New Derivatives on the Fractal Subset of Real-Line

Abstract

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on the fractals subset of real-line lies in the fact that they are better at modeling processes with memory effect.

Keywords

• fractal calculus
• triadic Cantor set
• non-local Laplace transformation
• memory processes
• generalized Mittag-Leffler function
• generalized gamma function
• generalized beta function