Can Fractional Order Models do more Justice to Anomalous Behavior than Familiar Calssical Order Models Do?
Abstract
The fractional derivatives (FD) are generalizations of familiar integer order derivatives. The concept of fractional derivative goes back to 1695 when Leibniz introduced the notation (d/n y)/(dx/n) to L Hospital. In a reply, L Hospital asked "What if n = 1/2?", which did not have a satisfactory answer at that time. Nowadays, it is easy to find applications of FDs in pure and applied mathematics, engineering, physics, chemistry, biology, medicine and even in psychology. The idea will enable researchers to find hidden beauties in the nature that cannot otherwise be identified using the classical derivatives. Some of the striking ideas of FDs include the concept of Fractional Brownian Motion, which, for example, governs the stock market, functions in our brain or anomalous diffusion that can be found in many porous medium such as the surface of the sun or inside battery cells. The objective of this project is to explore the applicability of fractional order models over the familiar classical order models in understanding the dynamics of fluid flows, especially in their anomalous nature. Emphasis will be given to anomalous diffusion and fractional generalizations of Burgers equation with external forces. This study will utilize a particular recently-introduced fractional derivative, now known as the Katugampola fractional derivative, which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives. It will analyze the fractional diffusion and Burgers equations for possible generalization to model anomalous behavior in fluids. The controllability, stability analysis and bifurcation study will also be conducted to strengthen and guide the modeling effort. Finally, the proposed research also focuses on some combinatorial aspects of a class of sequences of generalized Stirling numbers of the 2nd kind (A223523-A223532 on Sloane s OEIS) and generalized Fourier transform which, if successful, will redefine the Nuclear Magnetic Resonance (NMR) techniques. Thus, the work on FD will enhance the profound understanding of the field of fractional calculus and related fields and enables further applications in real-world situations by redefining the true nature of the realm which once was only available for pure mathematicians.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 12, 2017
- Source ID
- W911NF1510537
Entities
People
- Udita Katugampola
Organizations
- Army Contracting Command
- United States Army
- University of Delaware