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- MakaleAçık Erişim2025The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method
Şahin, Yasin; Merdan, Mehmet; Açıkgöz, Pınar
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.
- MakaleAçık Erişim2025New Numerical Solutions for Caputo–Fabrizio Fractional Differential Multidimensional Diffusion Problems
Şahin, Yasin; Merdan, Mehmet
This study examined the multidimensional fractional diffusion equations that characterize the density dynamics in a diffusing medium utilizing the exact and analytical Aboodh transform decomposition method (ATDM). The Caputo–Fabrizio (CF) derivative is utilized to account for the fractional derivative that is being employed here. The suggested method handles nonlinear terms by combining the Adomian decomposition method (ADM) with the Aboodh transform (AT) and Adomian polynomials. Since the AT can only be applied to linear equations, ADM is a helpful technique for estimating solutions to nonlinear differential equations. In nonlinear systems, multidimensional diffusion problems which account for the production of stripes in two-dimensional systems are important. Additionally, we used MATLAB to compute numerical and graphical data that illustrate how the close-form analytical solution compares to the precise solution. Promising and appropriate for addressing multidimensional diffusion problems with fractional derivatives are the obtained results. The key benefit is that our devised approach does not depend on presumptions or constraints on variables that skew the actual issue. Overcoming fluctuating constraints that can make it difficult to discover the answer and represent the problem depends much on this technique. In the domains of science and technology, the approach presented in this work offers exceptional computational accuracy and convenience of use for analyzing and resolving intricate phenomena associated with CF fractional nonlinear partial differential equations.