Asymptotic behaviours of the solutions of neutral type Volterra integro-differential equations and some numerical solutions via differential transform method

Year 2021, , 49 - 57, 30.04.2021

Yener Altun

https://doi.org/10.51354/mjen.878066

Cited By: 1

Abstract

 In this manuscript, we consider the first order neutral Volterra integro-differential equation (NVIDE) with delay argument. Firstly, we obtain novel sufficient conditions to establish the asymptotic behaviours of solutions of considered NVIDE using the Lyapunov method and present an example to demonstrate the applicability of proposed method. Secondly, we get some numerical solutions for a particular case of considered NVIDE via the differential transformation method (DTM). The results of this manuscript are novel and they improve some existing ones in the literature.

Keywords

Asymptotic behavior, Lyapunov functional, NVIDE, DTM

References

• Agarwal, R. P., Grace, S. R., “Asymptotic stability of certain neutral differential equations”, Math. Comput. Modelling, 31(8-9), (2000), 9–15.
• Altun, Y., “A new result on the global exponential stability of nonlinear neutral volterra integro-differential equation with variable lags”, Math. Nat. Sci., 5, (2019), 29–43.
• Altun, Y., “Further results on the asymptotic stability of Riemann–Liouville fractional neutral systems with variable delays”, Adv. Difference Equ., 437,(2019), 1-13.
• Altun, Y., “Improved results on the stability analysis of linear neutral systems with delay decay approach”, Math Meth Appl Sci., 43, (2020), 1467–1483.
• Altun, Y., Tunç C., “On the global stability of a neutral differential equation with variable time-lags”, Bull. Math. Anal. Appl., 9(4), (2017), 31-41.
• Hale, J., Verduyn Lunel, S.M., “Introduction to functional-differential equations”, Springer Verlag, (1993), New York.
• Kolmanovskii, V., Myshkis, A., “Applied Theory of Functional Differential Equations”, Kluwer Academic Publisher Group, (1992), Dordrecht.
• Kulenovic, M., Ladas, Meimaridou, A., “Necessary and sufficient conditions for oscillations of neutral differential equations”, J. Aust. Math. Soc. Ser. B, 28, (1987), 362-375.
• Park, J. H., Kwon, O. M., “Stability analysis of certain nonlinear differential equation”, Chaos Solitons Fractals, 37, (2008), 450-453.
• Raffoul, Y., “Boundedness in nonlinear functional differential equations with applications to Volterra integro differential equations”, J. Integral Equ. Appl., 16(4), (2004), 375–388.
• Rama Mohana Rao, M., Raghavendra, V., “Asymptotic stability properties of Volterra integro-differential equations”, Nonlinear Anal., 11(4), (1987), 475–480.
• Tunç, C., Altun, Y., “Asymptotic stability in neutral differential equations with multiple delays”, J. Math. Anal., 7(5), (2016), 40–53.
• Vanualailai, J, Nakagiri, S., “Stability of a system of Volterra integro-differential equations”, J. Math. Anal. Appl., 281(2), (2003), 602–619.
• Pukhov, G. E., “Differential Transformations and Mathematical Modelling of Physical Processes”, Naukova Dumka, (1986), Kiev.
• Zhou, J. K., “Differential Transformation and Its Application for Electrical Circuits”, Huazhong University Press, (1986), Wuhan.
• Arslan, D., “Approximate Solutions of Singularly Perturbed Nonlinear Ill-posed and Sixth-order Boussinesq Equations with Hybrid Method”, BEU Journal of Science, 8(2), (2019), 451-458.
• Arslan, D., “A Novel Hybrid Method for Singularly Perturbed Delay Differential Equations”, Gazi University Journal of Sciences, 32(1), (2019), 217-223.
• Arslan, D., “Numerical Solution of Nonlinear the Foam Drainage Equation via Hybrid Method”, New Trends in Mathematical Sciences, 8(1), (2020), 50-57.
• Ayaz, F., “Applications of Differential Transform Method to Differential-Algebraic Equations”, Applied Mathematics and Computation, 152, (2004), 649-657.
• Rebenda, J., Smarda, Z., Khan, Y., “A New Semi-analytical Approach for Numerical Solving of Cauchy Problem for Differential Equations with Delay”, Filomat, 31(15), (2017), 4725–4733.
• Arikoglu, A., Ozkol, I., “Solutions of integral and integro-differential equation systems by using differential transform method” Comput. Math. Appl., 56, (2008), 2411–2417.
• Zou, L., Wang, Z., Zong, Z., “Generalized differential transform method to differential-difference equation”, Phys. Lett. A, 373, (2009), 4142–4151.
• Chen, C. K., Ho, S. H., “Solving partial differential equations by two dimensional differentialtransform”, Appl. Math. Comput., 106, (1999), 171–179.
• Arikoglu, A., Ozkol, I., “Solution of fractional differential equations by using differential transform method”, Chaos Soliton. Fract., 34, (2007), 1473–1481.