Fatima M. Aboud, Department of mathematics, College of Sciences, University of Diyala, Iraq

Mathematical tools for partial differential equations analysis

Partial differential equations and their numerical simulation are essential tools in both industry and research. The objective of this course is to provide some essential tools for the analysis of partial differential equations (PDEs). The content brings together notions and results from the functional analysis, and the study of some PDEs using these tools.

A. K Alomari, Yarmouk University, Jordan

Recent methods for solving biological models using approximate analytic solutions

Approximate analytic solution for differential and integral equations become attractive topic in the last few years. In this lecture, resent methods for solving a biological model in standard and fractional case will be presented. Various methods to obtain analytic solution of the model will be handled. We outline the basic concept of these methods, and we provide the essential notions of fractional calculus that will be used. The study of special cases of the model with a specific initial conditions will be given. Finally, several properties of the given solutions will be discussed.

Sharifa Alsharif, Yarmouk University, Jordan

Conformable Fractional Derivative and Some Application

Studying fractional order differential equations have recently received a great attention. It plays a fundamental role in modelling real life problems and applications in many branches of science such as biology, populations dynamics, physics, engineering, and so on. There are many definitions of fractional derivatives, such as Caputo type, Hadamard type, Riemann-Liouville type, Caputo-Frabrizio type. Most of these definitions use integral forms. This lecture will focus on conformable fractional derivative, recently introduced by Roshdi Khalil et al. in 2014, that uses limit approach. In particular, we will focus on the presentation and use of the conformable fractional derivative to solve some fractional differential equations in modelling some biological systems.

Abdelhalim Larhlimi, Nantes-Atlantique Computer Science Laboratory,  Nantes University, France

Mathematical methods in metabolic engineering for strain design

Metabolic reactions play a fundamental role in sustaining cell growth. They import nutrients from the environment, and they convert them into molecules needed by the living organism. Metabolic reactions do not operate in isolation; they form large-scale metabolic networks. In this lecture, we will introduce the main mathematical methods that are mandatory for predicting the behaviour of metabolic networks using constraint-based modelling. We will then present some methods that are used in metabolic engineering to design new strains.

Abdeljalil Nachaoui, Jean Leray Mathematics Laboratory, Nantes University, France

Reconstruction of the potential on the brain surface from the potential data measured on the scalp

The framework of this course is in the topic of the inverse problems of sources in Electroencephalography (EEG). This is to determine the location of epileptogenic foci which are at the origin of measured EEG signals by means of electrodes placed on the scalp.

Several approaches have been applied to solve the inverse problem in EEG. As examples, we present the least squares method which minimizes the error between the measured data and those calculated by solving a direct problem. We also present another method which is to trace the potential data collected on the scalp to the cortex by iteratively solving several Cauchy problems. The implementation of the sequence the discrete problems is done using FreeFem.

Reconstruction of the potential on the brain surface from the potential data measured on the scalp

• Inverse problems in Electroencephalography

• Successive approach and Nachaoui's convergence acceleration algorithm

• Finite-element approximation

• Implementation, FreeFem++

Mathematical tools for partial differential equations analysis

• Basic function spaces

• Classification of Partial differential Equations

• Green formula and its applications

Mathematical methods in metabolic engineering for strain design

• Introduction to constraint-based modeling of metabolic networks

• Polyhedral theory and duality theory and their applications in metabolic engineering (minimal cut sets, objective prediction and strain design)

Fractional derivative in solving biological models 

• Some methods for solving ordinary and partial differential equations

• Introduction to fractional derivative theory

  • Definitions, properties and results for fractional derivatives
  • Physical interpretations of the fractional derivatives

• Applications in Biology

• Challenges in biological models