Stability And Convergence Of General Linear Methods For Nonlinear Neutral Delay-integro-differential Equations

Let Cd be a d dimensional Euclidian space with the inner product <·,·> and the corre-sponding norm ‖·‖. Consider the nonlinear neutral delay-integro-differential equations where r>0 is a real constant, N∈Cd×d is a constant matrix,‖N‖<1,φ:[-r,0]â†'Cd, f:[0,+∞)×Cd×Cd×Cdâ†'Cd and g:[0,+∞)×[-r,+∞)×Cdâ†'Cd are continuous mappings and satisfy the following conditions, respectively, and where α,β1,β2 and γ are constants.In this paper, we denote this problem class as R(α,β1,β2,γ) and study stability and the convergence of general linear methods for solving nonlinear neutral delay-integro-differential equations.First, the stability of general linear methods with unconstraint step-size to solve the R(α,β1,β2,γ) is studied and the sufficient conditions are given in order to guarantee the methods to be stable.Second, the convergent of general linear methods with constraint step-size for above problem class is consider. It is proved that, if the methods is algebraic stability and diago-nal stability and generalized stage order with p, the methods are convergent with order of min {p,q-1/2}, where q is the error order of the quadrature formula for approximations to integral term in equation.Finally, as an example, the numerical tests are given by used the multistep Runge-Kutta methods for the initial value problem equations and the numerical results verify correctness of the theoretical results.