An Observation on the Elzaki Transform and the Fractional Coupled System of PDEs

Mohammed E. A. RABIE; Tarig M. ELZAKI

doi:10.2478/ama-2025-0060

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An Observation on the Elzaki Transform and the Fractional Coupled System of PDEs

Acta Mechanica et Automatica

Volume 19 (2025): Issue 3 (September 2025)

By: Mohammed E. A. RABIE and Tarig M. ELZAKI

Open Access

|Sep 2025

Figures & Tables

Φ(τ)	E [Φ(τ)] = T(o)
1	o²
τ	o³
τⁿ	n! oⁿ⁺²
e^aτ	$\frac{o^{2}}{1 - a o}$
sin a τ	$\frac{a o^{3}}{1 + a^{2} o^{2}}$
cos a τ	$\frac{o^{2}}{1 + a^{2} o^{2}}$

(a) Comparing the provided method's 3D representations of the solutions to the exact solutions for Eq. (8), the outcome in comparison to the exact solutions, (b) represent the solutions at κ = 0.95,0.9,0.85

(a) Comparing the provided method's 3D representations of the solutions to the exact solutions for Eq. (11), P(*υ, ζ, τ*) the outcome in comparison to the exact solutions, (b) represent the solutions at κ = 0.95,0.9,0.85

(a) Comparing the provided method's 3D representations of the solutions to the exact solutions for Eq. (11), Q(*υ, ζ, τ*) the outcome in comparison to the exact solutions, (b) represent the solutions at κ = 0.95,0.9,0.85

(a) Comparing the provided method's 3D representations of the solutions to the exact solutions for Eq. (11), K(*υ, ζ, τ*) the outcome in comparison to the exact solutions, (b) represent the solutions at κ = 0.95,0.9,0.85

Tab. 1.

The comparison of the exact and approximate solutions yields the numerical result for example (1)

	τ	υ	κ = 0.85	κ = 0.9	κ = 0.95	Exact	Error
		0	0.	0.	0.	0.	0.
		$\frac{π}{6}$	0.489578	0.491835	0.493619	0.495025	0.00140562
P(υ, τ) Q(υ, τ)	0.01	$\frac{π}{3}$	0.847973	0.851883	0.854974	0.857408	0.00243461
		$\frac{π}{2}$	0.979155	0.98367	0.987239	0.99005	0.00281124

Tab. 2.

The comparison of the exact and approximate solutions yields the numerical result for example (2)

	ζ	τ	υ	κ = 0.85	κ = 0.9	κ = 0.95	Exact	Error
			0	1.08213	1.08712	1.09107	1.09417	0.0031069
			$\frac{π}{6}$	1.82674	1.83516	1.84182	1.84707	0.00524474
P(υ, ζ, τ)	0.1	0.01	$\frac{π}{3}$	3.08371	3.09793	3.10916	3.11802	0.0088536
			$\frac{π}{2}$	5.20558	5.22958	5.24855	5.2635	0.0149457

Tab. 3.

The comparison of the exact and approximate solutions yields the numerical result for example (2)

	ζ	τ	υ	κ = 0.85	κ = 0.9	κ = 0.95	Exact	Error
			0	0.885976	0.890061	0.89329	0.913931	0.0206408
			$\frac{π}{6}$	1.49561	1.50251	1.50796	1.5428	0.0348435
Q(υ, ζ, τ)	0.1	0.01	$\frac{π}{3}$	2.52473	2.53637	2.54557	2.60439	0.058819
			$\frac{π}{2}$	4.26197	4.28162	4.29715	4.39645	0.0992919

Tab. 4.

The comparison of the exact and approximate solutions yields the numerical result for example (2)

	ζ	τ	υ	κ = 0.85	κ = 0.9	κ = 0.95	Exact	Error
			0	1.08213	1.08712	1.09107	1.11628	0.0252107
			$\frac{π}{6}$	0.64104	0.643995	0.646332	0.661266	0.0149344
K(υ, ζ, τ)	0.1	0.01	$\frac{π}{3}$	0.379742	0.381493	0.382877	0.391724	0.00884693
			$\frac{π}{2}$	0.224954	0.225991	0.226811	0.232051	0.00524079

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DOI: https://doi.org/10.2478/ama-2025-0060 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 518 - 524

Submitted on: Feb 4, 2025

Accepted on: Jun 16, 2025

Published on: Sep 30, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

Fractional Coupled System of Nonlinear Partial Differential Equations,

Novel Analytical Method,

Elzaki Transform,

Mittag-Leffler

Related subjects:

Engineering,

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Mohammed E. A. RABIE, Tarig M. ELZAKI, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 3 (September 2025)

An Observation on the Elzaki Transform and the Fractional Coupled System of PDEs

Figures & Tables

Fig.1.

Fig. 2.

Fig. 3.

Fig.4.

Tab. 1.

Tab. 2.

Tab. 3.

Tab. 4.

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