Yazarlar
Yayınlar
- MakaleYalnız künye2026Development of a hybrid technique for solving the Newell–Whitehead–Segel equation using the Caputo–Fabrizio derivative
Şahin, Yasin; Merdan, Mehmet; Açıkgöz, Pınar
Purpose – The aim of the paper titled “Development of a hybrid technique for solving the Newell–Whitehead–Segel (NWS) equation using the Caputo-Fabrizio derivative” is to develop a new numerical method for solving the NWS equation using the Caputo-Fabrizio fractional derivative. This method involves the combination of the Caputo-Fabrizio Elzaki transformation and the q-homotopy analysis transformation method. The study aims to analyze the NWS equation that models the reaction-diffusion dynamics in biological and chemical systems. The convergence analysis of the proposed method is performed, and the obtained solutions are visualized with two- and three-dimensional graphics using the Maple software. The results show that the proposed method is more powerful and effective than the existing methods. Design/methodology/approach – In this study, a hybrid analytical-numerical method is developed to solve the NWS equation formulated using the Caputo-Fabrizio fractional derivative. The method is a combination of: q-homotopy analysis method (q-HATM) and Shehu transformation technique (ST). By combining these two powerful methods, a solution approach that does not require linearization or complex calculations for nonlinear fractional partial differential equations, converges quickly and gives high accuracy results, is presented. The obtained solutions are tested for numerical accuracy and are supported graphically. Findings – In the article titled “Development of a hybrid technique for solving the Newell–Whitehead–Segel equation using the Caputo–Fabrizio derivative”, the authors propose a new hybrid method for solving the time-fractional NWS equation. This method is a combination of the q-HATM and the ST. The proposed q-HSATM provides an efficient and lightweight approach for solving nonlinear fractional partial differential equations without requiring linearization, discretization or complex polynomials. This hybrid technique obtains the approximate analytical solution of the NWS equation using the Caputo–Fabrizio derivative. The method provides fast convergence and high accuracy with less memory and processing power requirements. It also stands out as a reliable tool for modeling complex systems in applied sciences such as physics, biology and engineering. In conclusion, this study provides an effective and applicable method for solving nonlinear differential equations formulated with the Caputo-Fabrizio derivative, making a significant contribution to the field of fractional calculus. Originality/value – This section of the article titled “Development of a hybrid technique for solving the Newell-Whitehead-Segel equation using the Caputo-Fabrizio derivative” can be summarized as follows: This study is one of the first hybrid q-HATM and ST-based methods developed to solve the NWS equation using the Caputo-Fabrizio fractional derivative operator. Its unique aspects: Unlike traditional methods, it provides models more suitable for physical systems by working with the Caputo-Fabrizio derivative, which includes the memory effect and whose kernel function is not exponential. The presented method presents a new and powerful analytical-numerical technique that can produce approximate solutions with high accuracy without requiring linearization or discretization. The developed technique is valuable not only theoretically but also with its applicability to reaction-diffusion systems in applied sciences.
- 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.