A new coupled natural transform and Adomian decomposition technique to solve some models of PDEs

Abdulhadi, Ahmed Najah; Altaie, Huda Omran

doi:10.47974/JIM-2542

Open Access Research Article

A new coupled natural transform and Adomian decomposition technique to solve some models of PDEs

^*Ahmed Najah AbdulhadiCorresponding authorahmed.abd2203@ihcoedu.uobaghdad.edu.iqDepartment of Mathematics College of Education for Pure Science (Ibn Al-Haitham) University of BaghdadBaghdad, 10071, IraqView full profile → , Huda Omran Altaiehuda.h.o@ihcoedu.uobaghdad.edu.iqDepartment of Mathematics College of Education for Pure Science (Ibn Al-Haitham) University of BaghdadDepartment of Mathematics College of Education for Pure Science (Ibn Al-Haitham) University of Baghdad Baghdad, 10071, IraqView full profile →

* Corresponding author · click or hover a name for details

pp. 1519–1524Vol. 29Issue 5-BMay 2026DOI: 10.47974/JIM-2542XML

Received:: 01 Nov 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2542
Pages:: 1519–1524

Abstract

Coupled NT and Decomposition technique (NTADM), is a powerful methodology for dealing with PDEs. The technique used NT to eliminate the time of the driving equation, whereas ADM tackles nonlinear variables methodically without resorting to linearization or numerical discretization. This work employs NTADM to solve a fourth-order NPDE and obtains approximate solutions up to the fifth iteration. The features of the solutions emerging from this show effortless, steady behavior in both space and time. The numerical results show that accuracy improves as the approximation order increases. The absolute error also remains below 1 across most of the computational domain, indicating very strong convergence. These results show that NTADM is a strong and reliable technique for analyzing complex high-order nonlinear PDE models.

Keywords

Natural transform Semi-analytical method Approximate solutions Fourth-order PDE

Subject Classifications

35R1135R60

References

[1] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, Series on Advances in Statistical Mechanics, vol. 35. Singapore: World Scientific Publishing Co. (2000), doi: 10.1142/3779.
[2] W. A. Strauss, Partial Differential Equations: An Introduction, 2nd ed., Hoboken, NJ, USA: John Wiley & Sons, pp. 464 (2007).
[3] I. Ateş and P. A. Zegeling, “A homotopy perturbation method for fractional-order advection-diffusion-reaction boundary-value problems,” Applied Mathematical Modelling, vol. 47, pp. 425–441 (2017).
[4] A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory. Berlin/Heidelberg, Germany: Springer Science & Business Media (2009), doi: 10.1007/978-3-642-00251-9.
[5] H. O. Altaie, “Study the Stability for Ordinary Differential Equations Using New Techniques via Numerical Methods,” in Proceedings of First International Conference on Mathematical Modeling and Computational Science: ICMMCS 2020, Singapore: Springer Singapore, pp. 509–516 (May 2021).
[6] A. M. Shukur, “Adomian decomposition method for certain space-time fractional partial differential equations,” IOSR Journal of Mathematics, vol. 11, no. 1, pp. 55–65 (2015).
[7] A. M. Mohammed and H. O. Altaie, “Analytical solutions via coupled Elzaki Adomian decomposition method for some applications,” Journal of Interdisciplinary Mathematics, vol. 27, no. 4, pp. 761–766 (2024).
[8] B. K. Singh and P. Kumar, “Fractional variational iteration method for solving fractional partial differential equations with proportional delay,” International Journal of Differential Equations, vol. 2017, Art. no. 5206380, pp. 1–11 (2017), doi: 10.1155/2017/5206380.
[9] A. M. Mohammed and H. O. Altaie, “Elzaki transform decomposition approach to solve Riccati matrix differential equations,” Journal of Interdisciplinary Mathematics, vol. 27, no. 4, pp. 767–773 (2024).
[10] F. Evirgen, “Analyze the optimal solutions of optimization problems by means of a fractional gradient-based system using VIM,” An International Journal of Optimization and Control, vol. 6, no. 2, p. 75 (2016).

Views: 22Downloads: 12Citations: 0