Fractional Calculus is the study of integrals and derivatives of arbitrary order. We are all at least acquainted with the idea of a regular derivative or integral, and, by iteration, second, third, and even arbitrary integer-order integrals or derivatives can be found with the tools of regular calculus. Fractional Calculus seeks to find the integrals and derivatives of real or even complex order using the Gamma function, Euler’s generalization of the factorials. Over the summer, I surveyed the literature on Fractional Calculus and worked on a simplification of the Fractional Integral equation using repeated Integration by Parts. I also reviewed some of Fractional Calculus’ many applications, including modeling movement with a dynamic rate of flow for time and heat transfer. In this presentation I intend to review the Fractional Integral equation, how it is derived and why we say that it produces integrals and derivatives of arbitrary order.

**John Klingner, ’12 **Littleton, CO

Major: Computer Science

Sponsor: Stephen Bean