Some applications of nonlinear fractional differential equations by modified Elzaki transform

Altaie, Huda Omran; Khalaf, Hatam Y.; Shihab, Buthyna Najad

doi:10.47974/JIM-2540

Open Access Research Article

Some applications of nonlinear fractional differential equations by modified Elzaki transform

^*Huda Omran AltaieCorresponding authorhuda.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 → , Hatam Y. Khalafhatim.y.k@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 BaghdadBaghdad, 10071, IraqView full profile → , Buthyna Najad Shihabbothaina.n.s@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 BaghdadBaghdad, 10071, IraqView full profile →

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pp. 1509–1513Vol. 29Issue 5-BMay 2026DOI: 10.47974/JIM-2540XML

Received:: 01 Nov 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2540
Pages:: 1509–1513

Abstract

An major goal for this work is to propose a modified technique, the Elzaki Adomain Decomposition Method (ETADM), an iterative analytical approach that yields approximate analytical solutions for certain nonlinear problems. This technique relies on applying the Adomain Decomposition Method (ADM) to a nonlinear term that is amenable to solution using Adomain polynomials. ETADM has been used to break down the solution to a partial differential equation into an infinite number of components. Moreover, illustrative examples are provided, and the results obtained demonstrate the proposed method’s accuracy, efficiency, and reliability. Results like this demonstrate how effective and efficient this method is in resolving problems of this kind. Therefore, using fractional differential equations as a model, our suggested approach can be used to analyze solutions to a broad range of genuine problems arising in the natural sciences and engineering.

Keywords

Decomposition method Elzaki transform Analytical method Partial differential equations Fractional differential equations

Subject Classifications

35R1135R60

References

[1] A. A. Alderremy, R. Shah, N. A. Shah, S. Aly, and K. Nonlaopon, “Comparison of two modified analytical approaches for the systems of time fractional partial differential equations,” AIMS Mathematics, vol. 8, pp. 7142–7162 (2023).
[2] A. S. Alshehry, R. Shah, N. A. Shah, and I. Dassios, “A reliable technique for solving fractional partial differential equations,” Axioms, vol. 11, p. 574 (2022).
[3] H. Altaie, “Performance of two-way nesting techniques for shallow water models,” International Journal of Mechanical Engineering and Technology, vol. 7, no. 6, pp. 425–434 (2016).
[4] H. O. Altaie, “Comparison the solutions for some kinds of differential equations using iterative methods,” Journal of Interdisciplinary Mathematics, vol. 24, no. 5, pp. 1113–1118 (2021).
[5] M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress in Fractional Differentiation & Applications, vol. 2, no. 1, pp. 73–85 (2015).
[6] T. M. Elzaki, “The new integral transform Elzaki transform,” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, pp. 57–64 (2011).
[7] R. Hilfer, Applications of Fractional Calculus in Physics. Singapore: World Scientific (2000).
[8] H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488–497 (2006).
[9] A. Khalouta, “A new analytical series solution with convergence for non-linear fractional Lienard’s equations with Caputo fractional derivative,” Kyungpook Mathematical Journal, vol. 62, no. 3, pp. 583-593 (2022).
[10] X. Li and Y. Zhang, “A fractional gradient method for solving nonlinear partial differential equations with multi-point boundary conditions,” Journal of Scientific Computing, vol. 80, pp. 474–498 (2021).
[11] 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).
[12] 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).
[13] L. Wang and X. Zhou, “A multi-point differential inequality-based approach for solving fractional partial differential equations with variable coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 85, p. 104835 (2022).
[14] R. Ye, P. Liu, K. Shi, and B. Yan, “State damping control: A novel simple method of rotor UAV with high performance,” IEEE Access, vol. 8, pp. 214346–214357 (2020).
[15] S. A. Al-Ameedee and A. J. Obaid, “Sandwich results of meromorphic multivalent functions defined by multiplier transform,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 651–658 (2023), doi: 10.47974/JIM-1484.

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