Lyapunov-Schmidt reduction used to solve a nonlinear differential equation with temporal fractions

The present work focuses on the examination of bifurcation in periodic traveling wave solutions to a nonlinear fractional differential equation. Our methodology utilizes the Lyapunov-Schmidt reduction and He's fractional derivative techniques. An original fractional differential problem is transformed into a partial differential equation using the fractional complex transform, therefore facilitating the analysis. Consequently, we derive a simplified equation, which is formulated as a system of four nonlinear algebraic equations that align with the underlying complexity. Further, we explore the feasibility of obtaining linear approximation solutions for the nonlinear fractional differential equation.