Numerical treatment for multi-pantograph integro-differential equation via tau spectral method

Samer S. Ezz-Eldien; Ali H. Tedjani; Faizah A. H. Alomari; Amra Al Kenany; Samer S. Ezz-Eldien; Ali H. Tedjani; Faizah A. H. Alomari; Amra Al Kenany

doi:10.3934/math.20251290

AIMS Mathematics

2025, Volume 10, Issue 12: 29380-29405. doi: 10.3934/math.20251290

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Numerical treatment for multi-pantograph integro-differential equation via tau spectral method

1.
Department of Mathematics, Faculty of Science, New Valley University, El-Kharga, 72511, Egypt
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia
4.
Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca, KSA

Received: 01 June 2025 Revised: 07 November 2025 Accepted: 18 November 2025 Published: 12 December 2025
MSC : 65L05, 65R20, 65N35, 65L03

Pantograph integro-differential equations have many crucial applications in science and engineering. The presence of differential behavior, scaling, and memory effects makes pantograph integro-differential equations capable of describing complex systems in control theory and mathematical biology. In this paper, we provide a numerical approach to the multi-pantograph integro-differential equation. The Jacobi tau spectral approach is utilized with the help of differential, integral, and pantograph operational matrices to solve one- and two-dimensional linear multi-pantograph integro-differential equations. The high accuracy, convergence, and simplicity motivate one to apply the tau spectral approach to the problem studied. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.
- operational matrix,
- Jacobi polynomial,
- pantograph equation
Citation: Samer S. Ezz-Eldien, Ali H. Tedjani, Faizah A. H. Alomari, Amra Al Kenany. Numerical treatment for multi-pantograph integro-differential equation via tau spectral method[J]. AIMS Mathematics, 2025, 10(12): 29380-29405. doi: 10.3934/math.20251290

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Abstract

Pantograph integro-differential equations have many crucial applications in science and engineering. The presence of differential behavior, scaling, and memory effects makes pantograph integro-differential equations capable of describing complex systems in control theory and mathematical biology. In this paper, we provide a numerical approach to the multi-pantograph integro-differential equation. The Jacobi tau spectral approach is utilized with the help of differential, integral, and pantograph operational matrices to solve one- and two-dimensional linear multi-pantograph integro-differential equations. The high accuracy, convergence, and simplicity motivate one to apply the tau spectral approach to the problem studied. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.

References

[1]	Y. Kuang, Delay differential equations: With applications in population dynamics, USA, New York: Academic Press, 91 (1993).
[2]	S. S. Ezz-Eldien, A. Alalyani, Legendre spectral collocation method for one-and two-dimensional nonlinear pantograph Volterra-Fredholm integro-differential equations, Int. J. Mod. Phys. C, 37 (2025), 2550061. https://doi.org/10.1142/s0129183125500615 doi: 10.1142/s0129183125500615
[3]	M. Dehghan, F. Shakeri, The use of the decomposition procedure of adomian for solving a delay differential equation arising in electrodynamics, Phys. Scripta, 78 (2008), 065004. https://doi.org/10.1088/0031-8949/78/06/065004 doi: 10.1088/0031-8949/78/06/065004
[4]	L. Fox, D. F. Mayers, J. A. Ockendon, A. B. Tayler, On a functional differential equation, IMA J. Appl. Math., 8 (1971), 271–307. https://doi.org/10.1093/imamat/8.3.271 doi: 10.1093/imamat/8.3.271
[5]	J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. A, 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[6]	M. A. Zaky, I. G. Ameen, N. A. Elkot, E. H. Doha, A unified spectral collocation method for nonlinear systems of multi-dimensional integral equations with convergence analysis, Appl. Numer. Math., 161 (2021), 27–45. https://doi.org/10.1016/j.apnum.2020.10.028 doi: 10.1016/j.apnum.2020.10.028
[7]	W. G. Ajello, H. I. Freedman, J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855–869. https://doi.org/10.1137/0152048 doi: 10.1137/0152048
[8]	M. A. Elkot, E. H. Doha, I. G. Ameen, G. Ibrahem, A. S. Hendy, M. A. Zaky, A re-scaling spectral collocation method for the nonlinear fractional pantograph delay differential equations with non-smooth solutions, Commun. Nonlinear Sci., 118 (2023), 107017. https://doi.org/10.1016/j.cnsns.2022.107017 doi: 10.1016/j.cnsns.2022.107017
[9]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157. https://doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
[10]	G. A. Bocharov, F. A. Rihan, Numerical modelling in biosciences using delay differential equations, J. Comput. Appl. Math., 125 (2000), 183–199. https://doi.org/10.1016/S0377-0427(00)00468-4 doi: 10.1016/S0377-0427(00)00468-4
[11]	J. Zhao, Y. Cao, Y. Xu, Sinc numerical solution for pantograph Volterra delay-integro-differential equation, Int. J. Comput. Math., 94 (2017), 853–865. https://doi.org/10.1080/00207160.2017.1283030 doi: 10.1080/00207160.2017.1283030
[12]	M. M. Alsuyuti, E. H. Doha, B. I. Bayoumi, S. S. Ezz-Eldien, Robust spectral treatment for time-fractional delay partial differential equations, Comput. Appl. Math., 42 (2023), 159. https://doi.org/10.1007/s40314-023-02287-w doi: 10.1007/s40314-023-02287-w
[13]	M. A. Zaky, Finite difference/fractional Pertrov-Galerkin spectral method for the linear time-space fractional reaction-diffusion equation, Mathematics, 13 (2025), 1864. https://doi.org/10.3390/math13111864 doi: 10.3390/math13111864
[14]	J. Zhao, Y. Cao, Y. Xu, Tau approximate solution of linear pantograph Volterra delay-integro-differential equation, Comput. Appl. Math., 39 (2020), 46. https://doi.org/10.1007/s40314-020-1080-5 doi: 10.1007/s40314-020-1080-5
[15]	M. A. Zaky, I. G. Ameen, A novel Jacobi spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions, Eng. Comput.-Germany, 37 (2021), 2623–2631. https://doi.org/10.1007/s00366-020-00953-9 doi: 10.1007/s00366-020-00953-9
[16]	J. G. Liu, W. H. Zhu, Y. K. Wu, G. H. Jin, Application of multivariate bilinear neural network method to fractional partial differential equations, Res. Phys., 47 (2023), 106341. https://doi.org/10.1016/j.rinp.2023.106341 doi: 10.1016/j.rinp.2023.106341
[17]	R. F. Zhang, S. Bilige, Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation, Nonlinear Dynam., 95 (2019), 3041–3048. https://doi.org/10.1007/s11071-018-04739-z doi: 10.1007/s11071-018-04739-z
[18]	R. M. Zhong, Bilinear neural network method for obtaining the exact analytical solutions to nonlinear evolution equations and its application to KdV equation, Afr. Diaspora J. Math., 27 (2024), 4. https://doi.org/10.7813/s3x0tb07 doi: 10.7813/s3x0tb07
[19]	Y. Alkhezi, Y. H. R. Almubarak, A. Shafee, Neural-network-based approximations for investigating a Pantograph delay differential equation with application in Algebra, Int. J. Math. Comput. Sci., 20 (2025), 195–209. https://doi.org/10.69793/ijmcs/01.2025/ahmad doi: 10.69793/ijmcs/01.2025/ahmad
[20]	X. R. Xie, R. F. Zhang, Neural network-based symbolic calculation approach for solving the Korteweg–de Vries equation, Chaos Soliton. Fract., 194 (2025), 116232. https://doi.org/10.1016/j.chaos.2025.116232 doi: 10.1016/j.chaos.2025.116232
[21]	T. Ji, J. Hou, C. Yang, The operational matrix of Chebyshev polynomials for solving pantograph-type Volterra integro-differential equations, Adv. Contin. Discrete M., 2022 (2022), 57. https://doi.org/10.1186/s13662-022-03729-1 doi: 10.1186/s13662-022-03729-1
[22]	H. Jafari, N. Tuan, R. M. Ganji, A new numerical scheme for solving pantograph type nonlinear fractional integro-differential equations, J. King Saud Univ. Sci., 33 (2021), 101185. https://doi.org/10.1016/j.jksus.2020.08.029 doi: 10.1016/j.jksus.2020.08.029
[23]	Y. Wang, S. S. Ezz-Eldien, A. A. Aldraiweesh, A new algorithm for the solution of nonlinear two-dimensional Volterra integro-differential equations of high-order, J. Comput. Appl. Math., 364 (2020), 112301. https://doi.org/10.1016/j.cam.2019.06.017 doi: 10.1016/j.cam.2019.06.017
[24]	P. Darania, F. M. Sotoudehmaram, Numerical analysis of a high order method for nonlinear delay integral equations, J. Comput. Appl. Math., 374 (2020), 112738. https://doi.org/10.1016/j.cam.2020.112738 doi: 10.1016/j.cam.2020.112738
[25]	N. A. Elkot, M. A. Zaky, E. H. Doha, I. G. Ameen, On the rate of convergence of the Legendre spectral collocation method for multi-dimensional nonlinear Volterra–Fredholm integral equations, Commun. Theor. Phys., 73 (2021), 025002. https://doi.org/10.1088/1572-9494/abcfb3 doi: 10.1088/1572-9494/abcfb3
[26]	A. H. Tedjani, S. E. Alhazmi, S. S. Ezz-Eldien, An operational approach for one-and two-dimension high-order multi-pantograph Volterra integro-differential equation, AIMS Math., 10 (2025), 9274–9294. https://doi.org/10.3934/math.2025426 doi: 10.3934/math.2025426
[27]	M. A. Zaky, M. Babatin, M. Hammad, A. Akgül, A. S. Hendy, Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations, AIMS Math., 9 (2017), 15246–15262. https://doi.10.3934/math.2024740 doi: 10.3934/math.2024740
[28]	S. S. Ezz-Eldien, On solving fractional logistic populationmodels with applications, Comput. Appl. Math., 37 (2018), 6392-–6409. https://doi.org/10.1007/s40314-018-0693-4 doi: 10.1007/s40314-018-0693-4
[29]	E. H. Doha, A. H. Bhrawy, D. Baleanu, S. S. Ezz-Eldien, The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation, Adv. Differ. Equ., 231 (2014). https://doi.org/10.1186/1687-1847-2014-231 doi: 10.1186/1687-1847-2014-231
[30]	M. A. Zaky, A. A. Kenany, S. E. Alhazmi, S. S. Ezz-Eldien, Numerical treatment of tempered space-fractional Zeldovich–Frank–Kamenetskii equation, Rom. Rep. Phys., 2025.
[31]	J. Zhou, Z. Jiang, H. Xie, H. Niu, The error estimates of spectral methods for 1-dimension singularly perturbed problem, Appl. Math. Lett., 100 (2020), 106001. https://doi.org/10.1016/j.aml.2019.106001 doi: 10.1016/j.aml.2019.106001
[32]	C. T. Sheng, Z. Q. Wang, B. Y. Guo, A multistep Legendre–Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal., 52 (2014). https://doi.org/10.1137/130915200 doi: 10.1137/130915200

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