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

Yıl 2023, Cilt: 03 Sayı: 01, 29 - 35, 31.07.2023

Zekeriya Özkan , Duygu Aruğaslan Çinçin , Marat Akhmet

Öz

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.

Anahtar Kelimeler

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• [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), 521-538, 2018.
• [15] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific: Singapore, New Jersey, London, 1995.
• [16] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific: Singapore, New Jersey, London, Hong Kong, 1995.
• [17] I. Győri, “On approximation of the solutions of delay differential equations by using piecewise constant arguments,” Internat. J. Math. Math. Sci., 14(1), 111-126, 1991.
• [18] A. R. Aftabizadeh and J. Wiener, “Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument,” Appl. Anal., 26(4), 327-333, 1988.
• [19] I. Győri, and G. Ladas, “Linearized oscillations for equations with piecewise constant arguments,” Differential and Integral Equations, 2(2), 123-131, 1989.
• [20] A. R. Aftabizadeh, J. Wiener and J. Ming Xu, “Oscillatory and periodic solutions of delay differential equations with piecewise constant argument,” Proc. Amer. Math. Soc., 99(4), 673-679, 1987.
• [21] Y. K. Huang, “Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument,” J. Math. Anal. Appl., 149(1), 70-85, 1990.
• [22] H. Liang and G. Wang, “Existence and uniqueness of periodic solutions for a delay differential equation with piecewise constant arguments,” Port. Math., 66(1), 1-12, 2009.
• [23] R. Yuan, “The existence of almost periodic solutions of retarded differential equations with piecewise constant argument,” Nonlinear Anal., 48(7), Ser. A: Theory Methods, 1013-1032, 2002.
• [24] J. Wiener, “Boundary value problems for partial differential equations with piecewise constant delay,” Internat. J. Math. Math. Sci., 14(2), 363-379, 1991.
• [25] J. Wiener and L. Debnath, “A wave equation with discontinuous time delay,” Internat. J. Math. Math. Sci., 15(4), 781-788, 1992.
• [26] J. Wiener and L. Debnath, “Boundary value problems for the diffusion equation with piecewise continuous time delay,” Internat. J. Math. Math. Sci., 20(1), 187-195, 1997.
• [27] J. Wiener and W. Heller, “Oscillatory and periodic solutions to a diffusion equation of neutral type,” Internat. J. Math. Math. Sci., 22(2), 313-348, 1999.
• [28] J. Wiener and V. Lakshmikantham, “Complicated dynamics in a delay Klein-Gordon equation,” Nonlinear Anal., 38(1), Ser. B: Real World Appl., 75-85, 1999.
• [29] Q. Wang and J. Wen, “Analytical and numerical stability of partial differential equations with piecewise constant arguments,” Numer. Methods Partial Differential Equations, 30(1), 1-16, 2014.
• [30] T. Veloz and M. Pinto, “Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument,” J. Math. Anal. Appl., 426(1), 330-339, 2015.
• [31] H. Bereketoğlu and M. Lafci, “Behavior of the solutions of a partial differential equation with a piecewise constant argument,” Filomat, 31.19, 5931-5943, 2017.
• [32] Q. Wang, “Stability of numerical solution for partial differential equations with piecewise constant arguments,” Advances in Difference Equations, 2018(1), 1–13, 2018.
• [33] M. L. Büyükkahraman and H. Bereketoğlu, “On a partial differential equation with piecewise constant mixed arguments,” Iranian Journal of Science and Technology, Transactions A: Science, 44.6, 1791-1801, 2010.
• [34] J. Wiener, Generalized solutions of functional differential equations, World Scientific. Publishing Co., Inc, River Edge, NJ. xiv+410 pp. ISBN:981-02-1207-0, 1993.
• [35] S. M. Shah, H. Poorkarimi and J. Wiener, Bounded solutions of retarded nonlinear hyperbolic equations, Bull. Allahabad. Math. Soc., 1, 1-14, 1986.
• [36] H. Chi et al., “On the exponential growth of solutions to nonlinear hyperbolic equations,” Internat. J. Math. Math. Sci., 12(3), 539-545, 1989.
• [37] J. Wiener and L. Debnath, “A survey of partial differential equations with piecewise continuous arguments,” Internat. J. Math. Math. Sci., 18(2), 209-228, 1995.
• [38] H. Poorkarimi and J. Wiener, “Bounded solutions of nonlinear parabolic equations with time delay,” Proceedings of the 15th Annual Conference of Applied Mathematics (Edman, OK, 1999), 87-91 (electronic), Electron. J. Differ. Equ. Conf., 2, Southwest Texas State Univ., San Marcos, TX, 1999.