An investigation of a closed-form solution for non-linear variable-order fractional evolution equations via the fractional Caputo derivative

Abstract

Determining the non-linear traveling or soliton wave solutions for variable-order fractional evolution equations (VO-FEEs) is very challenging and important tasks in recent research fields. This study aims to discuss the non-linear space–time variable-order fractional shallow water wave equation that represents non-linear dispersive waves in the shallow water channel by using the Khater method in the Caputo fractional derivative (CFD) sense. The transformation equation can be used to get the non-linear integer-order ordinary differential equation (ODE) from the proposed equation. Also, new exact solutions as kink- and periodic-type solutions for non-linear space–time variable-order fractional shallow water wave equations were constructed. This confirms that the non-linear fractional variable-order evolution equations are natural and very attractive in mathematical physics.