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

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

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

Discontinuous Almost Automorphic Functions and Almost Automorphic Solutions of Differential Equations with Piecewise Constant Argument

In this article we introduce a class of discontinuous almost automorphic functions which appears naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument. Their fundamental properties are used to prove the almost automorphicity of bounded solutions of a system of differential equations with piecewise constant argument. Due to the strong discrete character of these equations, the existence of a unique discrete almost automorphic solution of a non-autonomous almost automorphic difference system is obtained, for which conditions of exponential dichotomy and discrete Bi-almost automorphicity are fundamental

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

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

Engineering Dynamics and Life Sciences

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”