Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations

Adedapo Alaje; Moruf Oyedunsi Olayiwola

doi:10.26650/ijmath.2025.00030

Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations

Abstract

The high-order pantograph-type delay differential equations represent a model for the dynamics of electric trains with respect to pantograph systems for power collection from overhead wires. Accurate solutions for these equations are of importance for engineers dealing with improvements in the pantograph design, wear forecasting, and maximization of efficiency in power transmission as well as minimizing mechanical stress in the pantograph and overhead wires. This paper applies the Laplace Adomian decomposition method to handle both linear and nonlinear higher-order pantograph equations with achievable accurate solutions. The numerical tabulations and graphical illustrations demonstrate that this method is very efficient, accurate, and particularly easy to apply, and they therefore open up prospects for working on problems of a similar type.

Keywords

References

Adomian, G., 1981. A review of the decomposition method and some recent results for nonlinear equations. Comput. Math. Appl., 21: 101-127. google scholar
Alaje, A. I., Olayiwola, M. O., 2023. A fractional-order mathematical model for examining the spatiotemporal spread of COVID-19 in the presence of vaccine distribution. Healthcare Analytics, 4. https://doi.org/10.1016/j.health.2023.100230. google scholar
Alaje, A. I., Olayiwola, M. O., Ogunniran, M. O., Adedeji, J. A., Adedokun, K. A., 2021. Approximate analytical methods for the solution of fractional order integro-differential equations. Nigerian Journal of Mathematics and Applications, 31. google scholar
Alaje, A. I., Olayiwola, M. O., Adedokun, K. A., Adedeji, J. A., Oladapo, A. O., 2022. Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg-de Vries equation. Beni-Suef University Journal of Basic and Applied Sciences, 11: 1-17. https://doi.org/10.1186/s43088-022-00317-w. google scholar
Babasola, O., Kayode, O., Peter, O. J., Onwuegbuche, F. C., Oguntolu, F. A., 2022. Time-delayed modelling of the COVID-19 dynamics with a convex incidence rate. Informatics Med. Unlocked, 35: 101124. google scholar
Bahşi, M. M., Çevik, M., 2015. Numerical solution of pantograph-type delay differential equations using perturbation-iteration algorithms. Journal of Applied Mathematics, 2015(1), 139821. google scholar
Das, S., 2014. Delay differential equations in single species dynamics. Springer Science & Business Media. google scholar
Dong, S., Xu, L., Lan, Z. Z., Xiao, D., Gao, B., 2023. Application of a time-delay SIR model with vaccination in COVID-19 prediction and its optimal control strategy. Nonlinear Dyn., 111(11): 10677-10692. google scholar

Fang, X., 2020. Neural network dynamics with time delays. Neural Computation, 32: 1906-1932. google scholar
Ghaffari, V., 2024. Robust predictive regulation of uncertain time-delay control systems with non-zero references and disturbances. Int. J. Syst. Sci., 55(11): 2239-2251. google scholar
He, J. H., 1999. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng., 178: 257-262. google scholar
Izadi, M., Srivastava, H. M., 2021. A novel matrix technique for multi-order pantograph differential equations of fractional order. Proceedings of the Royal Society A, 477(2253), 20210321. google scholar
Jafari, H., Mahmoudi, M., Noori Skandari, M. H., 2021. A new numerical method to solve pantograph delay differential equations with convergence analysis. Advances in Difference Equations, 2021(1), 129. google scholar
Kiselev, I. N., Akberdin, I. R., Kolpakov, F. A., 2021. A delay differential equation approach to model the COVID-19 pandemic. medRxiv (2021):09. google scholar
Liu, L., Kalmar-Nagy, T., 2010. High-dimensional harmonic balance analysis for second-order delay-differential equations. J. Vib. Control, 16: 1189-1208. google scholar
Ockendon, J. R., Tayler, A. B., 1971. The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. A, 322: 447-468. google scholar
Olayiwola, M. O., Alaje, A. I., Olarewaju, A. Y., Adedokun, K. A., 2023. A Caputo fractional order epidemic model for evalu-ating the effectiveness of high-risk quarantine and vaccination strategies on the spread of COVID-19. Healthcare Analytics, 3. https://doi.org/10.1016/j.health.2023.100179. google scholar
Polyanin, A. D., Sorokin, V. G., 2021. Nonlinear pantograph-type diffusion PDEs: Exact solutions and the principle of analogy. Mathematics, 9(5), 511. google scholar
Rudnyi, E. B., Korvink, J. G., 2004. Model order reduction for large scale engineering models developed in ANSYS. In International Workshop on Applied Parallel Computing: 349-356. google scholar
Shin, K. G., Cui, X., 1995. Computing time delay and its effects on real-time control systems. IEEE Trans. Control Syst. Technol., 3(2): 218-224. google scholar
Sipahi, R., Vyhlfdal, T., Niculescu, S. I., Pepe, P. (Eds.), 2012. Time delay systems: Methods, applications and new trends. Springer, Berlin Heidelberg, p. 423. google scholar
Smith, H. L., 2011. An introduction to delay differential equations with applications to the life sciences. Vol. 57, pp. 119-130; Springer: New York. google scholar
Tang, Y., Kurths, J., Lin, W., Ott, E., Kocarev, L., 2020. Introduction to focus issue: When machine learning meets complex systems: Networks, chaos, and nonlinear dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30. google scholar
Wang, X., Lee, Y., 2023. Enhancing stability and performance of networked control systems with time delays using delay differential equations. IEEE Trans. Autom. Control, 68: 232-245. google scholar
Xu, R., Zhang, S., 2013. Modelling and analysis of a delayed predator-prey model with disease in the predator. Appl. Math. Comput., 224: 372-386. google scholar
Yunus, A. O., Olayiwola, M. O., Adedokun, K. A., Adedeji, J. A., Alaje, A. I., 2022. Mathematical analysis of fractional-order Caputo’s derivative of coronavirus disease model via Laplace Adomian decomposition method. Beni-Suef University Journal of Basic and Applied Sciences, 11: 144. https://doi.org/10.1186/s43088-022-00306-0. google scholar
Zaidi, A. A., Van Brunt, B., & Wake, G. C., 2015, Solutions to an advanced functional partial differential equation of the pantograph type, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2179), 20140947. google scholar

Details

Primary Language

English

Subjects

Applied Mathematics (Other)

Journal Section

Research Article

Authors

Adedapo Alaje ^*
0000-0002-3590-3256
Nigeria

Moruf Oyedunsi Olayiwola This is me
0000-0001-6101-1203
Nigeria

Publication Date

December 16, 2025

Submission Date

September 29, 2024

Acceptance Date

August 1, 2025

Published in Issue

Year 2025 Volume: 3 Number: 2

DOI

https://doi.org/10.26650/ijmath.2025.00030

IZ

https://izlik.org/JA44YX32MF

Cite

RIS / Bibtex

APA

Alaje, A., & Olayiwola, M. O. (2025). Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations. Istanbul Journal of Mathematics, 3(2), 78-89. https://doi.org/10.26650/ijmath.2025.00030

AMA

1.Alaje A, Olayiwola MO. Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations. Istanbul Journal of Mathematics. 2025;3(2):78-89. doi:10.26650/ijmath.2025.00030

Chicago

Alaje, Adedapo, and Moruf Oyedunsi Olayiwola. 2025. “Laplace-Adomian Decomposition Method for Solving Higher-Order Pantograph-Type Delay Differential Equations”. Istanbul Journal of Mathematics 3 (2): 78-89. https://doi.org/10.26650/ijmath.2025.00030.

EndNote

Alaje A, Olayiwola MO (December 1, 2025) Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations. Istanbul Journal of Mathematics 3 2 78–89.

IEEE

[1]A. Alaje and M. O. Olayiwola, “Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations”, Istanbul Journal of Mathematics, vol. 3, no. 2, pp. 78–89, Dec. 2025, doi: 10.26650/ijmath.2025.00030.

ISNAD

Alaje, Adedapo - Olayiwola, Moruf Oyedunsi. “Laplace-Adomian Decomposition Method for Solving Higher-Order Pantograph-Type Delay Differential Equations”. Istanbul Journal of Mathematics 3/2 (December 1, 2025): 78-89. https://doi.org/10.26650/ijmath.2025.00030.

JAMA

1.Alaje A, Olayiwola MO. Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations. Istanbul Journal of Mathematics. 2025;3:78–89.

MLA

Alaje, Adedapo, and Moruf Oyedunsi Olayiwola. “Laplace-Adomian Decomposition Method for Solving Higher-Order Pantograph-Type Delay Differential Equations”. Istanbul Journal of Mathematics, vol. 3, no. 2, Dec. 2025, pp. 78-89, doi:10.26650/ijmath.2025.00030.

Vancouver

1.Adedapo Alaje, Moruf Oyedunsi Olayiwola. Laplace-Adomian Decomposition Method for solving higher-Order Pantograph-Type Delay Differential Equations. Istanbul Journal of Mathematics. 2025 Dec. 1;3(2):78-89. doi:10.26650/ijmath.2025.00030