On fractional linear multi-step methods for fractional order multi-delay nonlinear pantograph equation

Document Type : Research Paper

Authors

Mathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

Abstract

This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order $0 < \alpha \leq 1$. These $p$ FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed $p$-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach.

Keywords

Main Subjects


  • [1] M. Bahmanpour, M. Tavassoli Kajani, and M. Maleki, Solving Fredholm integral equations of the first kind using Muntz wavelets, Applied Numerical Mathematics, 143 (2019), 159–171.
  • [2] C. TH. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315–335.
  • [3] S. Davaeifar and J. Rashidinia, Solution of a system of delay differential equations of multi pantograph type, Journal of Taibah University for Science, 11 (2017), 1141-1157.
  • [4] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5 (1997) 1-6.
  • [5] D. J. Evans and K. R. Raslan, The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math., 82(1) (2005), 49-54.
  • [6] Z. Frazaneh Bonab and M. Javidi, Higher order methods for fractional differential equation based on fractional backward differentiation formula of order three, Mathematics and Computers in Simulation, (2020).
  • [7] L. Galeone and R. Garrappa, Second Order Multistep Methods for Fractional Differential Equations, Technical Report 20/2007, Department 19 of Mathematics, University of Bari, 2007.
  • [8] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Int. J. Comput. Math., 87(10) (2010), 2281-2290.
  • [9] S. Irandoust-Pakchina, S. Abdi-Mazraeha and Sh. Rezapour, Stability properties of fractional second linear multistep methods in the implicit form: Theory and applications, Filomat 37(21) (2023).
  • [10] A. Isah, C. Phang, and P. Phang, Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations, Hindawi International Journal of Differential Equations, (2017).
  • [11] D. Li and M. Z. Liu, Runge-Kutta methods for the multi-pantograph delay equation, Appl. Math. Comput., 163 (2005), 383–395.
  • [12] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17(3) (1986), 704-719.
  • [13] M. Maleki and M. Tavassoli Kajani, Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method, Applied Mathematical Modelling, 39(15) (2015), 4300– 4308.
  • [14] B. P. Moghaddam, Z. S. Mostaghim, A. A. Pantelous, and J. A. T. Machado, An integro quadratic spline-based scheme for solving nonlinear fractional stochastic differential equations with constant time delay, Communications in Nonlinear Science and Numerical Simulation, 92 105475.
  • [15] P. Mokhtary, B. P. Moghaddam, A. M. Lopes, and J. A. Tenreiro Machado, A computational approach for the nonsmooth solution of non-linear weakly singular Volterra integral equation with proportional delay, Numer. Algor., 83 (2020), 987-1006.
  • [16] Z. Moniri, B. P. Moghaddam, and M. Z. Roudbaraki, An Efficient and Robust Numerical Solver for Impulsive Control of Fractional Chaotic Systems, Journal of Function Spaces, Special Issue: Recent Advances of Fractional Calculus in Applied Science, (2023).
  • [17] Z. S. Mostaghim, B. P. Moghaddam, and H. S. Haghgozar, Computational technique for simulating variable-order fractional Heston model with application in US stock market, Mathematical Sciences, 12, 277–283.
  • [18] Y. Muroya, E. Ishiwata, and H. Brunner, On the attainable order of collocation methods for pantograph integro differential equations, J. Comput. Appl. Math., 152 (2003), 347–366.
  • [19] J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 322 (1971), 447–468.
  • [20] B. Parsa Moghaddam, A. Dabiri, Z. S. Mostaghim, and Z. Moniri, Numerical solution of fractional dynamical systems with impulsive effects, International Journal of Modern Physics C, 34(01) (2023).
  • [21] I. Podlubny, Fractional differential equations, San Diego, Academic Press, 1999.
  • [22] M. Shadia, Numerical solution of delay differential and neutral differential equations using Spline Methods, Ph.D Thesis, Assuit University, 1992.
  • [23] M. Tavassoli Kajani, Numerical solution of fractional pantograph equations via Muntz-Legendre polynomials, Math. Sci., (2023).
  • [24] M. Valizadeh, Y. Mahmoudi, and F. D. Saei, Application of natural transform method to fractional pantograph delay differential equations, Hindawi Journal of Mathematics, (2019).
  • [25] E. Yusufoglu, An efficient algorithm for solving generalized pantograph equations with linear functional argument, App. Math. Comp., 217 (2010), 3591–3595.