Program

This school will consist of 6 lecture-series and of at least 4 talks. More precisely, 2 lecture-series will be devoted to mathematical biology both for applications and mathematical theory; 1 lecture will be on hyperbolic conservation law with some example to traffic fluxes or fluid dynamics; 1 on optimal control; and the remaining two on numerical methods and implementation for solving differential equations. This last one will be complemented by practical experience in the use of free scientific computing software. Finally, mini-projects are planned. 

Lecture notes will be daily updated. The information of courses including lecturers are as the following.

SCHOOL ORGANIZATION

LIST OF LECTURES

Mathematical biology

Y. Mammeri: Introduction to structured differential equations to describe biological processes.

F. Costa and M. Grinfeld: Infinite-dimensional dynamical systems and coagulation- fragmentation problems. 

Scientific computing

B. Lucquin: Finite differences and finite elements for elliptic and parabolic equations.

M. Darbas: Scientific computing with python. 

Hyperbolic and stochastic equations

N. Bedjaoui and J. Correia: Conservation laws to describe the traffic flows.

J. Salazar: Introduction to optimal control, deterministic and stochastic cancels due to the snowstorm in Europe and replaced by Y. Mammeri: Numerical resolution of conservation laws.

Group mini-projects will be proposed to complement the course. They will include theoretical and practical aspects. These projects will serve as exercises support for the lectures.

ADVANCED TALKS

Jayrold P. ARCEDE, Caraga State University, Philippines

Two differential equation models on human exposure to meHg

Marilia PIRES, University of Évora, Portugal

Mathematical model of cardiovascular system.

Ludwig STREIT, Bielefeld University

Polymers and random walks: two Nobel prizes, two Fields medals, and the problem is still open.

Alexis Almocera, University of the Philippines, Diliman

A dynamical analysis of an aquatic food web model with parasitic fungi.

ABSTRACTS

Y. Mammeri (University of Picardie): Introduction to structured differential equations to describe biological processes.

Structured population models become an important tool for the understanding of biological processes. These models take into account the overall population dynamics but also individual demographic differences such individuals age, size...

The first part of the course is devoted to a review of the main unstructured and compartmental models described by ordinary differential equations (Malthus, Lotka-Volterra...).

In the second part, age structure is introduced. Starting with discrete (matrix model) we converge to continuous models (transport partial differential equations). We define the basic reproductive number R0, a significant population growth indicator.

The last part will deal with space structured models via reaction-diffusion equations and we will finish studying pattern formations and Turing instabilities. 

F. Costa (University Aberta) and M. Grinfeld (University of Strathclyde): Infinite-dimensional dynamical systems and coagulation-fragmentation equations.

In the first half of the course, we will start by reviewing relevant concepts from finite-dimensional dynamical systems: strategies for proving local and global existence of solutions, invariance, equilibria, stability, attractors, and Liapunov functions. Then we will consider infinite-dimensional dynamical systems, and will discuss the semigroup theory approach to proving existence of solutions, using Hille-Yosida and perturbation generation theorems. We will also discuss the idea of finite-dimensional truncations and the tools needed to prove existence of solutions of the full equations. Finally, we will see how Liapunov methods can be adapted to the infinite-dimensional setting. 

In the second half of the course we will apply the tools introduced in the first half to the study of equations of fragmentation, coagulation, and coagulation-fragmentation types. We start by introducing those equations as models coming from material sciences, colloidal sciences, chemistry, etc., and will proceed by addressing issues of existence and well posedness. The long time behavior of solutions to these equations is a difficult issue not yet completely understood. We will present the problems, the currently known results and its proofs in the easiest possible setting and will point out to current developments and several open problems. 

B. Lucquin (University of Paris 6): Finite differences and finite elements for elliptic and parabolic equations.

We present different numerical approximations for partial differential equations. The course is divided into two parts. 

In the first part, we present the finite element method, which is well adapted to the numerical approximation of elliptic boundary value problems, once these problems have been transformed into their variational form. We first recall the continuous problem and its variational formulation, then we introduce the discrete problem in a general frame. We analyze more precisely the method in the one-dimensional case (explicit construction of the discrete variational space, computation of the approximated solution and convergence analysis). We also describe the method in the two-dimensional case, using either rectangles or triangles. Last, we generalize to systems: the Lamé system and the Stokes system. 

The second part is devoted to the numerical approximation of some evolution problems of parabolic or hyperbolic type. We first study the case of the heat equation and describe a finite difference approximation method but also cover a mixed finite differences-finite elements approximation. 

M. Darbas (University of Picardie): Scientific computing with scilab or python.

Scilab and Python are open computing environments designed for engineering and scientific applications. We will give a thorough description of the software use, including how to master its environment as programming language and provide a tour of its numerous applications toolboxes to solve linear and nonlinear systems, differential equations... 

J. Salazar (University of Évora): Introduction to optimal control, deterministic and stochastic.

We will introduce the concepts of controllability and observability. A review of functional analysis will be presented to show the bang-bang principle. Some applications of the Pontryagin maximum principle are shown and we conclude with a brief introduction to stochastic optimal control. 

Cancel due to the snowstorm and replaced by Y. Mammeri (University of Picardie): numerical resolution of conservation laws.

N. Bedjaoui (University of Picardie) and J. Correia (University of Évora): Conservation laws to describe the traffic flows.

We begin the course by describing the transport phenomena with example of applications in the real life (fluid dynamics, trafic flow,...). We introduce the general model of scalar hyperbolic equations, and present the characteristic method used to solve this kind of equations in the regular case. Next, the notion of weak solutions is introduced to describe the discontinuous solutions, and we focus particularly on the shock wave phenomena. These solutions (weak solutions) are generally not unique and we need the entropy criteria to select the physical interesting solutions.

In the last part, the case of systems will be studied in some simple cases, and we begin with the wave equation and general linear systems.