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Article

The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method

1
Gümüşhane National Education Directorate, Gümüşhane 29000, Turkey
2
Mathematical Engineering Department, Gümüşhane University, Gümüşhane 29010, Turkey
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(8), 1219; https://doi.org/10.3390/sym17081219 (registering DOI)
Submission received: 10 June 2025 / Revised: 4 July 2025 / Accepted: 17 July 2025 / Published: 1 August 2025

Abstract

This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives.
Keywords: Caputo–Fabrizio fractional; Helmholtz equation; Aboodh transform method; Adomian Decomposition Method Caputo–Fabrizio fractional; Helmholtz equation; Aboodh transform method; Adomian Decomposition Method

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MDPI and ACS Style

Şahin, Y.; Merdan, M.; Açıkgöz, P. The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method. Symmetry 2025, 17, 1219. https://doi.org/10.3390/sym17081219

AMA Style

Şahin Y, Merdan M, Açıkgöz P. The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method. Symmetry. 2025; 17(8):1219. https://doi.org/10.3390/sym17081219

Chicago/Turabian Style

Şahin, Yasin, Mehmet Merdan, and Pınar Açıkgöz. 2025. "The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method" Symmetry 17, no. 8: 1219. https://doi.org/10.3390/sym17081219

APA Style

Şahin, Y., Merdan, M., & Açıkgöz, P. (2025). The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method. Symmetry, 17(8), 1219. https://doi.org/10.3390/sym17081219

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