The ZMA projected differential transform method for solving linear and non-linear partial differential equations

Alwan, Zainab Mohammed

doi:10.47974/JIM-2590

Open Access Research Article

The ZMA projected differential transform method for solving linear and non-linear partial differential equations

^*Zainab Mohammed AlwanCorresponding authorlionwight_2009@uomustansiriyah.edu.iqDepartment of MathematicsCollege of ScienceMustansiriyah UniversityBaghdad, IraqView full profile →

* Corresponding author · click or hover a name for details

pp. 1335–1342Vol. 29Issue 5-AMay 2026DOI: 10.47974/JIM-2590XML

Received:: 01 Jan 2026
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2590
Pages:: 1335–1342

Abstract

In this project, present a novel technique to solve linear and non-linear (L and NL) of partial differential equations (PDEs) with simple steps to present active and accurate results take less time compare with other researcher techniques to find the exact solutions (ES). These techniques performed by mixture of ZMA-transform (ZMAT) and projected differential transformed method (PDTM) demonstrated as ZMA projected differential transformed method (ZMAPDTM). Also, to more understand this technique inserted some properties assisted of greeted algorithm with steps to solve various problems.

Keywords

Projected differential transformed method ZMA-transform ZMA projected differential transformed method Integral transform Differential transform method PDEs

Subject Classifications

Primary 93A30Secondary 49K15

References

[1] M. D. Raisinghania, Advanced Differential Equations. New Delhi, India: S. Chand & Co. Ltd. (1995).
[2] A. Jeffrey, Advanced Engineering Mathematics. San Diego, CA, USA: Harcourt/Academic Press (2002).
[3] K. A. Stroud and D. J. Booth, Engineering Mathematics. New York, USA: Industrial Press, Inc. (2001).
[4] L. M. Upadhyaya, “Introducing the Upadhyaya integral transform,” Bull. Pure Appl. Sci., Sect. E: Math. Stat., vol. 38E, no. 1, pp. 471–510 (2019), doi: 10.5958/2320-3226.2019.00051.1.
[5] S. Aggarwal and R. Chauhan, “A comparative study of Mohand and Aboodh transforms,” International Journal of Research in Advent Technology, vol. 7, no. 1, pp. 520–529 (2019), doi: 10.32622/IJRAT.712019107.
[6] E. Kuffi, E. S. Abbas, and S. O. Mebayet, “Al-Tememe transform in solving developed missile automatic control problem,” International Journal of Engineering & Technology, vol. 7, no. 4, pp. 6565–6567 (2018), doi: 10.14419/ijet.
[7] H. Jafari, “A new general integral transform for solving integral equations,” Journal of Advanced Research, vol. 32, pp. 133–138 (2021).
[8] M. M. A. Mahgoub, “The new integral transform ‘Mohand Transform’,” Advances in Theoretical and Applied Mathematics, vol. 12, pp. 113–120 (2017).
[9] H. Jafari and S. Aggarwal, “Upadhyaya integral transform: A tool for solving non-linear Volterra integral equations,” Mathematics and Computational Sciences, vol. 5, no. 2, pp. 63–71 (2024).
[10] M. Higazy and S. Aggarwal, “Sawi transformation for system of ordinary differential equations with application,” Ain Shams Engineering Journal, vol. 12, pp. 3173–3182 (2021).
[11] Z. M. Alwan, “ZMA-transform method,” Journal of Interdisciplinary Mathematics, vol. 24, no. 7, pp. 1841–1849 (2021), doi: 10.1080/09720502.2021.1963520.
[12] Z. M. Alwan, “Solution homogeneous and non-homogeneous fractional ordinary differential equations by ZMA-transform,” Journal of Interdisciplinary Mathematics, vol. 27, no. 4, pp. 781–785 (2024), doi: 10.47974/JIM-1871.
[13] S. Kumar, “A new analytical modelling for fractional telegraph equation via Laplace transform,” Applied Mathematical Modelling, vol. 38, no. 13, pp. 3154–3163 (2014), doi: 10.1016/j.apm.2013.11.035.
[14] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60, Fundamental Theories of Physics. Dordrecht, Netherlands: Kluwer Academic Publishers (1994), doi: 10.1007/978-94-015-8289-6.
[15] J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, pp. 257–262 (1999).
[16] J. H. He, “Variational iteration method—a kind of nonlinear analytical technique: Some examples,” Int. J. Non. Linear. Mech., vol. 34, pp. 699–708 (1999).
[17] M. A. Gondal, A. S. Arife, M. Khan, and I. Hussain, “An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method,” World Applied Sciences Journal, vol. 15, pp. 1786–1791 (2011).
[18] S. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,” Journal of Applied Mathematics, vol. 1, pp. 141–155 (2001).
[19] Y. Khan, “An effective modification of the Laplace decomposition method for nonlinear equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, pp. 1373–1376 (2009).
[20] A. Yildirim, “HES homotopy perturbation method for solving the space- and time-fractional telegraph equations,” International Journal of Computational Mathematics, vol. 87, no. 13, pp. 2998–3006 (2010).

Views: 88Downloads: 83Citations: 0