Exact Solutions of M-fractional Complex Ginzburg-Landau Equation via Two Analytical Methods

This article investigates the M-fractional complex Ginzburg-Landau (CGL) equationby utilizing the modified simplest equation method and the modified G'G-expansion method. The CGL holds significant importance in the realm of oscillatory phenomena and optical fibers. By implementing a fractional complex transformation, the M-fractional CGL equation is transformed into an ordinary differential equation (ODE).By employing these methodologies, bright periodic solutions, periodic solutions, and dark wave solutions relevant to phenomena witnessed in nonlinear optics or plasma environments are ascertained.It is demonstrated that these methods are very powerful and efficient to solve many fractional differential equations.