On Solutions of A Parabolic Partial Differential Equation of Neutral Type Including Piecewise Continuous Time Delay

Year 2023, Volume: 03 Issue: 01, 29 - 35, 31.07.2023

Zekeriya Özkan Duygu Aruğaslan Çinçin Marat Akhmet

Abstract

There have been very few studies on partial differential equations including piecewise constant arguments and generalized piecewise constant arguments. However, as far as we know, there is no study conducted on neutral type partial differential equations including piecewise constant argument of generalized type. With this motivation, we discuss the solution and analysis of a parabolic partial differential equation of neutral type including generalized piecewise constant delay. The aim of this study is to investigate detailed and well-defined qualitative properties of this equation. The formal solution of the handled equation is obtained by using the separation of variables method. Since there exist the piecewise constant arguments, we get an ordinary differential equation with respect to the time variable on each consecutive intervals and then apply the Laplace transform method using the unit step function and method of steps. With the help of the qualitative properties of the solutions of the obtained differential equation, unboundedness and oscillations of the solutions of the issue problem can be investigated.

Keywords

partial differential equation, neutral type, piecewise constant argument, laplace transformation, oscillation, unboundedness

Parçalı Sürekli Zaman Gecikmesi İçeren Nötr Tip Bir Parabolik Kısmi Diferansiyel Denklemin Çözümleri Üzerine

Abstract

Parçalı sabit argümanlar ve genelleştirilmiş parçalı sabit argümanlar içeren kısmi diferansiyel denklemler üzerine çok az sayıda çalışma yapılmıştır. Ancak bildiğimiz kadarıyla genelleştirilmiş tipte parçalı sabit argüman içeren nötr tip kısmi diferansiyel denklemler üzerine yapılmış bir çalışma bulunmamaktadır. Bu motivasyonla, genelleştirilmiş parçalı sabit gecikme içeren nötr tipte bir parabolik kısmi diferansiyel denklemin çözümü ve analizi tartışılmıştır. Bu çalışmanın amacı, bu denklemin ayrıntılı ve iyi tanımlanmış niteliksel özelliklerini araştırmaktır. Ele alınan denklemin formel çözümü değişkenlere ayırma yöntemi kullanılarak elde edilmiştir. Parçalı sabit argümanlar mevcut olduğundan, ardışık her aralıkta zaman değişkenine göre bir adi diferansiyel denklem elde edilir ve ardından birim adım fonksiyonu ve adımlar yöntemi kullanılarak Laplace dönüşümü yöntemi uygulanır. Elde edilen diferansiyel denklemin çözümlerinin niteliksel özellikleri yardımıyla, söz konusu problemin çözümlerinin sınırsızlığı ve salınımları araştırılabilir.

Keywords

kısmi diferansiyel denklem, nötr tip, parçalı sabit argüman, laplace dönüşümü, salınım, sınırsızlık

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