Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

Highlights: • Fractal-fractional differentiation. • Fractal-fractional integration. • New numerical scheme for fractal-fractional operators. • New model of Darcy scale describing flow in a dual medium. - Abstract: New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.