Development of a hybrid technique for solving the Newell–Whitehead–Segel equation using the Caputo–Fabrizio derivative

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.