Abstract
This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.
References
Gurney WSC, Blythe SP, Nisbet RM. Nicholson's Blowflies revisited. Nature. 1980; 287: 17-21. https://doi.org/10.1038/287017a0
Busenberg S, Cooke KL. Periodic solutions of a periodic nonlinear delay differential equation. SIAM J Appl Math. 1978; 35: 704-21. https://doi.org/10.1137/0135059
Ockendon J, Tayler A. The dynamics of a current collection system for an electric locomotive. Proc R Soc Lond A, 1971; 322: 447-68. https://doi.org/10.1098/rspa.1971.0078
Cushing J. Integrodifferential equations and delay models in population dynamics. Vol. 20. Berlin Heidelberg New York: Springer Verlag; 1979.
Gopalsamy K. Stability and oscillations in delay differential equations of population dynamics. Dordrecht: Springer Netherlands; 1992. https://doi.org/10.1007/978-94-015-7920-9
Kuang Y. Delay differential equations with applications in population dynamics. Academic Press; 1993.
Baker CTH, Paul CAH, Willé DR. Issues in the numerical solution of evolutionary delay differential equations. Adv Comput Math. 1995; 3: 171-96. https://doi.org/10.1007/BF03028370
Bocharov GA, Rihan FA. Numerical modeling in biosciences using delay differential equations. J Comput Methods Appl Math. 2000; 125: 183-99. https://doi.org/10.1016/S0377-0427(00)00468-4
Szydłowski M, Krawiec A, Toboła J. Nonlinear oscillations in business cycle model with time lags. Chaos Solitons Fractals. 2001; 12: 505-17. https://doi.org/10.1016/S0960-0779(99)00207-6
Richard JP. Time-delay systems: an overview of some recent advances and open problems. Automatica. 2003; 39: 1667-94. https://doi.org/10.1016/S0005-1098(03)00167-5
Kyrychko YN, Hogan SJ. On the use of delay equations in engineering applications. J Vib Control. 2010; 16: 943-60. https://doi.org/10.1177/1077546309341100
Ji XA, Orosz G. Learning time delay systems with neural ordinary differential equations. IFAC-PapersOnLine. 2022; 55: 79-84. https://doi.org/10.1016/j.ifacol.2022.11.337
Yu C, Gao G. Some results on a class of fractional functional differential equations. Commun Appl Nonlinear Anal. 2004; 11: 67-75.
Zhou Y. Existence and uniqueness of fractional functional differential equations with unbounded delay. Int J Dyn Syst Differ Equ. 2008; 1: 239-44. https://doi.org/10.1504/IJDSDE.2008.022988
Zhou Y, Jiao F, Li J. Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal Theory Methods Appl. 2009; 71: 3249-56. https://doi.org/10.1016/j.na.2009.01.202
Zhou Y, Jiao F, Li J. Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Anal Theory Methods Appl. 2009; 71: 2724-33. https://doi.org/10.1016/j.na.2009.01.105
Zhang C, Chen H. Asymptotic stability of block boundary value methods for delay differential-algebraic equations. Math Comput Simul. 2010; 81: 100-8. https://doi.org/10.1016/j.matcom.2010.07.012
Wang J, Zhou Y. Existence of mild solutions for fractional delay evolution systems. Appl Math Comput. 2011; 218: 357-67. https://doi.org/10.1016/j.amc.2011.05.071
Morgado ML, Ford NJ, Lima PM. Analysis and numerical methods for fractional differential equations with delay. J Comput Appl Math. 2013; 252: 159-68. https://doi.org/10.1016/j.cam.2012.06.034
Luo D, Tian M, Zhu Q. Some results on finite-time stability of stochastic fractional-order delay differential equations. Chaos Solitons Fractals. 2022; 158: 111996. https://doi.org/10.1016/j.chaos.2022.111996
Du F, Lu JG. Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. Appl Math Comput. 2020; 375: 125079. https://doi.org/10.1016/j.amc.2020.125079
Huseynov IT, Mahmudov NI. Analysis of positive fractional-order neutral time-delay systems. J Frankl Inst. 2022; 359: 294-330. https://doi.org/10.1016/j.jfranklin.2021.07.001
Kumar K, Pandey RK, Yadav S. Finite difference scheme for a fractional telegraph equation with generalized fractional derivative terms. Physica A Stat Mech Appl. 2019; 535: 122271. https://doi.org/10.1016/j.physa.2019.122271
Al-Zhour Z, Al-Mutairi N, Alrawajeh F, Alkhasawneh R. Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative. Alexandria Eng J. 2019; 58: 1413-20. https://doi.org/10.1016/j.aej.2019.11.012
Thirumalai S, Seshadri R. Spectral solutions of fractional differential equation modelling electrohydrodynamics flow in a cylindrical conduit. Commun Nonlinear Sci Numer Simul. 2019; 79: 104931. https://doi.org/10.1016/j.cnsns.2019.104931
Sun Z, Wu X. A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math. 2006; 56: 193-209. https://doi.org/10.1016/j.apnum.2005.03.003
Wu G, Lee EWM. Fractional variational iteration method and its application. Phys Lett A, 2010; 374: 2506-9. https://doi.org/10.1016/j.physleta.2010.04.034
Jiang Y, Ma J. High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math. 2011; 235: 3285-90. https://doi.org/10.1016/j.cam.2011.01.011
Li C, Zeng F, Liu F. Spectral approximations to the fractional integral and derivative. Fract Calc Appl Anal. 2012; 15: 383-406. https://doi.org/10.2478/s13540-012-0028-x
Zaky MA, Doha EH, Tenreiro Machado JA. A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl Numer Math. 2018; 132: 51-72. https://doi.org/10.1016/j.apnum.2018.05.009
Zaky MA. Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J Comput Appl Math. 2019; 357: 103-22. https://doi.org/10.1016/j.cam.2019.01.046
Kukla S, Siedlecka U. A numerical‐analytical solution of multi‐term fractional‐order differential equations. Math Methods Appl Sci. 2020; 43: 4883-94. https://doi.org/10.1002/mma.6242
Srivastava HM, Gusu DM, Mohammed PO, Wedajo G, Nonlaopon K, Hamed YS. Solutions of general fractional-order differential equations by using the spectral tau method. Fractal Fract. 2021; 6: 7. https://doi.org/10.3390/fractalfract6010007
Pandit S, Mittal RC. A numerical algorithm based on scale-3 Haar wavelets for fractional advection dispersion equation. Eng Comput. 2020; 38: 1706-24. https://doi.org/10.1108/EC-01-2020-0013
Mittal RC, Pandit S. A numerical algorithm to capture spin patterns of fractional bloch nuclear magnetic resonance flow models. J Comput Nonlinear Dyn. 2019; 14: 081001. https://doi.org/10.1115/1.4043565
Mittal RC, Pandit S. Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems. Eng Comput. 2018; 35: 1907-31. https://doi.org/10.1108/EC-09-2017-0347
Hafshejani M, Vanani S, Hafshejani J. Numerical solution of delay differential equations using Legendre wavelet method. World Appl Sci J. 2011; 13: 27-33.
Ghasemi M, Fardi M, Khoshsiar Ghaziani R. Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl Math Comput. 2015; 268: 815-31. https://doi.org/10.1016/j.amc.2015.06.012
Saeed U, Rehman M ur, Iqbal MA. Modified Chebyshev wavelet methods for fractional delay-type equations. Appl Math Comput. 2015; 264: 431-42. https://doi.org/10.1016/j.amc.2015.04.113
Rabiei K, Ordokhani Y. Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials. Eng Comput. 2019; 35: 1431-41. https://doi.org/10.1007/s00366-018-0673-8
Chen Z, Gou Q. Piecewise Picard iteration method for solving nonlinear fractional differential equation with proportional delays. Appl Math Comput. 2019; 348: 465-78. https://doi.org/10.1016/j.amc.2018.10.058
Yang C, Hou J, Lv X. Jacobi spectral collocation method for solving fractional pantograph delay differential equations. Eng Comput. 2022; 38: 1985-94. https://doi.org/10.1007/s00366-020-01193-7
Singh BK, Agrawal S. Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations. Math Methods Appl Sci. 2022; 45: 8305-28. https://doi.org/10.1002/mma.8335
Elkot NA, Doha EH, Ameen IG, Hendy AS, Zaky MA. A re-scaling spectral collocation method for the nonlinear fractional pantograph delay differential equations with non-smooth solutions. Commun Nonlinear Sci Numer Simul. 2023; 118: 107017. https://doi.org/10.1016/j.cnsns.2022.107017
Bhrawy AH, Al-Zahrani AA, Alhamed YA. A new generalized laguerre-gauss collocation scheme for numerical solution of generalized fractional pantograph equations. Romanian J Phys. 2014; 59: 646-57.
Rahimkhani P, Ordokhani Y, Babolian E. A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations. Numer Algorithms. 2017; 74: 223-45. https://doi.org/10.1007/s11075-016-0146-3
Yang C. Modified Chebyshev collocation method for pantograph-type differential equations. Appl Numer Math. 2018; 134: 132-44. https://doi.org/10.1016/j.apnum.2018.08.002
Nemati S, Lima P, Sedaghat S. An effective numerical method for solving fractional pantograph differential equations using modification of hat functions. Appl Numer Math. 2018; 131: 174-89. https://doi.org/10.1016/j.apnum.2018.05.005
Podlubny I. Fractional differential equations, San Diego: Academic Press; 1999.
Odibat ZM, Shawagfeh NT. Generalized Taylor’s formula. Appl Math Comput. 2007; 186: 286-93. https://doi.org/10.1016/j.amc.2006.07.102
Cheng Q. Applied functional analysis. Beijing: Higher Education Press; 2008.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2023 Xueying Cui, Yuqiang Feng, Jun Jiang