Project Info


Investigating Discrete Fractional Calculus

Kevin Ahrendt | kahrendtmines.edu

In a typical calculus course, students learn about first order derivatives, second order derivatives, and so on. This may lead one to ask the question, is there such a thing as a half order derivative or a π’th order derivative? Fractional calculus seeks to answer this question in a way that extends the basic results from continuous calculus to non-integer ordered operators. This project would investigate difference equations on a discrete domain that involve fractional derivatives. In particular, we will investigate the Caputo fractional difference, it’s properties, and how to solve various fractional difference equations involving the Caputo difference.

More Information

See Discrete Fractional Calculus, by Goodrich and Peterson, for an in depth introduction to the area.

Grand Engineering: Not applicable

Student Preparation


Qualifications

A successful student for this project should have completed real analysis and have an interest in the broad area of differential equations.

Time Commitment

30 hours/month

Skills/Techniques Gained

The student will gain valuable experience on understanding what mathematical research consists of and how mathematicians undertake it. They will learn how to apply the theorems and results from real analysis in an application setting. Finally, they will leave the project with a strong understanding of fractional difference equations, as well as will be on the forefront of this young research area.

Mentoring Plan

We would engage in a weekly meeting to advance through the project. Initially, the student will go through the necessary background materials, while completing various exercises to get a strong understanding of how typical proofs go in this setting. These early weekly meetings will be a discussion on their progress along with addressing any issues with material. After one or two months of this, the student will have been exposed to many different open problems, and we will proceed with one that particularly interests the student. We will continue these meetings where I will help guide them through their understanding of the open problem and encourage specific questions to investigate.