An extended $\frac{G^{\prime }}{G}$-expansion method for the conformable space-time fractional Newell-Whitehead-Segel equation: Exact traveling-wave solutions and regularity features

This paper employed an extended $\frac{G^{\prime }}{G}$-expansion method to investigate the conformal fractional Newell-Whitehead-Segel equation. By applying a traveling-wave transformation, we derive three types of analytical solutions that describe this fractional equation, thereby providing a useful theoretical foundation and intuitive interpretation for understanding its dynamical behavior. Analysis of the obtained solutions revealed that when both spatiotemporal orders approach $1$, the regularity of the solutions becomes highly complex in a specific region. However, when one of the parameters $\lambda$ or $\mu$ is taken as $1$ and the other as $0$, the regularity region is significantly improved. Furthermore, the comparison showed that the regularity becomes worse as $\frac{\lambda +\mu }{2}$ increases. The corresponding contour plots serve as a clear supplement to the complex variations in regularity. We also considered the influence of the parameter $k$ on the solutions: larger values of $k$ lead to more singularities in the solutions and increase the likelihood of blow-up, while smaller values of $k$ reduce singularities and promote the global existence of the solutions.