Kemal Aydin

Asymptotic Stability of Solutions to Perturbed Linear Difference Equations with Periodic Coefficients

We consider perturbed linear systems of difference equations with periodic coefficients. The zero solution of a nonperturbed system is assumed asymptotically stable, i.e., all eigenvalues of the monodromy matrix belong to the unit disk {|λ|<1}. We obtain conditions on the perturbation of this system under which the zero solution of the system is asymptotically stable and also establish continuous dependence of one class of numeric characteristics of asymptotic stability of solutions on the coefficients of the system.

A new variable step size algorithm for Cauchy problem

Abstract In this study, we have obtained a step size strategy based on Picard–Lindelof theorem and error analysis for the numeric integration of Cauchy problems in a region where the solutions of Cauchy problems exist and are unique. We have given an algorithm which calculates step size based on Picard–Lindelof theorem and error analysis and obtains approximate solutions. The strategy and algorithm are modifications of the strategy and algorithm in [G. Celik Kizilkan, K. Aydin, Step size strategy based on error analysis, Selcuk University Science and Art Faculty Journal of Science 25 (2005) 79–86 (in Turkish)].

Iterative decreasing dimension algorithm

In this study, we have given an iterative decreasing dimension method (IDDM) which decreases by one dimension at every step for solution of a system of linear algebraic equations without any pre-processing and an iterative decreasing dimension algorithm (IDDA) which is based on this method. We have also given numerical examples using this algorithm.

Sensitivity of Schur stability of monodromy matrix

For Schur stable linear difference equation system with periodic coefficients, we prove continuity theorems on monodromy matrix which show how much change is permissible without disturbing the Schur stability, and some examples illustrating the efficiency of the theorems are given.

Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients

For Hurwitz stable linear differential equation system with constant coefficients, we have proved continuity theorems which show how much change is permissible without disturbing the Hurwitz stability and the κ∗-Hurwitz stability. The results have been applied to the scalar–linear differential equations with order k and some examples illustrating the efficiency of the theorems have been given.

Computation of the monodromy matrix in floating point arithmetic with the Wilkinson Model

In this study, results have been obtained that compute the monodromy matrix in floating point arithmetic using the Wilkinson Model. These results have been applied to the asymptotic stability of systems of linear difference equations with periodic coefficients. Also the effect of floating point arithmetic has been investigated on numerical examples.

Step size strategies for the numerical integration of systems of differential equations

In this study, the step size strategies are obtained such that the local error is smaller than the desired error level in the numerical integration of a type of nonlinear equation system in interval t 0 , T ] . The algorithms are given for calculating step sizes and numerical solutions according to these strategies. The algorithms are supported by the numerical examples.