jzs-10833

On the Boundedness of Solution of the Parabolic Differential Equation with Time Involution

Allaberen Ashyralyev1, 2, 3 and Amer Ahmed*1

1 Department of Mathematics, Near East University, Nicosia, Mersin 10, Turkey.

2 Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St,Moscow 117198, Russian Federation.

3 Institute of Mathematics and Mathematical Modeling, 050010, Almaty,Kazakhstan.

*Corresponding author’s e-mail: amermzory81@gmail.com and allaberen.ashyralyev@neu.edu.tr*

Original: 7 August 2020        Revised: 14 septembre 2020        Accepted: 26 September 2020        Published online: 20 December 2020  

Doi Link:

Abstract

In the present paper, the initial value problem for the parabolic type involutory partial differential equation is investigated. Applying Green’s function of space operator generated by the differential problem, we get formula for solution of this problem. The theorem on the existence and uniqueness of bounded solution of the nonlinear problem with involution is established.

Key Words: Hilbert space, Boundedness, Involution, Parabolic equation

References 

[1] C. E. Falbo. "Idempotent differential equations". Journal of Interdisciplinary Mathematics 6(3), 279–289 (2003).

[2] R. Nesbit. "Delay Differential Equations for Structured Populations". Structured Population Models in Marine,Terrestrial and Freshwater Systems. (1997).

[3] D. Przeworska-Rolewicz. "Equations with Transformed Argument". Algebraic Approach, Amsterdam, Warszawa. (1997).

[4] J. Wiener. "Generalized Solutions of Functional Differential Equations". Singapore New Jersey, London Hong Kong. (1993).

[5] A.Cabada and F. Tojo. "Differential Equations with Involution’s". Atlantis Press. (2015).

[6] A. Ashyralyev and A. S. Erdogan."Well-Posedness of a Parabolic Equation with Involution". Numerical Functional Analysis and Optimization. Vol. 38, No. 10, pp. 1295–1304. (2017).

[7] A. Ashyralyev and A. M. Sarsenbi. "Well- Posedness of an Elliptic Equation with Involution". Electronic Journal of Differential Equations, Vol. 284, No. 8, pp. 1072–6691. (2014).

[8] A. Ashyralyev, B. Karabaeva, and A. M. Sarsenbi. "Green’s function of the second order differential operator with involution". AIP Conf. Proc 1759, 1063 (2016).

[9] A. Ashyralyev and A.Sarsenbi. "Stability of a Hyperbolic Equation with the Involution".Springer Proceedings in Mathematics and Statistics. Vol. 216, pp. 204–212. (2017).

[10] A.Sarsenbi. "Well-Posedness of Mixed Problems for Parabolic type with Involution". PhD Thesis, CSU, Kazakhstan. (2019).

[11] A. Ashyralyev and T. Abbas. "A numerical algorithm for the involutory Schrödinger type problem". AIP Conf. Proc. 2183, No. 3. (2019).

[12] A. Ashyralyev, A. Erdogan, and A. Sarsenbi. "A note on the parabolic identification problem with involution and Dirichlet condition". Bullition of the Karaganda Univesity-Mathematics Vol. 99, No. 3, pp. 130–139. (2020).

[13]M. Ashyraliyev, M. Ashyralyyeva, and A. Ashyralyev. "A note on the hyperbolic-parabolic identification problem with involution and Dirichlet boundary condition". Bullition of the Karaganda Univesity-Mathematics. Vol. 99, No. 3, pp. 120–129. (2020).

[14]A. Ashyralyev and P. E. Sobolevskii. "New Difference Schemes for Partial Differential Equations". Birkhäuser Verlag, Basel, Boston, Berlin. (2004).

View Full Article