Two New Modifications of the Exp-Function Method for Solving the Fractional-Order Hirota-Satsuma Coupled KdV

Two novel modifications of the exp-function method have been suggested to solve the nonlinear system of the partial differential equation of the fractional order for the first time. The fractional-order Hirota-Satsuma coupled KdV system has been solved precisely; as a result, some exact solutions, which include soliton-type and rational solutions, will be derived. Eventually, the graphs of the obtained results have been illustrated at the end of the article. The newly used methods are highly accurate, flexible, effective, and programmable to solve fractional-order devices.