Wave Effects of the Fractional Shallow Water Equation and the Fractional Optical Fiber Equation

The non-linear space-time fractional Estevez-Mansfield-Clarkson (EMC) equation and the non-linear space-time fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation showed the motion of waves in the shallow water equation and the optical fiber equation, respectively. The process used to solve these equations is to transform the non-linear fractional partial differential equations (PDEs) into the non-linear ordinary differential equations by using the Jumarie's Riemann-Liouville derivative and setting the solution in the finite series combined with the simple equation (SE) method with the Bernoulli equation. The new traveling wave solutions were the exponential functions resulting in the physical wave effects are produced in the form of kink waves and represented by the two-dimensional graph, three-dimensional graph, and contour graph. In addition, the comparison of the solutions revealed that the new solutions have a more convenient and easier format.