Speaker/Presentation

Time in CET

Prof. Dr. Jonathan M. Blackledge

Chair Introduction

2:00 – 2:10 pm

Prof. Dr. Juan E. Nápoles Valdes

From L´Hopital to Atangana: Some Applications of Fractional Calculus

Abstract:
In this presentation, we will touch on some historical aspects of Fractional Calculus, its main exponents, and some of the most recent applications, mainly linked to Integral Inequalities.

2:10 – 2:40 pm

Q&A

2:40 – 2:50 pm

Prof. Dr. Jorge E. Macias-Diaz

Numerical Solution of the Fractional Fermi-Pasta-Ulam-Tsingou Problem

Abstract:
In this talk, we consider a partial differential equation that extends the well-known Fermi-Pasta-Ulam-Tsingou chains from nonlinear dynamics. The continuous model under consideration includes the presence of both a damping term and a polynomial function in terms of Riesz space-fractional derivatives. Initial and boundary conditions on a closed and bounded interval are considered in this work. The mathematical model has a fractional Hamiltonian which is conserved when the damping coefficient is equal to zero, and dissipated otherwise. Motivated by these facts, we propose a finite-difference method to approximate the solutions of the continuous model. The method is an explicit scheme which is based on the use of fractional centre differences to approximate the fractional derivatives of the model. A discretized form of the Hamiltonian is also proposed in this work, and we prove analytically that the method is capable of conserving or dissipating the discrete energy under the same conditions that guarantee the conservation or dissipation of energy of the continuous model. We show that solutions of the discrete model exist and are unique under suitable regularity conditions on the reaction function. We rigorously establish the properties of consistency, stability and convergence of the method. To that end, novel technical results are mathematically proved. Computer simulations that assess the capability of the method to preserve energy are also provided for illustration purposes.

2:50 – 3:20 pm

Q&A

3:20 – 3:30 pm

Prof. Dr. Jonathan M. Blackledge

Fractional Calculus and Fractal Geometry: A Unification using Einstein’s Evolution Equation

Abstract:
The presentation focuses on demonstrating how certain random self-affine fields can be cast in terms of the solution(s) to various classes of stochastic fractional differential equations. This is undertaken using a fundamental field equation in statistical mechanics (the Evolution Equation, first derived by Einstein in 1905) in order to provide a unifying theme. It is shown how the application of a Levy distribution to this equation yields the spatial Fractional Diffusion Equation (FDE) and how the application of certain memory functions (applied to the Generalized Kolmogorov-Feller Equation) generates a temporal FDE. Example fundamental solutions to these equations are provided, and investigated in terms of the visualization of various random fractal fields. These include the Mandelbrot surface (for the time independent case), and time dependent fields that are relevant to the analysis of time series with a wide range of applications.

3:30 – 4:00 pm

Q&A

4:00 – 4:10 pm

Prof. Dr. Jonathan M. Blackledge

General Discussion, Further Questions to all the Speakers, a Vote of Thanks and Closure of the Webinar

4:10 – 4:30 pm