Mittag-Leffler wavelet-based numerical method for fractional pantograph delay differential equations

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

2 Department of Mathematics and Statistics, Mississippi State University, MS, USA

Abstract

This paper proposes a robust numerical framework for solving fractional pantograph delay differential equations. The approach leverages the Riemann–Liouville fractional integral operator, represented through Mittag-Leffler wavelet functions within a collocation-based scheme. To facilitate computation, an operational matrix is constructed, enabling the transformation of the fractional differential system into a system of algebraic equations. The proposed method’s accuracy, stability, and convergence are rigorously validated through comprehensive numerical experiments.

Keywords

Main Subjects


[1] A.B. Albidah, N.E. Kanaan, A. Ebaid, H.K. Al-Jeaid, Exact and numerical analysis of the pantograph delay
differential equation via the homotopy perturbation method, Math. 11 (2023) 944.
[2] H. Dehestani, Y. Ordokhani, M. Razzaghi, A numerical technique for solving various kinds of fractional
partial differential equations via Genocchi hybrid functions, Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A
Mat. 113 (2019) 3297–3321.
[3] P. Das, S. Rana, Theoretical prospects of fractional order weakly singular Volterra integro-differential equa
tions and their approximations with convergence analysis, Math. Methods Appl. Sci. 44 (2021) 9419–9440.
[4] P. Das, S. Natesan, A uniformly convergent hybrid scheme for singularly perturbed system of reaction
diffusion Robin type boundary-value problems, J. Appl. Math. Comput. 41 (2013) 447–471.
[5] P. Das, S. Rana, H. Ramos, On the approximate solutions of a class of fractional order nonlinear Volterra
integro-differential initial value problems and boundary value problems of first kind and their convergence
analysis, J. Comput. Appl. Math. 404 (2022) 113116.
[6] S. Dhama, A reliable scheme for nonlinear delay differential equations of pantograph-type, J. Comput. Sci.
75 (2024) 102206.
[7] A. Ghasempour, Y. Ordokhani, S. Sabermahani, Fractional-order Mittag–Leffler functions for solving multi
dimensional fractional pantograph delay differential equations, Iran. J. Sci. 47 (2023) 885–898.
[8] J. Hajishafieiha, S. Abbasbandy, T. Allahviranloo, A new numerical approach for solving the fractional non
linear multi-pantograph delay differential equations, Iran. J. Sci. 47 (2023) 825–835.
[9] N.A. Khan, M. Ali, A. Ara, M.I. Khan, S. Abdullaeva, M. Waqas, Optimizing pantograph fractional dif
ferential equations: a Haar wavelet operational matrix method, Partial Differ. Equ. Appl. Math. 11 (2024)
100774
[10] M.A. Iqbal, M. Shakeel, S.T. Mohyud-Din, M. Rafiq, Modified wavelets-based algorithm for nonlinear delay
differential equations of fractional order, Adv. Mech. Eng. 9(4) (2017).
[11] M. Izadi, S¸. Y¨ uzbas¸ı, K.J. Ansari, Application of Vieta–Lucas series to solve a class of multi-pantograph
delay differential equations with singularity, Symmetry 13 (2021) 2370.
[12] H. Jafari, M. Mahmoudi, M.H. Noori Skandari, A new numerical method to solve pantograph delay differen
tial equations with convergence analysis, Adv. Differ. Equ. 2021 (2021) 129.
[13] R.P. Meilanov, R.A. Magomedov, Thermodynamics in fractional calculus, J. Eng. Phys. Thermophys. 87
(2014) 1521–1531.
[14] S. Mashayekhi, M. Razzaghi, M. Wattanataweekul, Analysis of multi-delay and piecewise constant delay
systems by hybrid functions approximation, Differ. Equ. Dyn. Syst. 24 (2016) 1–20.
[15] N. Negero, and G. Duressa, An efficient numerical approach for singularly perturbed parabolic convection
diffusion problems with large time-lag, J. Math. Model. 10(2) (2022) 173-190.
[16] S. Nemati, P. Lima, S. Sedaghat, An effective numerical method for solving fractional pantograph differential
equations using modification of hat functions, Appl. Numer. Math. 131 (2018) 174–189.
[17] Z. Odibat, V.S. Erturk, P. Kumar, V. Govindaraj, Dynamics of generalized Caputo type delay fractional
differential equations using a modified predictor-corrector scheme, Phys. Scr. 96 (2021) 125213.
[18] Z. Odibat, V.S. Erturk, P. Kumar, A. Ben Makhlouf, V. Govindaraj, An implementation of the generalized
differential transform scheme for simulating impulsive fractional differential equations, Math. Probl. Eng.
2022 (2022).
[19] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[20] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation.
arXiv preprint math/0110241. 2001
[21] K. Rabiei, Y. Ordokhani, Solving fractional pantograph delay differential equations via fractional-order
Boubaker polynomials, Eng. Comput. 35 (2019) 1431–1441.
[22] P. Rahimkhani, Y. Ordokhani, E. Babolian, A new operational matrix based on Bernoulli wavelets for solving
fractional delay differential equations, Numer. Algorithms 74 (2017) 223–245.
[23] P. Rahimkhani, Y. Ordokhani, A computational method based on Legendre wavelets for solving distributed
order fractional diffrential equations, J. Math. Model. 9(3) (2021) 501-516.
[24] S. Santra, J. Mohapatra, P. Das, D. Choudhuri, Higher order approximations for fractional order integro
parabolic partial differential equations on an adaptive mesh with error analysis, Comput. Math. Appl. 150
(2023) 87–101.
[25] D. Sarkar, S. Kumar, P. Das, H. Ramos, Higher-order convergence analysis for interior and boundary layers
in a semi-linear reaction-diffusion system networked by a k-star graph with non-smooth source terms, Netw.
Heterog. Media 19 (2024).
[26] M. Senol, M. Gencyigit, M.E. Koksal, S. Qureshi, New analytical and numerical solutions to the (2+1)
dimensional conformable cpKP–BKP equation, Opt. Quant. Electron. 56 (2024) 352.
[27] P. Vichitkunakorn, T.N. Vo, M. Razzaghi, A numerical method for fractional pantograph differential equa
tions based on Taylor wavelets, Trans. Inst. Meas. Control 42 (2020) 1334–1344.
[28] B. Yuttanan, M. Razzaghi, T.N. Vo, Legendre wavelet method for fractional delay differential equations,
Appl. Numer. Math. 168 (2021) 127–142.
[29] S¸. Yuzbas¸ı, N. Ismailov, A Taylor operation method for solutions of generalized pantograph type delay dif
ferential equations, Turk. J. Math. 42 (2018) 395–406