Study on the variable coefficient space-time fractional Korteweg de Vries equation

In this paper, the fractional Riccati method is modified for solving nonlinear variable coefficients fractional differential equations involving modified Riemann-Liouville derivative. This approach is successfully applied to the variable coefficient space-time fractional Korteweg de Vries (vcSTFKdV) equation. Variety of analytical solutions are obtained. The validity of this approach is discussed. The arbitrariness of the non-integer derivative order α possesses much richer structures. The amplitude increases when the non-integer derivative order increases such that 0.1<α⩽1. While it decreases the non-integer derivative order increases such that 0<α<0.1. From the graphical presentation of the obtained solutions it is observed that changing the non-integer derivative order value α affects the soliton behavior in a fundamental way; therefore, the non-integer derivative order can modify the wave shape without changing the properties of the medium, the nonlinearity and the dispersive effects.