Novel fractional solutions and singular kink effect of some fluid mechanics equation and long wave propagation equation via simple equation method

The simple equation method with Bernoulli equations can yield fractional analytical solutions for nonlinear evolution equations in mathematical physics. We present a simple equation method based on the Bernoulli equation for figuring out nonlinear evolution equations. This lets us get analytical solutions with parameters using the space-time fractional Calogero-Degasperis equation and the space-time fractional modified Benjamin-Bona-Mahony equation. The fractional solutions and the resulting 2D, 3D, and contour graphical effects of those equations have the characteristics of exponential functions and the singular kink wave, respectively. As demonstrated, the use of the simple equation method, in conjunction with the Bernoulli equation, provides a robust mathematical tool for generating a wider range of analytical solutions. Furthermore, the Runge-Kutta method’s estimation of the solutions to both equations results in the same direction.