Prof. J. A. Tenreiro Machado
Institute of Engineering, Polytechnic of Porto, Portugal
Title: The Logical Song
Abstract: Data analytics is penetrating in all areas of the human activity. The availability of data measuring the behavior of complex phenomena allows a new quantitative perspective not conceivable with classical mathematical tools. This presentation addresses 3 distinct areas of key impact in society and that reflect the behavior of men kind. We consider music, artistic painting and soccer as manifestations of the human spirit that can be processed and analyzed, since large volumes of data are presently available in digital form. The data records are studied using several mathematical and computational tools, such as, fractional calculus, entropy, multidimensional scaling, hierarchical clustering and scientific visualization. The embedding of mathematical, computational and algorithmic modeling leads to the emergence of patterns that are analyzed and interpreted.
Prof. Sverre Holm
University of Oslo, Norway
Title: Fractional Wave Equations and Complex Acoustic Media
Abstract: Wave equations with non-integer derivative operators can describe attenuation which increases with frequency with other powers than two, unlike ordinary wave equations. Such attenuation is found in many complex media. Both shear and compressional waves in media as diverse as biological tissue, rocks, and sub bottom sediments are examples of this. These wave modes are central in applications such as medical ultrasound, diagnostic shear wave imaging in elastography, seismics, and underwater acoustics. These equations can be divided into two classes depending on whether they can be derived from more fundamental principles or not. In the first class one can find the fractional Kelvin-Voigt and fractional Zener wave equations, while several fractional Laplacian wave equations are in the second category. Such examples as well as the properties of their solutions will be presented. In many cases just having such a wave equation is enough to model a phenomenon.
In [Holm, S. (2019). Waves with power-law attenuation. Springer and ASA (Acoustical Society of America) Press] I also wanted to understand what it is about complex media that gives rise to power law behavior. The main attenuation mechanisms of standard acoustics are heat conduction and relaxation, structural relaxation, and chemical relaxation. They have fractional parallels and the first one is heat relaxation described by fractional Newton cooling due to anomalous diffusion. The most important mechanism is however the second one, the fractional parallel to structural relaxation. Instead of one there are multiple relaxation processes with a distribution of relaxation times that follows a power-law distribution, possibly indicating that the material has fractal properties. This distribution also has a relationship to the Arrhenius equation, indicating a link to chemical relaxation, albeit a quite speculative one.
Other sources of power-law behavior can be non-Newtonian rheology with time-varying viscosity and propagation when there is a fractal distribution of scatterers in an otherwise lossless medium. Existing models in sediment acoustics such as the grain shearing model and the Biot poroelastic model can also be reformulated with fractional operators. These approaches are presented in the hope of coming one step closer to answering if fractional wave equations give clues to some deeper reality, or if they are just a compact phenomenological description.
Prof. Dumitru Baleanu
Cankaya University, Turkey
Title: On Singular and Non-Singular Fractional Operators and Their Applications in Mathematical Biology
Abstract: The fractional calculus and its applications is a hot topic for researchers from many branches of science and engineering. Real world applications started to be investigated with a great sucees within this very helpful mathematical tool. In my talk I will concentrate on the sucessful applications of both singular and non-singular fractional operators to the complex dynamics of some mathematical biology systems. Besides, some new aspects of the classifcation of fractional operators will be presented. Illustratives examples will be provided.
INVITED SPEAKERSProf. Jordan Hristov
University of Chemical Technology and Metallurgy, Bulgaria
Title: Fractional Operators with Non-Singular Memories in Viscoelasticity: Basic Concepts Applicable to Linear and Non-Linear Viscoelasticity
Abstract: The fractional operators with non-singular memory kernel described by exponential (Caputo-Fabrizio derivative) and generalized Mittag-Leffler function (Atangana-Baleanu derivative) raise many questions about their properties and mainly about their physical relevance and applications.
This lecture focuses on basic principles in description relaxation behaviors of linear and non-linear viscoelastic materials and the adequate selection of the memory kernels of the fractional operators leading to both derivatives with singular or non-singular memories.
The targets are non-aging (linear and non-linear) viscoelastic materials with behaviors beyond the power-law limit related to the Caputo fractional derivative. The stress-strain response functions are the main physical objects allowing selecting the corresponding memories of the fractional operator sand their constructions. As a consequence of the memory kernel selection, the causality of both the constitutive equations and the frequently used rheological equations are discussed.
Prof. Carla Pinto
School of Engineering, Polytechnic of Porto, Portugal
Title: Tackling specificities of different diseases using within-host models
Abstract: Epidemics make exciting news. They are often presented with dramatic headlines, and the pictures accompanying them are of healthcare workers dressed with protective equipment or working at labs. People often forget about the behind scenes work of mathematicians, who, with more or less simplified models, help on the understanding and prediction of infections spread. In this lecture I will focus on several within-host models useful for a deeper knowledge of virus dynamics with different specificities, namely HIV, HCV, HSV-2, etc.
Prof. Huseyin Merdan
TOBB University of Economy and Technology, Turkey
Title: Nonlinear dynamics of a ratio-dependent prey-predator model: Stability, bifurcations and chaos
Abstract: Nonlinear dynamical behaviors of a prey-predator system with Leslie type will be presented. First, the dynamics of its continuous form will be analyzed; the local and global stabilities and bifurcations will be discussed. Second, the dynamical behavior of its discrete form will be analyzed; bifurcations and chaotic behavior will be shown. Numerical simulations will be given to support and extend the theoretical results. Finally, we will compare the results that we obtained.