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Mathematics Webinar | Fractional Calculus and Its Applications

25 Nov 2022, 14:00 (CET)

Fractional Calculus, History and More Recent Applications, Fractional Differential Equations, Nonlinear Dynamics, Stochastic Fields, Analytical and Numerical Solutions, Fractal Geometry
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Welcome from the Chair

4th Mathematics Webinar

Fractional Calculus and Its Applications

The idea of a fractional derivative and a fractional integral (the fractional calculus) goes back to the time when (integer) calculus was being developed in the late Seventeenth Century. In a letter from Guillaume l’Hospital to Gottfried von Leibnitz written in 1695, l’Hospital asked the following question: “Given that dnf(x)/dxn exists for all integer n, what if n be ½”? The reply from Leibnitz was all the more interesting: “It will lead to a paradox ... From this paradox, one day useful consequences will be drawn”. The foundations of Fractional Calculus were constructed later in the Nineteenth Century by mathematicians such as Joseph Liouville, and, in particular, Niels Henrik Abel, whose work led to the re-birth of the subject. During this period, many of the fundamental elements of the subject were conceived including the mutual inverse relationships between fractional-order integrals and differentials in terms of the same generalized operations.

Although both the theory and applications of fractional calculus continued to be developed over the Twentieth Century, relatively few papers, journals, conferences, and books were devoted to the subject compared to other areas of mathematics. However, since the start of the current century, there has been a considerable expansion of interests in the subject area, coupled with a growing range of applications. It is in this context that this webinar is being provided which will cover aspects of the history of the subject, its more recent developments, and some of the modern applications to which it is being applied.

Date: 25 November 2022

Time: 2:00 pm CET | 8:00 am EST | 9:00 pm CST Asia

Webinar ID: 881 3441 2882

Webinar Secretariat: mathematics.webinar@mdpi.com

Chair

SFI Stoke Professor School of Electrical and Electronic Engineering, Technological University Dublin, Ireland

DETAILS
Talk
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.
Bio
Prof. Dr. Jonathan M. Blackledge obtained a PhD degree in Theoretical Physics from London University in 1984, and a second PhD in Mathematics from the Jyvaskyla University in 2010. After a parallel career in the scientific civil service, academia, and industry, in 2008, he was elected to the position of the Science Foundation Ireland's Stokes Professor (named after the Irish Mathematician and Physicist, George Gabriel Stokes). From 2013-2016, he was Deputy Vice-Chancellor for Research at the University of KwaZulu-Natal and currently holds professorships with research institutes and universities in the UK and overseas while directing R & D programs for the UK defense and intelligence sectors and in industry, focusing on post-quantum cryptography. He is Director of Cyber Security for Smart Data and Assets Limited in the City of London and a Fellow of the Institute of Mathematics and its Applications.

Invited Speakers

Facultad de Ciencias Exactas y Naturales y Agrimensura (FaCENA), Universidad Nacional del Nordest (UNNE), Argentina,
Facultad Regional Resistencia (FRRE), Universidad Tecnológica Nacional (UTN), Argentina

DETAILS
Talk
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.
Bio
Prof. Dr. Juan E. Nápoles Valdes graduated with a Bachelor of Education, specializing in Mathematics in 1983. He studied two specialties and completed a Doctorate in Mathematical Sciences in 1994, at the Universidad de Oriente (Santiago de Cuba). In 1997 he was elected President of the Cuban Mathematical Society and Computing until 1998 when he established a residence in the Argentine Republic. He has directed postgraduate courses in Cuba and Argentina and held management positions in several Cuban and Argentine universities. He has published different works in scientific journals specializing in the topics on qualitative theory of ordinary differential equations, mathematics education, problem solving, and the history and philosophy of mathematics. He has received several awards and distinctions for his teaching and research work.

Department of Mathematics and Didactics of Mathematics, Tallinn University, Estonia,
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Mexico

DETAILS
Talk
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.
Bio
Prof. Dr. Jorge E. Macías-Díaz obtained a PhD degree in mathematics from Tulane University, and a second PhD in physics from the Louisiana State university system. Since 2005, he has been a Professor at the Department of Mathematics and Physics of the Autonomous University of Aguascalientes, Mexico, and an associate professor at the Department of Mathematics and Didactics of Mathematics of Tallinn University, Estonia, since 2020. He is a member of the Mexican Academy of Sciences since 2012, and an associate editor of various journals, including Applied Numerical Mathematics, the International Journal of Computer Mathematics, Advances in Mathematical Physics, and Axioms. His current areas of research include fractional calculus, numerical analysis, nonlinear science and mathematical epidemiology.

Webinar Content

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Program

Click on the presenter's name to download their slides.

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

Relevant SIs

Abstract Fractional Integro-Differential Equations and Fixed Point Theory with Applications
Guest Editors: Prof. Dr. Wei-Shih Du, Prof. Dr. Marko Kostić, Prof. Dr. Vladimir E. Fedorov & Prof. Dr. Manuel Pinto
Deadline for manuscript submissions: 31 December 2022

Stability Analysis of Fractional Systems-II
Guest Editors: Prof. Dr. Andrey Zahariev & Prof. Dr. Hristo Kiskinov
Deadline for manuscript submissions: 31 December 2022

Fractional Differential Equations and Control Problems
Guest Editors: Prof. Dr. Ivan Matychyn
Deadline for manuscript submissions: 31 December 2022

Fractional-Order Systems: Control, Modeling and Applications 2022
Guest Editors: Prof. Dr. Yiming Chen, Prof. Dr. Aimin Yang, Dr. Jiaquan Xie & Dr. Yanqiao Wei
Deadline for manuscript submissions: 15 January 2023

Applications of Fractional Calculus in Option Pricing
Guest Editors: Dr. Jan Korbel & Dr. Jean-Philippe Aguilar
Deadline for manuscript submissions: 31 March 2023

Fractional Calculus in Kinetic Theory
Guest Editors: Prof. Dr. Richard L. Magin & Prof. Dr. Ervin K. Lenzi
Deadline for manuscript submissions: 30 April 2023

Fractional Calculus and Mathematical Applications
Guest Editors: Prof. Dr. Juan Eduardo Nápoles Valdes & Dr. Miguel Vivas-Cortez
Deadline for manuscript submissions: 31 May 2023

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