Chengjian Zhang

LDG method for reaction–diffusion dynamical systems with time delay

Abstract In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction–diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.

Boundary value methods for Volterra integral and integro-differential equations

Abstract New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.

Split Newton iterative algorithm and its application

Abstract Inspired by some implicit–explicit linear multistep schemes and additive Runge–Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction–diffusion equations, such as Burger’s–Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.

An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations

This paper deals with stability of the extended Runge-Kutta methods for nonlinear neutral delay-integro-differential equations. The stability results in the reference [Y. Yu, L. Wen, S. Li, Nonlinear stability of Runge-Kutta methods for neutral delay integro-differential equations, Appl. Math. Comput. 191 (2007) 543-549] are improved. With this improvement, several new numerical stability criteria are obtained, it is proven that the extended Runge-Kutta methods are globally and asymptotically stable under the suitable conditions.

Hopf bifurcation analysis of integro-differential equation with unbounded delay

Abstract The complexity of a nonlinear dynamical system is controllable via a selection of system parameters. One representative behavior of such a complex system can be illustrated by Hopf bifurcation. This paper presents a Hopf bifurcation analysis of a kind of integro-differential equations with unbounded delay. Based on the Hopf bifurcation principle, a set of relationships among system parameters are obtained when a periodic orbit exists in the system. A numerical analysis is applied to solve the integro-differential delay equation. This paper proves the existence of Hopf bifurcation in the corresponding difference equations under the same system parameters as that in the integro-differential delay equations.

A computational analysis for mean exit time under non-Gaussian Lévy noises

Abstract Complex dynamical systems are often subject to non-Gaussian random fluctuations. The exit phenomenon, i.e., escaping from a bounded domain in state space, is an impact of randomness on the evolution of these dynamical systems. The existing work is about asymptotic estimate on mean exit time when the noise intensity is sufficiently small. In the present paper, however, the authors analyze mean exit time for arbitrary noise intensity, via numerical investigation. The mean exit time for a dynamical system, driven by a non-Gaussian, discontinuous (with jumps), α -stable Levy motion, is described by a differential equation with nonlocal interactions. A numerical approach for solving this nonlocal problem is proposed. A computational analysis is conducted to investigate the relative importance of jump measure, diffusion coefficient and non-Gaussianity in affecting mean exit time.

Stable solutions of one-leg methods for a class of nonlinear functional-integro-differential equations

Abstract This paper deals with stable solutions of one-leg methods for a class of nonlinear functional-integro-differential equations (FIDEs). A type of extended one-leg methods are suggested for the FIDEs. The (weak) global stability results of the methods are presented. In particular, it is shown under suitable condition that a G-stable extended BDF method is globally and asymptotically stable for the problems of class FID ( α , β , γ , η , + ∞ ) . Numerical experiments further illustrate the theoretical results and the methodical effectiveness. In the end, a connection and comparison between the obtained results and the existed ones is given.

The extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations

This paper deals with the extended Pouzet–Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. Nonlinear stability and numerical implementation of the methods are investigated. It is proven under the suitable conditions that the extended Pouzet–Runge–Kutta methods are globally and asymptotically stable for problems of the class

The extended generalized Störmer-Cowell methods for second-order delay boundary value problems

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