Numerical methods in fractional calculus

We will examine how we can use Caputo’s definition of a fractional derivative to develop a method to numerically evaluate a fractional derivative of a function at a point. A fractional derivative can be thought as an instantaneous fractional average. In addition we will take some concepts from Complex Step Differentiation to help improve the method. We will also examine how to improve the Riemann-Liouville fractional integral. We will find that we can get solutions to these operators with relative error of between 0 and 10−7, with an average error of 10−12. We will see by reformulating the different operators we can control singularities in the operators. It will also be shown that these operators smoothly deform a function to its nth order derivative or integral.