Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations

Abstract

In this paper, we focus on the error behavior of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (NVDIDEs) with constant delay. The convergence properties of the RungeKutta methods with two classes of quadrature technique, compound quadrature rule and Pouzet type quadrature technique, are investigated.

Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

“…Enright and Hu [25] proved that the method converges assuming that the delay argument τ is constant, the exact solution is smooth throughout the integration interval, the integrals are computed analytically, and τ L y < 1. This latter requirement is quite strong but is typical of a condition required to guarantee that (2) has a unique solution (see, for example, [17,26]). We will present a convergence theorem which applies to the more general time-dependent class of problems (with arbitrary delay arguments) and also accounts for discontinuity points and the quadrature errors associated with the evaluation of (10) and δ(t).…”

“…Collocation approximate solutions of nonlinear systems of VIDEs with delay arguments of the general type, N 1 (t) = f 1 (N 1 (t), N 2 (t)) + t t−τ F 1 (t − s)G 1 (N 1 (t), N 2 (s))ds, N 2 (t) = f 2 (N 1 (t), N 2 (t)) + t t−τ F 2 (t − s)G 2 (N 2 (t), N 1 (s))ds, (3) which includes system (1) as a special case were analyzed in [14] based on a constant stepsize strategy. Discrete RK methods and their numerical stability for VIDEs with constant delay have been investigated in [15,16,17] assuming a fixed stepsize strategy. These methods are mainly extensions of the classical Pouzet and Beltyukov RK methods where the discretization is a part of the method and no continuous approximation is produced.…”

Convergence of Runge–Kutta methods for neutral delay integro-differential equations

“…These important convergence results are based on the classical Lipschitz conditions. The studies about convergence of the numerical method for nonlinear NDIDEs based on a one-sided Lipschitz condition are proceed only by Wang [38,39]. However, up to now, the convergence results, which concern the numerical method for nonlinear NDIDEs (…”

A novel numerical method and its convergence for nonlinear delay Volterra integro‐differential equations

“…Wei and Chen used the spectral Galerkin method to solve the nonlinear Volterra integral equations of the second kind. In previous works, Runge‐Kutta method was used to solve Volterra integro‐differential equations with delay. Gan dealt with an numerical method for nonlinear delay VIDEs.…”