Dynamical behavior of soliton solutions to the fractional phi-four model via two analytical techniques

This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein-Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it's correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations.