Authors : Yasir Abeid Hussin; Safa khider Malik; Abir Khalil Salibi

Volume/Issue : Volume 10 - 2025, Issue 6 - June

Google Scholar : https://tinyurl.com/ycx87n9v

DOI : https://doi.org/10.38124/ijisrt/25jun1520

Note : A published paper may take 4-5 working days from the publication date to appear in PlumX Metrics, Semantic Scholar, and ResearchGate.

Note : Google Scholar may take 30 to 40 days to display the article.

Abstract : This paper presents a comparative analysis of the Laplace Transform Method and the Adomian Decomposition Method (ADM) for solving the three-dimensional heat equation. The Laplace Transform Method converts equations into the frequency domain, enabling precise solutions for linear systems but struggling with asymmetric and nonlinear cases. In contrast, ADM decomposes the solution into an infinite series computed recursively, making it suitable for complex nonlinear applications. Through comparative analysis, this paper demonstrates that the Laplace Transform Method offers high accuracy for linear cases, while ADM is more flexible and better suited for handling complex boundary conditions. Consequently, the Laplace Transform is preferable for simple linear problems, whereas ADM proves to be more effective for complex and nonlinear cases.

Keywords : Adomian, Laplace, Transform, Decomposition.

References :

1. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method. New York, NY, USA: Springer, 1994
2. M. Wazwaz, “Adomian decomposition method for solving partial differential equations: A review,” Int. J. Comput. Math., vol. 86, no. 10–11, pp. 1634–1642, 2009.
3. M. Wazwaz, “A new method for solving nonlinear integral equations,” Appl. Math. Comput., vol. 116, no. 2–3, pp. 123–133, 2000.
4. G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation. Berlin, Germany: Springer-Verlag, 1974.
5. Churchill and J. W. Brown, Operational Mathematics, 3rd ed. New York, NY, USA: McGraw-Hill, 2009.
6. Betty Subartini, Ira Sumiati, Sukono, Riaman, Ibrahim Mohammed Sulaiman, "Combined Adomian Decomposition Method with Integral Transform," Mathematics and Statistics, Vol. 9, No. 6, pp. 976 - 983, 2021. DOI: 10.13189/ms.2021.090613.
7. J. S. Duan, “Further discussion on the convergence of Adomian decomposition method,” Appl. Math., 2012.
8. Y. Cherruault and G. Adomian, “Decomposition methods: A new proof of convergence,” Math. Comput. Model., 1990.
9. Saeed, N.A., Pachpatte, D.B. A modified fuzzy Adomian decomposition method for solving time-fuzzy fractional partial differential equations with initial and boundary conditions. Bound Value Probl 2024, 82 (2024). https://doi.org/10.1186/s13661-024-01885-9
10. S. A. Khuri, “A new approach to Bratu’s problem,” Appl. Math., 2001.
11. Biazar and H. Ghazvini, “Exact solutions for nonlinear Schrödinger equations by He’s variational iteration method and Adomian decomposition method,” Appl. Math. Comput., 2006.
12. M. E. Bakr, A. A. El‑Toony, A. Almohaimeed, and A. M. Gadallah, “Advancements in Laplace transform techniques: Performing non‑parametric hypothesis testing on real‑world data through statistical analysis,” AIP Advances, vol. 14, no. 3, art. 035118, Mar. 2024, https://doi.org/10.1063/5.0190624.
13. Garima Agarwala, Man Mohana, Athira M. Menonb, Amit Sharmac, Tikam Chand Dakalb, and Sunil Dutt Purohit, Analysis of the Adomian decomposition method to estimate the COVID-19 pandemic, 2022 Elsevier Inc., 2022 Elsevier Inc.
14. Obeidat NA, Rawashdeh MS, Al Smadi MN. Theoretical analysis of J-transform decomposition method with applications of nonlinear ordinary differential equations. Science Progress. 2024;107(2). doi:10.1177/00368504241256864.
15. Kot, V.A. Integral Method with the Adomian Decomposition: Approximate Solution of the Liouville–Bratu–Gelfand Problem for a Cylindrical Space. J Eng Phys Thermophy 97, 1356–1382 (2024). https://doi.org/10.1007/s10891-024-03008-8