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Higher order numerical methods for fractional delay differential equations

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Abstract

In this paper, we present a new family of higher-order numerical methods for solving non-linear fractional delay differential equations (FDDEs) along with the error analysis. Further, we solve various non-trivial systems of FDDEs to illustrate their applicability and utility. By using the proposed numerical methods, computational time is reduced drastically. These methods take only 5 to 10 percent of the time required for other methods such as the fractional Adams method (FAM). Furthermore, these methods converge for very small values of fractional derivative while FAM and the new predictor-corrector method (NPCM) introduced by Daftardar-Gejji et al. [1] do not converge. The order of convergence of the proposed methods is \(r+\alpha \), where r is the order of fractional backward difference formulae and \(\alpha \) denotes the order of the fractional derivative. Thus these methods have a higher order of accuracy than FAM or NPCM.

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References

  1. V. Daftardar-Gejji, Y. Sukale, and S. Bhalekar, “Solving fractional delay differential equations: a new approach,” Fractional Calculus and Applied Analysis, vol. 18, no. 2, pp. 400–418, 2015.

    Article  MathSciNet  Google Scholar 

  2. L. Liu, S. Zhang, L. Zhang, G. Pan, and J. Yu, “Multi-uuv maneuvering counter-game for dynamic target scenario based on fractional-order recurrent neural network,” IEEE Transactions on Cybernetics, 2022.

  3. L. Liu, J. Wang, L. Zhang, and S. Zhang, “Multi-auv dynamic maneuver countermeasure algorithm based on interval information game and fractional-order de,” Fractal and Fractional, vol. 6, no. 5, p. 235, 2022.

    Article  Google Scholar 

  4. M. S. Mahmoud, Robust control and filtering for time-delay systems. CRC Press, 2018.

  5. R. Garrappa and E. Kaslik, “On initial conditions for fractional delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 90, p. 105359, 2020.

    Article  MathSciNet  Google Scholar 

  6. I. R. Epstein and Y. Luo, “Differential delay equations in chemical kinetics. nonlinear models: The cross-shaped phase diagram and the oregonator,” The Journal of chemical physics, vol. 95, no. 1, pp. 244–254, 1991.

    Article  Google Scholar 

  7. Y. Kuang, Delay differential equations: with applications in population dynamics, vol. 191. Academic press, 1993.

  8. L. Fridman, E. Shustin, and E. Fridman, “Steady modes in the relay control systems with time delay and periodic disturbances,” in 1997 1st International Conference, Control of Oscillations and Chaos Proceedings (Cat. No. 97TH8329), vol. 1, pp. 75–78, IEEE, 1997.

  9. L. Davis, “Modifications of the optimal velocity traffic model to include delay due to driver reaction time,” Physica A: Statistical Mechanics and its Applications, vol. 319, pp. 557–567, 2003.

    Article  Google Scholar 

  10. V. Daftardar-Gejji, “Fractional calculus: Theory and applications,” Narosa, New Delhi, 2013.

    Google Scholar 

  11. Z. Wang, X. Wang, Y. Li, and X. Huang, “Stability and hopf bifurcation of fractional-order complex-valued single neuron model with time delay,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, p. 1750209, 2017.

    Article  MathSciNet  Google Scholar 

  12. S. Chen, J. Cao, and I. Stamova, “Persistence of traveling waves to the time fractional keller-segel system with a small parameter,” Authorea Preprints, 2023.

  13. H.-Y. Jin and Z.-A. Wang, “Boundedness, blowup and critical mass phenomenon in competing chemotaxis,” Journal of Differential Equations, vol. 260, no. 1, pp. 162–196, 2016.

    Article  MathSciNet  Google Scholar 

  14. B. Tao, M. Xiao, Q. Sun, and J. Cao, “Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model,” Neurocomputing, vol. 275, pp. 677–686, 2018.

    Article  Google Scholar 

  15. S. Liu, R. Yang, X.-F. Zhou, W. Jiang, X. Li, and X.-W. Zhao, “Stability analysis of fractional delayed equations and its applications on consensus of multi-agent systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 73, pp. 351–362, 2019.

    Article  MathSciNet  Google Scholar 

  16. X. Wang, Z. Wang, and J. Xia, “Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders,” Journal of the Franklin Institute, vol. 356, no. 15, pp. 8278–8295, 2019.

    Article  MathSciNet  Google Scholar 

  17. F. Rihan, Q. Al-Mdallal, H. AlSakaji, and A. Hashish, “A fractional-order epidemic model with time-delay and nonlinear incidence rate,” Chaos, Solitons & Fractals, vol. 126, pp. 97–105, 2019.

    Article  MathSciNet  Google Scholar 

  18. H. Li, R. Peng, and Z.-a. Wang, “On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms,” SIAM Journal on Applied Mathematics, vol. 78, no. 4, pp. 2129–2153, 2018.

    Article  MathSciNet  Google Scholar 

  19. Z. Lin and H. Wang, “Modeling and application of fractional-order economic growth model with time delay,” Fractal and Fractional, vol. 5, no. 3, p. 74, 2021.

    Article  Google Scholar 

  20. V. P. Latha, F. A. 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, vol. 10, no. 08, p. 1750111, 2017.

    Article  MathSciNet  Google Scholar 

  21. M. Xiao, W. X. Zheng, J. Lin, G. Jiang, L. Zhao, and J. Cao, “Fractional-order pd control at hopf bifurcations in delayed fractional-order small-world networks,” Journal of the Franklin Institute, vol. 354, no. 17, pp. 7643–7667, 2017.

    Article  MathSciNet  Google Scholar 

  22. M. Das and G. Samanta, “A delayed fractional order food chain model with fear effect and prey refuge,” Mathematics and Computers in Simulation, vol. 178, pp. 218–245, 2020.

    Article  MathSciNet  Google Scholar 

  23. F. Rihan and G. Velmurugan, “Dynamics of fractional-order delay differential model for tumor-immune system,” Chaos, Solitons & Fractals, vol. 132, p. 109592, 2020.

    Article  MathSciNet  Google Scholar 

  24. V. Lakshmikantham and A. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.

    Article  MathSciNet  Google Scholar 

  25. M. Benchohra, J. Henderson, S. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008.

    Article  MathSciNet  Google Scholar 

  26. X. Zhang, “Some results of linear fractional order time-delay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407–411, 2008.

    Article  MathSciNet  Google Scholar 

  27. J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 2091–2105, 2009.

    Article  MathSciNet  Google Scholar 

  28. C. Liao and H. Ye, “Existence of positive solutions of nonlinear fractional delay differential equations,” Positivity, vol. 13, no. 3, pp. 601–609, 2009.

    Article  MathSciNet  Google Scholar 

  29. J. K. Hale and S. M. V. Lunel, Introduction to functional differential equations, vol. 99. Springer Science & Business Media, 2013.

  30. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 3–22, 2002.

    Article  MathSciNet  Google Scholar 

  31. 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, vol. 1, no. 5, pp. 1–9, 2011.

    Google Scholar 

  32. B. P. Moghaddam and Z. S. Mostaghim, “A numerical method based on finite difference for solving fractional delay differential equations,” Journal of Taibah University for Science, vol. 7, no. 3, pp. 120–127, 2013.

    Article  Google Scholar 

  33. Z. Wang, “A numerical method for delayed fractional-order differential equations,” Journal of Applied Mathematics, vol. 2013, 2013.

  34. R. K. Pandey, N. Kumar, and R. Mohaptra, “An approximate method for solving fractional delay differential equations,” International Journal of Applied and Computational Mathematics, vol. 3, no. 2, pp. 1395–1405, 2017.

    Article  MathSciNet  Google Scholar 

  35. K. Diethelm, N. J. Ford, A. D. Freed, and Y. Luchko, “Algorithms for the fractional calculus: a selection of numerical methods,” Computer methods in applied mechanics and engineering, vol. 194, no. 6-8, pp. 743–773, 2005.

    Article  MathSciNet  Google Scholar 

  36. C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.

    Article  MathSciNet  Google Scholar 

  37. Z. M. Odibat and S. Momani, “An algorithm for the numerical solution of differential equations of fractional order,” Journal of Applied Mathematics & Informatics, vol. 26, no. 1_2, pp. 15–27, 2008.

  38. V. Daftardar-Gejji, Y. Sukale, and S. Bhalekar, “A new predictor–corrector method for fractional differential equations,” Applied Mathematics and Computation, vol. 244, pp. 158–182, 2014.

  39. A. Jhinga and V. Daftardar-Gejji, “A new finite-difference predictor-corrector method for fractional differential equations,” Applied Mathematics and Computation, vol. 336, pp. 418–432, 2018.

    Article  MathSciNet  Google Scholar 

  40. A. Jhinga and V. Daftardar-Gejji, “A new numerical method for solving fractional delay differential equations,” Computational and Applied Mathematics, vol. 38, no. 4, p. 166, 2019.

    Article  MathSciNet  Google Scholar 

  41. M. Kumar, “An efficient numerical scheme for solving a fractional-order system of delay differential equations,” International Journal of Applied and Computational Mathematics, vol. 8, no. 5, p. 262, 2022.

    Article  MathSciNet  Google Scholar 

  42. M. Kumar and V. Daftardar-Gejji, “A new family of predictor-corrector methods for solving fractional differential equations,” Applied Mathematics and Computation, vol. 363, p. 124633, 2019.

    Article  MathSciNet  Google Scholar 

  43. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198. Academic press, 1998.

  44. S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional integrals and derivatives,” Theory and Applications, Gordon and Breach, Yverdon, 1993.

    Google Scholar 

  45. R. Wu, H. Ding, and C. Li, “Determination of coefficients of high-order schemes for riemann-liouville derivative,” The Scientific World Journal, vol. 2014, 2014.

  46. V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006.

    Article  MathSciNet  Google Scholar 

  47. C. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986.

    Article  MathSciNet  Google Scholar 

  48. C. Li and F. Zeng, “The finite difference methods for fractional ordinary differential equations,” Numerical Functional Analysis and Optimization, vol. 34, no. 2, pp. 149–179, 2013.

    Article  MathSciNet  Google Scholar 

  49. R. Lin and F. Liu, “Fractional high order methods for the nonlinear fractional ordinary differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 856–869, 2007.

    Article  MathSciNet  Google Scholar 

  50. V. Daftardar-Gejji, Fractional Calculus and Fractional Differential Equations. Springer, 2019.

  51. A. Bellen and M. Zennaro, Numerical methods for delay differential equations. Oxford university press, 2013.

  52. M. Umeki, “Chaos in the battisti-hirst original model for el ninõ southern oscillation,” Theoretical and Applied Mechanics Japan, vol. 60, pp. 21–27, 2012.

    Google Scholar 

  53. A. Uçar, “On the chaotic behaviour of a prototype delayed dynamical system,” Chaos, Solitons & Fractals, vol. 16, no. 2, pp. 187–194, 2003.

    Article  Google Scholar 

  54. S. Bhalekar, “Dynamical analysis of fractional order uçar prototype delayed system,” Signal, Image and Video Processing, vol. 6, no. 3, pp. 513–519, 2012.

    Article  Google Scholar 

  55. X. Liu, X. S. Shen, and H. Zhang, “Multi-scroll chaotic and hyperchaotic attractors generated from chen system,” International Journal of Bifurcation and Chaos, vol. 22, no. 02, p. 1250033, 2012.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National Defence Academy, Khadakwasala, Pune, 411023, India

    Manoj Kumar

  2. Department of Mathematics and Computer Science, West University of Timişoara, 300223, Timişoara, Romania

    Aman Jhinga

  3. Department of Scientific Computing, Modeling and Simulation, Savitribai Phule Pune University, Pune, 411007, India

    Varsha Daftardar-Gejji

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  1. Manoj Kumar
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Correspondence to Varsha Daftardar-Gejji.

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Communicated by NM Bujurke.

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Kumar, M., Jhinga, A. & Daftardar-Gejji, V. Higher order numerical methods for fractional delay differential equations. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00579-3

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