The Fractal Lie Derivative: Theory and Applications

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as k-fractal jet space, and algorithms for k-th fractal prolongation. The symmetries of the fractal nonlinear $n$-th $\alpha$-order differential equation are examined, followed by a discussion of the symmetries of the fractal linear $n$-th $\alpha$-order differential equation. Additionally, the symmetries of the fractal linear first $\alpha$-order differential equation are derived. Several examples are provided to illustrate and highlight the details of these concepts.