Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases

Troesch Probleminin Perturbasyon İterasyon Yöntemi İle Analizi

Konferans Bildirisi-- İstanbul Teknik Üniversitesi, Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2017Conference Paper -- İstanbul Technical University, Theoretical and Applied Mechanical Turkish National Committee, 2017Bu çalışmada Troesch denkleminin sayısal çözümü için perturbasyon iterasyon yöntemi (PIA) kullanılmıştır. Yöntem Taylor seri açılımındaki türevin mertebesi ve yaya açılımındaki düzeltme teriminin sayısına bağlı olarak geliştirilmiştir. Yöntemin diğer bilinen perturbasyon yöntemlerine göre önemli avantajlarından birisi küçük perturbasyon parametresi zorunluluğu olmamasıdır. Yöntemin bu özelliği problemin incelenmesi için en önemli nedendir. Yöntemin bir diğer önemli avantajı ise iterasyon ve perturbasyon yöntemlerinin birleşimi olmasından dolayı uygulanılan problemin sayısal çözümlerinin etkili ve hızlı bulunmasını sağlamasıdır. Perturbasyon iterasyon yöntemi ile bulunan sayısal sonuçlar literatürde bilinen diğer sonuçlar ile karşılaştırılmıştır. Bulunan sonuçlar yardımıyla yöntemin Troesch denkleminin çözümündeki etkinliği incelenmiştir. Yöntemin uygulanmasında ve sayısal çözümlerin bulunmasında MATLAB sembolik yazılımı kullanılmıştır.In this study, perturbation iteration method (PIA) is used for numerical solution of Troesch equation. The method was developed based on the number of Taylor series derivation and the number of correction terms on straightforward opening. One of the important advantages of the method over other known perturbation methods is that it does not require a small perturbation parameter. This feature of the method is the most important reason for studying the problem. Another important advantage of the method is that because it is a combination of iteration and perturbation methods, numerical solutions of the applied problem can be found effective and fast. The numerical results obtained by the perturbation iteration method are compared with other results known in the literature. The effect of the method on the solution of the Troesch equation was investigated with the help of the results. MATLAB symbolic software is used to implement the method and to find numerical solutions

A new approach for solving multi-pantograph type delay differential equations

In this paper, a modified procedure based on the residual power series method (RPSM) was implemented to achieve approximate solution with high degree of accuracy for a system of multi-pantograph type delay differential equations (DDEs). This modified procedure is considered as a hybrid technique used to improve the curacy of the standard RPSM by combining the RPSM, Laplace transform and Pade approximant to be a powerful technique that can be solve the problems directly without large computational work, also even enlarge domain and leads to very accurate solutions or gives the exact solutions which is consider the best advantage of this technique. Some numerical applications are illustrated and numerical results are provided to prove the validity and the ability of this technique for this type of important differential equation that appears in different applications in engineering and control system