New Results on a Partial Differential Equation with General Piecewise Constant Argument

Year 2023, Volume: 15 Issue: 2, 237 - 246, 31.12.2023

Marat Akhmet Duygu Aruğaslan Çinçin Zekeriya Özkan

https://doi.org/10.47000/tjmcs.1166651

Abstract

There have been very few analyses on partial differential equations with piecewise constant arguments and as far as we know, there is no study conducted on heat equation with piecewise constant argument of generalized type. Motivated by this fact, this study aims to solve and analyse heat equation with piecewise constant argument of generalized type. We obtain formal solution of heat equation with piecewise constant argument of generalized type by separation of variables. We apply the Laplace transform method using unit step function and method of steps on each consecutive intervals. We investigate stability, oscillation, boundedness properties of solutions.

Keywords

Partial differential equation, piecewise constant argument, Laplace transformation, heat equation, zeros of solutions, oscillation

References

• Aftabizadeh, A.R.,Wiener, J., Ming Xu, J., Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc., 99(4)(1987), 673–679.
• Aftabizadeh, A.R., Wiener, J., Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument, Appl. Anal., 26(4)(1988), 327–333.
• Akhmet, M.U., Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal, 66(2007), 367–383.
• Akhmet, M.U., Stability of differential equations with piecewise constant arguments of generalized type, Nonlinear Anal., 68(2008), 794–803.
• Akhmet, M.U., Almost periodic solutions of the linear differential equation with piecewise constant argument, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 16(2009), 743–753.
• Akhmet, M.U., Nonlinear Hybrid Continuous Discrete-Time Models, Atlantis Press: Amsterdam-Paris, 2011.
• Akhmet, M.U., Functional Differential Equations with Piecewise Constant Argument, In: Regularity and Stochasticity of Nonlinear Dynamical Systems, Springer, 2018.
• Akhmet, M.U., Aruğaslan, D., Yılmaz, E., Stability in cellular neural networks with a piecewise constant argument, Journal of Computational and Applied Mathematics, 233(2010), 2365–2373.
• Akhmet, M.U., Büyükadalı, C., Differential equations with state-dependent piecewise constant argument, Nonlinear Analysis: Theory, methods and applications, 72(11)(2010), 4200–4211.
• Akhmet, M., Dauylbayev, M., Mirzakulova, A., A singularly perturbed differential equation with piecewise constant argument of generalized type, Turkish Journal of Mathematics, 42(1)(2018), 1680–1685.
• Akhmet, M.U., Yılmaz, E., Neural Networks with Discontinuous/Impact Activations, Springer: New York, 2013.
• Aruğaslan, D., Cengiz, N., Green’s function and periodic solutions of a Spring-Mass system in which the forces are functionally dependent on piecewise constant argument, S¨uleyman Demirel University Journal of Natural and Applied Sciences, 21(1)(2017), 266–278.
• Aruğaslan, D., Cengiz, N., Existence of periodic solutions for a mechanical system with piecewise constant forces, Hacet. J. Math. Stat., 47(3)(2018), 521–538.
• Bainov, D.D., Simeonov, P.S., Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific: Singapore, New Jersey, London, 1995.
• Bereketoğlu, H., Lafcı, M., Behavior of the solutions of a partial differential equation with a piecewise constant argument, Filomat, 31(19)(2017), 5931–5943.
• Busenberg, S., Cooke, K.L., Models of vertically transmitted diseases with sequential-continuous dynamics, in Nonlinear Phenomena in Mathematical Sciences, Lakshmikantham, V. (editor), Academic Press, New York, (1982), 179–187.
• Büyükahraman, M.L., Bereketoğlu, H., On a partial differential equation with piecewise constant mixed arguments, Iranian Journal of Science and Technology, Transactions A: Science 44(6)(2020), 1791–1801.
• Chi, H., Poorkarimi, H., Wiener, J., Shah, S.M., On the exponential growth of solutions to nonlinear hyperbolic equations, Internat. J. Math. Math. Sci., 12(3)(1989), 539–545.
• Cooke, K.L., Wiener, J., Retarded differential equations with piecewise constant delays, J. Math. Anal. and Appl., 99(1)(1984), 265–297.
• Farlow, S.J., Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, 1982.
• Györi, I., On approximation of the solutions of delay differential equations by using piecewise constant arguments, Internat. J. Math. Math. Sci., 14(1)(1991), 111–126.
• Györi, I., Ladas, G., Linearized oscillations for equations with piecewise constant arguments, Differential and Integral Equations, 2(2)(1989), 123–131.
• Huang, Y.K., Oscillations and asymptotic stability of solutions of first order neutral differential equations with piecewise constant argument, J. Math. Anal. Appl., 149(1)(1990), 70–85.
• Liang, H., Wang, G., Existence and uniqueness of periodic solutions for a delay differential equation with piecewise constant arguments, Port. Math., 66(1)(2009), 1–12.
• Muroya, Y., New contractivity condition in a population model with piecewise constant arguments, J. Math. Anal. Appl., 346(1)(2008), 65–81.
• Pinto, M., Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments, Math. Comput. Modelling, 49(9-10)(2009), 1750–1758.
• Poorkarimi, H., Wiener, J., 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.
• Samoilenko, A.M., Perestyuk, N.A., Impulsive Differential Equations, World Scientific: Singapore, New Jersey, London, Hong Kong, 1995.
• Shah, S., Poorkarimi, M H., Wiener, J., Bounded solutions of retarded nonlinear hyperbolic equations, Bull. Allahabad. Math. Soc., 1(1986), 1–14.
• Veloz, T., Pinto, M., Existence, computability and stability for solutions of the diffusion equation with general piecewise constant argument, J. Math. Anal. Appl., 426(1)(2015), 330–339.
• Wang, Q., Stability of numerical solution for partial differential equations with piecewise constant arguments, Advances in Difference Equations, 2018(1)(2018), 1–13.
• Wang, Q., Wen, J., Analytical and numerical stability of partial differential equations with piecewise constant arguments, Numer. Methods Partial Differential Equations, 30(1)(2014), 1–16.
• Wiener, J., Boundary value problems for partial differential equations with piecewise constant delay, Internat. J. Math. Math. Sci., 14(2)(1991), 363–379.
• Wiener, J., Generalized Solutions of Functional Differential Equations, World Scientific, Publishing Co., Inc, River Edge, NJ, 1993.
• Wiener, J., Debnath, L., A wave equation with discontinuous time delay, Internat. J. Math. Math. Sci. 15(4)(1992), 781–788.
• Wiener, J., Debnath, L., A survey of partial differential equations with piecewise continuous arguments, Internat. J. Math. Math. Sci., 18(2)(1995), 209–228.
• Wiener, J., Debnath, L., Boundary value problems for the diffusion equation with piecewise continuous time delay, Internat. J. Math. Math. Sci. 20(1)(1997), 187–195.
• Wiener, J., Heller, W., Oscillatory and periodic solutions to a diffusion equation of neutral type, Internat. J. Math. Math. Sci. 22(2)(1999), 313–348.
• Wiener, J., Lakshmikantham, V., Complicated dynamics in a delay Klein-Gordon equation, Nonlinear Anal. 38(1), Ser. B:Real World Appl., (1999), 75–85.
• Yuan, R., The existence of almost periodic solutions of retarded differential equations with piecewise constant argument, Nonlinear Anal. Ser. A:Theory Methods, 48(7)(2002), 1013–1032.