The analysis of delay differential equations with the perturbation-iteration method

In this thesis, perturbation-iteration method, which is a new method for delay differential equations, has been introduced. The method is presented with two types of algorithms depending on the order of the derivative in the Taylor series expansion and on the number of correction terms in the straightforward expansion. Perturbation-iteration algorithms are obtained separately for three different classes of delay differential equations. Firstly, perturbation-iteration algorithms are developed for time delayed differential equations. Three different numerical examples for this class of equations are discussed. Then, perturbation-iteration algorithms are developed for pantograph type delay differential equations and six different problems are considered for this class. Finally, perturbation-iteration algorithm is developed and applied to three different numerical examples for time delayed differential equations based on the history function which emerged as the mathematical model of many engineering problems. Perturbation-iteration solutions of all these numerical examples are presented with graphs and tables. The method is compared with many well-known methods such as variational iteration method, collocation method and homotopy perturbation method. In addition, the mathematical model of the physical systems of delayed Mathieu and delayed damped Mathieu equations are solved and the effect of the delay term in the equations is demonstrated. Stable and unstable solutions of the system are obtained by the method applied to the mathematical equation of the milling cutting process.