Florida Polytechnic University

Applied Mathematics & Science Seminar

Tues 3/31 in IST 1068 12:00-1:00pm

Speaker: Dr. Alberto Condori - Florida Gulf Coast University

Title: A problem of Erdos on the area of polynomial lemniscates

Abstract: The Caputo fractional derivative is a generalization of the classical derivative that arises naturally in models of anomalous diffusion, viscoelasticity, and other physical systems exhibiting memory effects. In practice, one often needs to compute this derivative from data that are only approximately known, for instance, from noisy measurements. Unfortunately, this is an ill-posed problem: small errors in the data can produce large errors in the computed derivative, a difficulty already familiar from ordinary numerical differentiation.

In this talk, we describe a two-step regularization method for the stable evaluation of the Caputo fractional derivative of any positive order. The approach draws on a range of mathematical ideas: Volterra integral equations provide the problem's structure, functional analysis (compactness, Sobolev regularity) explains its instability, Hardy spaces and the Laplace transform reveal the method's behavior in the frequency domain, and classical tools from numerical linear algebra — including Toeplitz systems and the Hilbert matrix — underpin its implementation. We present convergence results, discuss the role of a filtering polynomial, and illustrate the method's performance through numerical examples.