A new approach for solving delay differential equations with time varying delay

Document Type : Research Paper

Authors

1 Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

2 Center of Excellence on Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Mashhad, Iran

10.22075/ijnaa.2023.31609.4681

Abstract

In this paper, Artificial Neural Networks are used to solve Delay Differential Equations. We have suggested an appropriate approximation function based on ANN and then by solving an optimization problem of error function, the neural network is trained. The advantage of this technique is that the proposed approximation functions, with a slight modification, can be used for most types of delay differential equations, including DDE with constant delay, time-dependent delay and pantograph delay. To demonstrate the effectiveness of the method, various examples have been tested and the validity and efficiency of the method have been shown.

Keywords

[1] S. Abbasbandy, M. Otadi, and M. Mosleh, Numerical solution of a system of fuzzy polynomials by fuzzy neural network, Inf. Sci. 178 (2008), no. 8, 1948–1960.
[2] I. Aziz and R. Rohul Amin, Numerical solution of a class of delay differential and delay partial differential equations via haar wavelet, Appl. Math. Modell. 40 (2016), no. 23-24, 10286–10299.
[3] C. Baker, C. Paul, and D. Wille, A bibliography on the numerical solution of delay differential equations, Technical Report 269, University of Manchester, 1995.
[4] M. Behroozifar and S.A. Yousefi, Numerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials, Comput. Meth. Differ. Equ. 1 (2013), no. 2, 78–95.
[5] A. Bellenand M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, 2013.
[6] G. Bhagya Raj and K.K. Kshirod K Dash, Comprehensive study on applications of artificial neural network in food process modeling, Critic. Rev. Food Sci. Nutr. 62 (2022), no. 10, 2756–2783.
[7] R.D. Driver, Ordinary and Delay Differential Equations, volume 20. Springer Science & Business Media, 2012.
[8] S. Effati, M. Mansoori, and M. Eshaghnezhad, Linear quadratic optimal control problem with fuzzy variables via neural network, J. Exper. Theor. Artific. Intell. 33 (2021), no. 2, 283–296.
[9] S. Effati and M. Pakdaman, Artificial neural network approach for solving fuzzy differential equations, Inf. Sci. 180 (2010), no. 8, 1434–1457.
[10] S. Effati and M. Pakdaman, Optimal control problem via neural networks, Neural Comput. Appl. 23 (2013), no. 7-8, 2093–2100.
[11] A. El-Safty, M.S. Salim, and M.A. El-Khatib, Convergence of the spline function for delay dynamic system, Int. J. Comput. Math. 80 (2003), no. 4, 509–518.
[12] D.J. Evans and K.R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math. 82 (2005), no. 1, 49–54.
[13] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Springer, 2014.
[14] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74, Springer Science & Business Media, 2013.
[15] G. Gybenko, Approximation by superposition of sigmoidal functions, Math. Control Signals Syst. 2 (1989), no. 4, 303–314.
[16] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, vol. 6, Elsevier, 1966.
[17] C. Hwang and M.Y. Chen, Analysis of time-delay systems using the Galerkin method, Int. J. Control, 44 (1986), no. 3, 847–866.
[18] A. Jafarian, S. Measoomy, and S. Abbasbandy, Artificial neural networks based modeling for solving Volterra integral equations system, Appl. Soft Comput. 27 (2015), 391–398.
[19] A. Kheirabadi, A.M. Vaziri, and S. Effati, Numerical solution of time-delay systems by Hermite wavelet, Int. J. Dyn. Syst. Differ. Equ. 11 (2021), no. 1, 1–17.
[20] I. Lagaris, A. Likas, and D. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Networks 9 (1998), no. 5, 987–1000.
[21] H.R. Marzban and M. Razzaghi, Solution of time-varying delay systems by hybrid functions, Math. Comput. Simul. 64 (2004), no. 6, 597–607.
[22] S.T. Mohyud-Din and A. Yildirim, Variational iteration method for delay differential equations using he’s polynomials, Z. Naturfor. A 65 (2010), no. 12, 1045–1048.
[23] F. Rihan, Delay Differential Equations and Applications to Biology, Springer, 2021.
[24] A. Saadatmandi and M. Dehghan, Variational iteration method for solving a generalized pantograph equation, Comput. Math. Appl. 58 (2009), no. 11-12, 2190–2196.
[25] J. Sabouri, S. Effati, and M. Pakdaman, A neural network approach for solving a class of fractional optimal control problems, Neural Process. Lett. 45 (2017), no. 1, 59–74.
[26] M. Shadia, Numerical solution of delay differential and neutral differential equations using spline methods, Ph.D. Thesis, Assuit University, Asyut, Egypt, 1992.
[27] F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Modell. 48 (2008), no. 3-4, 486–498.
[28] L.F. Shampine, S. Thompson, and J. Kierzenka, Solving delay differential equations with dde23, available at http://www.runet.edu/~{}thompson/webddes/tutorial.pdf.
[29] T.L. Yookesh, E.D. Boobalan, and T.P. Latchoumi, Variational iteration method to deal with time delay differential equations under uncertainty conditions, Int. Conf. Emerg. Smart Comput. Inf. (ESCI), IEEE, 2020, pp. 252–256.

Articles in Press, Corrected Proof
Available Online from 07 December 2023
  • Receive Date: 09 August 2023
  • Revise Date: 27 September 2023
  • Accept Date: 29 November 2023