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The space time fractional modified regularized long wave equation is a model equation to the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, the hydromagnetic waves in cold plasma, the phonetic waves in dissident quartz and phonetic gravitational waves in contractible liquids. In nonlinear science and engineering, the mentioned equation is applied to analyze the one way tract of long waves in seas and harbors. In this study, the closed form traveling wave solutions to the above equation are evaluated due to conformable fractional derivatives through double (G'⁄G,1⁄G)-expansion method and the Exp-function method. The existence of chain rule and the derivative of composite function permit the nonlinear fractional differential equations (NLFDEs) converted into ODEs using wave transformation. The obtain solutions are very much effective to analyze the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, one way tract of long waves in seas and harbors. These two methods are efficient, convenient, and computationally attractive.

Analytical wave solutions of the space time fractional modified regularized long wave equation involving the conformable fractional derivative