An Efficient Numerical Scheme for the Solution of Integer-Order Delays Differential Equations

Ibrahim M.D.¹ , M.S. Adamu² , Isah Abdullahi³

Section:Research Paper, Product Type: Journal-Paper
Vol.11 , Issue.6 , pp.23-31, Dec-2024

Online published on Dec 31, 2024

Copyright © Ibrahim M.D., M.S. Adamu, Isah Abdullahi . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

Abstract :
This study presents the MNIM approach, which integrates the El-Kalla polynomial with a new iterative method. The linear components of delay differential equations (DDEs) are first addressed using the new iterative method, followed by a secondary iterative process to manage the complexities introduced by nonlinear terms, as detailed in Section 2.3. Essential definitions and concepts related to DDEs, the iterative methodology, and the El-Kalla polynomial are also discussed. The effectiveness and reliability of MNIM are demonstrated through three notable test cases, with absolute errors evaluated both graphically and numerically for different values of the time variable xxx. The findings emphasize MNIM`s capability to produce highly accurate approximations that closely match exact solutions with minimal error. This highlights MNIM`s potential as a powerful and efficient tool for solving nonlinear DDEs. Additionally, the study lays the groundwork for future research, showcasing the method`s potential in addressing complex problems across various scientific and engineering fields where nonlinear DDEs play a key role in modeling.

Key-Words / Index Term :
Delay differential equations; Fractional delay differential equations; El-kalla polynomials; Adomian polynomials

References :
[1] Jhinga, V., & Daftardar-Gejji, V., "An innovative predictor-corrector technique for nonlinear fractional delay differential equations," 2019.
[2] Srivastava, V., "Approximate Analytical Solution of Linear and Nonlinear Fractional Delay Differential Equations using Variational Iteration Method," Open Science Journal, vol. 5, no. 4, pp. 1–11, 2020.
[3] Du, M. J., "A Specific Method for Solving Fractional Delay Differential Equation via Fraction Taylor’s Series," Mathematical Problems in Engineering, vol. 2022, pp. 1–6, 2022.
[4] El-Kalla, I. L., Elgaber, K. M. A., Elmahdy, A. R., & Sayed, A. Y., "Solution of a Nonlinear Delay Differential Equation Using Adomian Decomposition Method with Accelerated Formula of Adomian Polynomial," American Journal of Computational Mathematics, vol. 9, no. 4, pp. 221–233, 2019.
[5] Avc?, D., "Numerical solutions for fractional delay differential equations using Caputo fractional derivatives," 2022.
[6] Diethelm, K., Ford, N. J., & Freed, A. D., "Detailed error analysis for a fractional Adams method," Numerical Algorithms, vol. 36, pp. 31–52, 2004.
[7] Diethelm, K., Ford, N. J., Freed, A. D., & Luchko, Y., "Algorithms for the fractional calculus: A selection of numerical methods," Computers and Mathematics with Applications, vol. 48, no. 10, pp. 1047–1061, 2002.
[8] Bhalekar, S., & Daftardar-Gejji, V., "A fractional Adams method for initial value problems with delay terms," Journal of Computational and Applied Mathematics, vol. 235, no. 12, pp. 3442–3450, 2011.
[9] Daftardar-Gejji, V., Jafari, H., & Bhalekar, S., "Numerical Predictor-Corrector Method for Solving Fractional Delay Differential Equations," Journal of Computational Analysis and Applications, vol. 16, pp. 1–11, 2014.
[10] Daftardar-Gejji, V., & Jafari, H., "An iterative method for solving nonlinear functional equations," Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 753–763, 2006.
[11] Bhalekar, S., & Daftardar-Gejji, V., "Solving fractional boundary value problems using a new iterative method," Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1801–1809, 2010.
[12] Daftardar-Gejji, V., & Kumar, R., "Error analysis for numerical methods in fractional delay differential equations," Applied Numerical Mathematics, vol. 126, pp. 60–72, 2018.
[13] Daftardar-Gejji, V., Bhalekar, S., & Kumar, R., "Improved Predictor-Corrector Method for Fractional Delay Differential Equations," Computational Mathematics and Modeling, vol. 26, no. 3, pp. 345–360, 2015.
[14] Kumar, R., & Methi, S., "Application of Banach Contraction Principle to Fractional Delay Differential Equations," Mathematical Problems in Engineering, vol. 2021, pp. 1–10, 2021.
[15] Daftardar-Gejji, V., & Bhalekar, S., "Banach Contraction Method for Fractional Differential Equations," 2009.