the global error satisfies a bound of the form where the functions C1(t),C2(t) and the maximum step size ho depend only on the method, some of the bounds Mi,Ni, the parameters ?,?,?,L,? and the matrix N. This inequality means the one-leg methods to be convergent of order at least min {p, q+1/2}.The numerical tests are given by used the one-leg ? methods and second order BDF methods and the numerical results verify correctness of the theoretical results.