Existence and uniqueness theorems for fractional differential equations with proportional delay

Document Type : Research Paper

Authors

1 Department of Mathematics, Shivaji University, Kolhapur - 416004, India.

2 Department of Mathematics, Jaysingpur College, Jaysingpur (Affiliated to Shivaji University, Kolhapur) - 416101, India.

Abstract

In this paper, we apply the successive approximation method (SAM) to solve nonlinear differential equations (DEs) with proportional delay. Utilizing SAM, we establish results on existence and uniqueness. Differential equations (DEs) with proportional delay represent a particular case of time-dependent delay differential equations (DDEs). We demonstrate that the equilibrium solution of time-dependent DDEs is asymptotically stable over finite time intervals. We obtained a series solution for the pantograph and Ambartsumian equations and proved their convergence. Furthermore, we prove that the zero solution of the pantograph and Ambartsumian equations is asymptotically stable. The outcomes of integer order obtained for DEs with proportional delay and time-dependent DDEs have been extended to the initial value problem (IVP) for fractional DDEs and a system of fractional DDEs involving the Caputo fractional derivative. Finally, we illustrate SAMs efficacy using particular non-linear DEs with proportional delay. The results obtained for non-linear DEs with proportional delay by SAM are compared with exact solutions and other iterative methods. It is noted that SAM is easier to use than other techniques, and the solutions obtained using SAM are consistent with the exact solution.

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References

  • [1] K. Ali, M. El Salam, and E. Mohamed, Chebyshev operational matrix for solving fractional order delay-differential equations using spectral collocation method, Arab Journal of Basic and Applied Sciences, 26(1) (2019), 342–353.
  • [2] A. Alomari, M. Noorani, and R. Nazar, Solution of delay differential equation by means of homotopy analysis method, Acta Applicandae Mathematicae, 108 (2009), 395–412.
  • [3] V. Ambartsumian, On the fluctuation of the brightness of the milky way, Dokl. Akad. Nauk SSSR, 44 (1944), 244–247.
  • [4] T. Apostol and C. Ablow, Mathematical analysis, Physics Today, 11(7) (1958), 32.
  • [5] S. Bhalekar and V. Daftardar-Gejji, A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, Journal of Fractional Calculus and Applications,1(5) (2011), 1–9.
  • [6] S. Bhalekar, Stability analysis of a class of fractional delay differential equations, Pramana, 81(2) (2013), 215–224.
  • [7] S. Bhalekar, Stability and bifurcation analysis of a generalized scalar delay differential equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(8) (2016), 084306.
  • [8] S. Bhalekar, Analysis of 2-term fractional-order delay differential equations, in: Fractional Calculus and Fractional Differential Equations, Springer, (2019), 59–75.
  • [9] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, Fractional bloch equation with delay, Computers & Mathematics with Applications, 61(5) (2011), 1355–1365.
  • [10] S. Bhalekar and J. Patade, Series solution of the pantograph equation and its properties, Fractal and Fractional, 1(1) (2017), 16.
  • [11] S. Bhalekar and J. Patade, Analytical solutions of nonlinear equations with proportional delays, Appl. Comput. Math., 15(3) (2016), 331–345.
  • [12] M. Buhmann and A. Iserles, Stability of the discretized pantograph differential equation, Mathematics of Computation, 60(202) (1993), 575–589.
  • [13] E. Coddington, An introduction to ordinary differential equations, Courier Corporation, 2012.
  • [14] C. Cuvelier, A. Segal, and A. Steenhoven, Finite element methods and Navier-Stokes equations, Springer Science & Business Media, 22 (1986).
  • [15] V. Daftardar-Gejji, Y. Sukale, and S. Bhalekar, Solving fractional delay differential equations:a new approach, Fractional Calculus and Applied Analysis, 18(2) (2015), 400–418.
  • [16] W. Deng, Y. Wu, and C. Li, Stability analysis of differential equations with time-dependent delay, International Journal of Bifurcation and Chaos, 16(02) (2006), 465–472.
  • [17] D. Evans and K. Raslan, The Adomian decomposition method for solving delay differential equation, International Journal of Computer Mathematics, 82(1) (2005), 49–54.
  • [18] M. Iqbal, U. Saeed, and S. Mohyud-Din, Modified laguerre wavelets method for delay differential equations of fractional-order, Egyptian journal of basic and applied sciences, 2(1) (2015), 50–54.
  • [19] A. Kilbas, H. Srivastava, and J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006).
  • [20] E. Kaslik and S. Sivasundaram, Analytical and numerical methods for the stability analysis of linear fractional delay differential equations, Journal of Computational and Applied Mathematics, 236(16) (2012), 4027–4041.
  • [21] V. Latha, F. Rihan, R. Rakkiyappan, and G. Velmurugan, A fractional-order delay differential model for ebola infection and cd8+ t-cells response: stability analysis and hopf bifurcation, International Journal of Biomathematics, 10(08) (2017), 1750111.
  • [22] J. Ockendon and A. Tayler, The dynamics of a current collection system for an electric locomotive, Proceedings of the Royal Society of London, A. Mathematical and Physical Sciences, 322(1551) (1971), 447–468.
  • [23] J. Patade and S. Bhalekar, On analytical solution of ambartsumian equation, National Academy Science Letters, 40 (2017), 291–293.
  • [24] J. Patade and S. Bhalekar, Analytical solution of pantograph equation with incommensurate delay, Physical Sciences Reviews, 2(9) (2017), 20165103.
  • [25] J. Patade, Series solution of system of fractional order ambartsumian equations: Application in astronomy, arXiv preprint, arXiv:2008.04904.
  • [26] Y. Rangkuti and M. Noorani, The exact solution of delay differential equations using coupling variational iteration with Taylor series and small term, Bulletin of Mathematics, 4(01) (2012), 1–15.
  • [27] N. Anakira, A. Alomari, and I. Hashim, Optimal homotopy asymptotic method for solving delay differential equations, Mathematical Problems in Engineering, 2013.
  • [28] F. Rihan, Fractional-order delay differential equations with predator-prey systems, in:Delay Differential Equations and Applications to Biology, Springer, (2021), 211–232.
  • [29] F. Rihan and G. Velmurugan, Dynamics of fractional-order delay differential model for tumor immune system, Chaos, Solitons & Fractals, 132 (2020), 109592.
  • [30] S. Sabermahani, Y. Ordokhani, and S. Yousefi, Fractional-order fibonacci-hybrid functions approach for solving fractional delay differential equations, Engineering with Computers,36(2) (2020), 795–806.
  • [31] S. Saker, S. Selvarangam, S. Geetha, E. Thandapani, and J. Alzabut, Asymptotic behavior of third order delay difference equations with a negative middle term, Advances in Difference Equations, 1 (2021), 1–12.
  • [32] J. Senecal and W. Ji, Approaches for mitigating over-solving in multiphysics simulations, International Journal for Numerical Methods in Engineering, 112(6) (2017), 503–528.
  • [33] V. Uchaikin, Fractional derivatives for physicists and engineers, Berlin: Springer, 2 (2013).
  • [34] K. Vidhyaa, E. Thandapani, J. Alzabut, and A. Ozbekler, Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term, Electronic Journal of Differential Equations, 1 (2023), 45–12.
Computational Methods for Differential Equations
Volume 13, Issue 3
July 2025
Pages 904-918

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History

  • Receive Date: 22 July 2023
  • Revise Date: 10 January 2024
  • Accept Date: 06 May 2024

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