A new computational approach for solving a boundary-value problem for DEPCAG

Year 2023, , 362 - 376, 23.07.2023

Zhazira Kadirbayeva , Anar Turmaganbetkyzy Assanova , Elmira Bakirova

https://doi.org/10.31197/atnaa.1202501

Abstract

In this paper, a new computational approach is presented to solve a boundary-value problem for a differential equation with piecewise constant argument of generalized type (DEPCAG). The presented technique is based on the Dzhumabaev parametrization method. A useful numerical algorithm is developed to obtain the numerical values from the problem. Numerical experiments are conducted to demonstrate the accuracy and efficiency.

Keywords

two-point boundary value problem, differential equation, piecewise-constant argument of generalized type, Dzhumabaev parametrization method, numerical solution

Supporting Institution

Institute of Mathematics and Mathematical Modeling

Project Number

Grant No. AP19174331

Thanks

The authors would like to thank the professors Erdal Karapinar, Haydar Akca and anonymous reviewers for carefully reading the article and for their comments and suggestions which have improved the article.

References

• [1] M.U. Akhmet, Nonlinear hybrid continuous/discrete time models, Atlantis, Amsterdam-Paris, 2011.
• [2] S. Kartal, Mathematical modeling and analysis of tumor-inmune system interaction by using Lotka-Volterra predator-prey like model with piecewise constant arguments, Periodical of Engeneering and Natural Sciences, 2 (2014) 7-12.
• [3] L. Dai, Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments, World Scientific Press Publishing Co, Singapore, 2008.
• [4] F. Bozkurt, Modeling a tumor growth with piecewise constant arguments, Discrete Dynamics in Nature and Society, 2013, (2013), Article ID 841764, 8 p.
• [5] M. Akhmet, E. Yilmaz, Neural Networks with Discontinuous/Impact Activations, Springer, New York, 2014.
• [6] M.U. Akhmet, Almost periodic solution of differential equations with piecewise-constant argument of generalized type, Nonlinear Analysis-Hybrid Systems, 2, (2008) 456-467.
• [7] M.U. Akhmet, On the reduction principle for differential equations with piecewise-constant argument of generalized type, J. Math. Anal. Appl., 1, (2007) 646-663.
• [8] M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal., 66, (2007) 367-383.
• [9] S. Castillo, M. Pinto, R. Torres, Asymptotic formulae for solutions to impulsive differential equations with piecewise constant argument of generalized type, Electronic Journal of Differential Equations, 2019, (2019) 1-22.
• [10] A.T. Assanova, Zh.M. Kadirbayeva, Periodic problem for an impulsive system of the loaded hyperbolic equations, Electronic Journal of Differential Equations, 72, (2018) 1-8.
• [11] A.T. Assanova, Zh.M. Kadirbayeva, On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations, Comp. and Applied Math., 37, (2018) 4966–4976.
• [12] Zh.M. Kadirbayeva, S.S. Kabdrakhova, S.T.Mynbayeva, A Computational Method for Solving the Boundary Value Problem for Impulsive Systems of Essentially Loaded Differential Equations, Lobachevskii J. of Math., 42, (2021) 3675-3683.
• [13] K.-S. Chiu, M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applica- tions, Electron. J. Qual. Theory Differ., 2010, (2010) 1-19.
• [14] K.-S.Chiu, Global exponential stability of bidirectional associative memory neural networks model with piecewise alter- nately advanced and retarded argument, Comp. and Applied Math., 40, (2021) Article number: 263.
• [15] D.S. Dzhumabaev, Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, USSR Comput. Math. Math. Phys., 29, (1989) 34-46.
• [16] A.T.Assanova, E.A. Bakirova, Zh.M.Kadirbayeva, R.E. Uteshova, A computational method for solving a problem with parameter for linear systems of integro-differential equations, Comp. and Applied Math., 39, (2020) Article number: 248.
• [17] E.A. Bakirova, A.T. Assanova, Zh.M. Kadirbayeva, A Problem with Parameter for the Integro-Differential Equations, Mathematical Modelling and Analysis, 26, (2021) 34-54.
• [18] S.M. Temesheva, D.S. Dzhumabaev, S.S. Kabdrakhova, On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem, Lobachevskii J. of Math., 42, (2021) 606-612.
• [19] A.M. Nakhushev A.M., Loaded equations and their applications, Nauka, Moscow, (2012) (in Russian).
• [20] A.M. Nakhushev, An approximation method for solving boundary value problems for differential equations with applications to the dynamics of soil moisture and groundwater, Differential Equations, 18, (1982) 72-81.
• [21] V.M. Abdullaev, K.R. Aida-zade, Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations, Comput. Math. Math. Phys., 54, (2014) 1096-1109.
• [22] M.T. Dzhenaliev, Loaded equations with periodic boundary conditions, Differential Equations, 37, (2001) 51-57.
• [23] A.T. Assanova, A.E. Imanchiyev, Zh.M. Kadirbayeva, Numerical solution of systems of loaded ordinary differential equa- tions with multipoint conditions, Comput. Math. Math. Phys., 58, (2018) 508-516.
• [24] D.S. Dzhumabaev, Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro-differential equations, Math. Methods Appl. Sci., 41, (2018) 1439-1462.
• [25] G.-C. Wu, D. Baleanu, W.-H. Luo, Lyapunov functions for Riemann–Liouville-like fractional difference equations, Appl. Math. Comput., 314, (2017) 228–236.
• [26] S. Muthaiah M. Murugesan, N.Thangaraj, Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Adv. Theory Nonlinear Anal. Appl., 3(3), (2019) 162-173.
• [27] A. Hamoud, Existence and Uniqueness of Solutions for Fractional Neutral Volterra-Fredholm Integro Differential Equations, Adv. Theory Nonlinear Anal. Appl., 4(4), (2020) 321 - 331.
• [28] A. Hamoud, N. Mohammed, K. Ghadle, Existence and Uniqueness Results for Volterra-Fredholm Integro Differential Equations, Adv. Theory Nonlinear Anal. Appl., 4(4), (2020) 361-372.
• [29] F. Al-Saar, K. Ghadle, Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods, Adv. Theory Nonlinear Anal. Appl., 5(2), (2021) 260-276.
• [30] R. Nedjem Eddine, S. Pinelas, Solving nonlinear integro-differential equations using numerical method, Turkish Journal of Mathematics, 46 (2022) 675-687.
• [31] D.S. Dzhumabaev, E.A. Bakirova. S.T. Mynbayeva, A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation, Math. Methods Appl. Sci., 43, (2020) 1788-1802.
• [32] M. Song, M.Z. Liu, Stability of Analytic and Numerical Solutions for Differential Equations with Piecewise Continuous Arguments, Abstract and Applied Analysis, 2012, (2012): Article ID 258329.
• [33] P. Hammachukiattikul, B. Unyong, R. Suresh, G. Rajchakit, R. Vadivel, N. Gunasekaran, Chee Peng Lim, Runge-Kutta Fehlberg Method for Solving Linear and Nonlinear Fuzzy Fredholm Integro-Differential Equations, Appl. Math. Inf. Sci., 15, (2021) 43-51.