This paper employed an extended $ \frac{G'}{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.
Citation: Chunyan Zhao, Jie Wu, Zheng Yang. An extended $ \frac{G'}{G} $-expansion method for the conformable space-time fractional Newell-Whitehead-Segel equation: Exact traveling-wave solutions and regularity features[J]. AIMS Mathematics, 2026, 11(4): 11347-11371. doi: 10.3934/math.2026467
This paper employed an extended $ \frac{G'}{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.
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Chunyan Zhao, Jie Wu, Zheng Yang. An extended $ \frac{G'}{G} $-expansion method for the conformable space-time fractional Newell-Whitehead-Segel equation: Exact traveling-wave solutions and regularity features[J]. AIMS Mathematics, 2026, 11(4): 11347-11371. doi: 10.3934/math.2026467
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