Stability and Oscillation of θ-methods for Differential Equation with Piecewise Constant Arguments

This paper studies the numerical properties of θ-methods for the alternately advanced and retarded differential equation u′(t) = au(t)+bu(2[(t+1)/2]). Using two classes of θ-methods, namely the linear θ-method and the one-leg θ-method, the stability regions of numerical methods are determined, and the conditions of oscillation for the θ-methods are derived. Moreover, we give the conditions under which the numerical stability regions contain the analytical stability regions. It is shown that the θ-methods preserve the oscillation of the analytic solution. In addition, the relationships between stability and oscillation are presented. Several numerical examples are given

Stability of Runge–Kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments

AbstractThis paper deals with the numerical properties of Runge–Kutta methods for the solution of u′(t)=au(t)+a0u([t+12]). It is shown that the Runge–Kutta method can preserve the convergence order. The necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained. It is interesting that the θ-methods with 0⩽θ<12 are asymptotically stable. Some numerical experiments are given

Oscillation of θ-methods for the Lasota-Wazewska model

The aim of this paper is to discuss the oscillation of numerical solutions for the Lasota-Wazewska model. Using two θ-methods (the linear θ-method and the one-leg θ-method), some conditions under which the numerical solutions oscillate are obtained for different range of θ. Furthermore, it is shown that every non-oscillatory numerical solution tends to the fixed point of the original continuous equation. Numerical examples are given

Variational Iteration Method for Partial Differential Equations with Piecewise Constant Arguments

In this paper, the variational iteration method is applied to solve the partial differential equations with piecewise constant arguments. This technique provides a sequence of functions which converges to the exact solutions of the problem and is based on the use of Lagrangemultipliers for identification of optimal value of a parameter in a functional. Employing this technique, we obtain the approximate solutions of the above mentioned equation in every interval [n, n + 1) (n = 0, 1, · · ·). Illustrative examples are given to show the efficiency of themethod

Exponential periodic attractor of impulsive Hopfield-type neural network system with piecewise constant argument

© 2018, University of Szeged. All rights reserved. In this paper we study a periodic impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Under general conditions, existence and uniqueness of solutions of such systems are established using ergodicity, Green functions and Gronwall integral inequality. Some sufficient conditions for the existence and stability of periodic solutions are shown and a new stability criterion based on linear approximation is proposed. Examples with constant and non-constant coefficients are simulated, illustrating the effectiveness of the results

Exponential periodic attractor of impulsive Hopfield-type neural network system with piecewise constant argument

In this paper we study a periodic impulsive Hopfield-type neural network system with piecewise constant argument of generalized type. Under general conditions, existence and uniqueness of solutions of such systems are established using ergodicity, Green functions and Gronwall integral inequality. Some sufficient conditions for the existence and stability of periodic solutions are shown and a new stability criterion based on linear approximation is proposed. Examples with constant and nonconstant coefficients are simulated, illustrating the effectiveness of the results

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments of advanced type

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results