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This study proposes a new method, called the Fractal Yang transform method (F$\mathcal{𝒴}$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 ($\mathcal{𝒴}$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.

This study presents the modified form of the homotopy perturbation method (HPM), and the Yang transform is adopted to simplify the solving process for the Kuramoto–Sivashinsky (KS) problem with fractal derivatives. This scheme is established by combining the two-scale fractal scheme and Yang transform, which is very helpful to evaluate the approximate solution of the fractal KS problem. Initially, we transfer the fractal problem into its partners using the two-scale fractal approach, and then we use the Yang transform (${𝒴}$T) to obtain the recurrent relation. Second, the HPM is then introduced to deal with the nonlinear elements of the fractal model. The numerical example demonstrates how the suggested technique is incredibly straightforward and precise for nonlinear fractal models. In addition, the graphical error of the proposed fractal model is compared with the calculated results of our suggested approach and the exact results. This graphical error displays the strength and authenticity of our proposed scheme.