ABSTRACT
Calculus is the formal method by which scientists carry out quantitative reasoning. In the analysis of complex phenomena, particularly those that change over time, the traditional guideposts such as identifying causality and making verifiable predictions become problematic. One way to anticipate the improbable and prepare for unintended consequences is through the systematic handling of fractal operators, using the fractional calculus and fractional differential equations. But we also need to incorporate the more traditional notions of complexity, such as randomness into the discussion. Therefore we start our tour of the tools required to think newly about complexity with a brief introduction to random walks using fractional difference equations. Such fractional random walks have been used to model complex phenomena from fluctuations in the financial market [24] to climate change [17].
