Fractal geometry, introduced by Benoit Mandelbrot, provides a framework for analyzing natural patterns, emphasizing scale invariance and intricate structures crucial to studying complex systems. This work explores fractal geometry’s mathematical foundations via Felix Hausdorff’s framework, incorporating fractional dimensions and mass distribution metrics, making fractal analysis useful for physical and biological systems. Fractal calculus extends traditional calculus by integrating Hausdorff's mass distribution, supporting fractal differential equations like the Fokker-Planck Equation (FPE) for modelling complex systems. Fractional derivatives, modelling anomalous diffusion, serve as continuous approximations of fractal derivatives. The fractal FPE’s approximation yields the Plastino-Plastino Equation (PPE), linking to Tsallis statistics, relevant in fields like urban scaling, high-energy physics, and deforestation, showing fractal models' value in complex phenomena.
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