A Fractal Solution of Camassa-Holm and Degasperis-Procesi Models under Two Scale Dimension Approach

This study proposes a new method, called the Fractal Yang transform method (FY TM), for obtaining the fractal solution of the modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) models with fractal derivatives. The authors use the two-scale fractal approach to convert the fractal problem into its differential components and implement the Yang transform (Y T) to achieve the recurrence iteration. We then apply the homotopy perturbation method (HPM) to overcome the difficulty of nonlinear elements in the recurrence iteration, which makes it simple to acquire further iterations. The most advantage of this fractal approach is that it has no restriction on variables and provides successive iterations. The fractal results are presented in the sense of a series that converges to the exact solution only after a few iteration. Graphical behavior demonstrates that this fractal approach is a very fast and remarkable solution, particularly with fractal derivatives.