Pell Collocation Approach for the Nonlinear Pantograph Differential Equations

Abstract

Pantograph equations, which we encounter in the branches of pure and applied mathematics such as electrodynamics, control systems and quantum mechanics, are essentially a particular form of the functional differential equations characterized with proportional delays. This study focuses on exploring the approximate solution to the Pantograph differential equation. Since there is no analytic solutions for this equation class, only the approximate solutions can be obtain. For this purpose, Pell Collocation Method which is one of the numerical solution methods is chosen. As the result of applying the method to the equation, an algebraic equation system has been gained and then the approximate solution has been found by using MATHEMATICA via the given initial conditions. The method is applied to the some test examples and then the results are presented by both graphically and by table. The error estimations show that the method works accurately and efficiently.

Keywords

• Approximate solution
• pantograph differential equation
• collocation method

Lineer Olmayan Pantograf Diferansiyel Denklemleri İçin Pell Sıralama Yaklaşımı

Abstract

Elektrodinamik, kontrol sistemleri ve kuantum mekaniği gibi teorik ve uygulamalı matematiğin dallarında karşılaştığımız Pantograf denklemleri, orantısal gecikmeli fonksiyonel diferansiyel denklemlerin özel bir türüdür. Bu çalışmada, Pantograf diferansiyel denklemin yaklaşık çözümleri üzerine çalışılmıştır. Bu denklem sınıfı için analitik çözüm olmadığından sadece yaklaşık çözümleri bulunabilir. Bu amaçla sayısal çözüm yöntemlerinden biri olan Pell sıralama yöntemi seçilmiştir. Yöntemin denkleme uygulanması sonucunda bir cebirsel denklem sistemi elde edilmiş ve MATHEMATICA programı kullanılarak verilen başlangıç koşulları ile yaklaşık çözüm bulunmuştur. Bu yöntem bazı test örneklerine uygulanmış ve sonuçlar hem grafiksel olarak hem de tablo olarak ifade edilmiştir. Hata analizleri bu yöntemin doğru ve etkili çalıştığını göstermiştir.

Keywords

• Yaklaşık çözüm
• pantograf diferansiyel denklemi
• sıralama yöntemi

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