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Journals / Researcher / 2023 / Cilt: 03 Sayı: 01

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

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

Zekeriya ÖZKAN
Duygu ARUĞASLAN ÇİNÇİN
Marat AKHMET
Journal
Researcher
Pages
29–35
Publication Date
2023-07-31
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:There have been very few studies on partial differential equations including piecewise constant arguments and generalized piecewise constant arguments.
📊
Key Findings & Data:However, as far as we know, there is no study conducted on neutral type partial differential equations including piecewise constant argument of generalized type.
💡
Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:kısmi diferansiyel denklem · nötr tip · parçalı sabit argüman

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.

Özet

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.

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Cite This Article

ÖZKAN, Z., ÇİNÇİN, D. A., AKHMET, M. (2023). On Solutions of A Parabolic Partial Differential Equation of Neutral Type Including Piecewise Continuous Time Delay. Researcher, Cilt: 03 Sayı: 01, 29–35. https://ajindex.com/articles/957763

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References (14)

  1. S. Busenberg and K. L. Cooke, “Models of vertically transmitted diseases with sequential-continuous dynamics,” in Nonlinear Phenomena in Mathematical Sciences, Lakshmikantham, V. (editor), Academic Press, New York, 179-187, 1982
  2. K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” J. Math. Anal. and Appl., 99(1), 265-297, 1984
  3. Y. Muroya, “New contractivity condition in a population model with piecewise constant arguments,” J. Math. Anal. Appl., 346(1), 65-81, 2008
  4. M. U. Akhmet, D. Aruğaslan, and E. Yılmaz, “Stability in cellular neural networks with a piecewise constant argument,” Journal of Computational and Applied Mathematics, 233, 2365-2373, 2010
  5. M. U. Akhmet and E. Yılmaz, Neural Networks with Discontinuous/Impact Activations, Springer: New York, 2013
  6. M. U. Akhmet, “Integral manifolds of differential equations with piecewise constant argument of generalized type,” Nonlinear Anal, 66, 367-383, 2007
  7. M. U. Akhmet, Nonlinear Hybrid Continuous Discrete-Time Models, Atlantis Press: Amsterdam-Paris, 2011
  8. M. U. Akhmet, “Functional Differential Equations with Piecewise Constant Argument,” In: Regularity and Stochasticity of Nonlinear Dynamical Systems, Springer, 79-109, 2018
  9. M. Akhmet, M. Dauylbayev, and A. Mirzakulova, “A singularly perturbed differential equation with piecewise constant argument of generalized type,” Turkish Journal of Mathematics, 42(1), 1680-1685 (IF 0.614), 2018
  10. M. U. Akhmet, “Almost periodic solutions of the linear differential equation with piecewise constant argument,” Discrete and Impulsive Systems, Series A, Mathematical Analysis, 16, 743-753, 2009
  11. M. U. Akhmet and C. Büyükadalı, “Differential equations with state-dependent piecewise constant argument,” Nonlinear Analysis: Theory, methods and applications, 72 (11), 4200-4211 (IF 2.012), 2010
  12. M. U. Akhmet, “Stability of differential equations with piecewise constant arguments of generalized type,” Nonlinear Anal., 68, 794-803, 2008
  13. D. Aruğaslan and N. Cengiz, “Green’s Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument,” Süleyman Demirel University Journal of Natural and Applied Sciences, 21 (1), 266-278, 2017
  14. D. Aruğaslan, and N. Cengiz, “Existence of periodic solutions for a mechanical system with piecewise constant forces,” Doi: 10.15672/HJMS.2017.469 Hacet. J. Math. Stat., 47, (no. 3), 52