A general definition of the fractal derivative: Theory and applications

In this paper, we introduce a general definition of the fractal derivative with respect to a function $\psi$, in the context of the order $0<\alpha \le 1$ and the function $\psi (\mathrm{\Theta })$. This novel definition generalizes the classical fractal derivative, which is recovered when $\psi (\mathrm{\Theta })=\mathrm{\Theta }$, as described in previous works by Chen et al. ^[1,2]. We explored key properties of the $\psi$-fractal derivative, including the $\psi$-fractal Laplace transform, which provides a powerful tool for solving complex differential equations in fractal domains. We also derived a generalized $\psi$-chain rule, extending classical calculus into the fractal domain, and presented fundamental operations related to this unique derivative. We give some applications.