Fractional Operators -- An Overview

Sammanfattning

While the classical integer-order derivative is a standard tool in analysis, the extension to non-integer orders offers a new way to model certain systems. However, how would one formalize the concept of half of one of these operators? In this thesis we explore the concept of fractional operators and how they can be used to solve the fractional differential equation Dαf(t) = λf(t). In contrast to the classical derivative, there are numerous distinct definitions of the fractional differential operator. Given this diversity, this thesis investigates four commonly used definitions: Grünwald-Letnikov, Riemann-Liouville, Caputo, as well as the more recent conformable derivative. In the process of solving the previously mentioned equation we also get some insight regarding each of the different operators and how they work. Finally, we also state some real-world applications of how fractional operators can be used, demonstrating their capability involving memory effects and hereditary properties in certain materials.