## Topic outline

### Teacher presentation

Hello,

my name is Lisl Weynans, I am assistant professor in Applied Mathematics at Bordeaux University, and I am in charge of the course "Integration and differential equations" in the mathematics minor of the international licence.

If you have any questions about the course or the exercices, you can send me an email. I will answer you by email too, or propose you an appointment.

To join me:

Lisl Weynans, lisl.weynans@math.u-bordeaux.fr

INRIA Bordeaux Sud-Ouest, 200 Avenue de la Vieille Tour, 33405 Talence

### Course presentation

The objective of this course, called

**"Integration and Differential Equations"**, is to learn the tools allowing you to solve differential equations. The course and the exercises will be illustrated with examples coming from**mechanics, biology...**In some cases differential equations can be solved exactly. In many others, it is only possible to integrate or solve them in an approximate way. Therefore, in this course we will study how to deduce some properties of ODE's solutions without solving them explicitely, and learn numerical techniques to solve them approximately.

I will give on a regular basis some personal work to prepare, and we will meet on a regular basis to talk about the course and answer your questions with more details. Feel free to ask questions or make (constructive!) remarks about the course and the exercises.

The previsionnal program is the following:

- 3th week (01/16 - 01/22) : Geometrical study of ordinary differential equations
- 4th week (01/23 - 01/29) : Resolution of first order differential equations
- 5th week (01/30 - 02/05) : Qualitative study of differential equations: Kepler's laws
- 6th week (02/06 - 02/12) : Qualitative study of differential equations: Epidemiology
- 7th week (02/13 - 02/19) : Break
- 8th week (02/20 - 02/26) : Maximal solutions
- 9th week (02/27 - 03/05) : Autonomous equations
- 10th week (03/06 - 03/12) : Class test/Homework
- 11th week (03/13 - 03/19): Numerical schemes for differential equations (1)
- 12th week (03/20 - 03/26): Numerical schemes for differential equations (2)
- 13th week (03/27 - 04/02): Practical application with Python/Scilab (1)
- 14th week (04/03 - 04/09): Practical application with Python/Scilab (2)
- 15th week (04/10 - 04/16) : Lotka-Volterra system

Evaluation will be performed through:

- Class test (mid-semester) or homework
- Project
- Terminal exam

### Week 3: Geometrical study of ODE's

This week we will study how to obtain informations about the solutions of ODE's with geometrical reasonings.

Here's a link to a course about the subject:

And below you will find some short notes about the geometrical study of ODE's.

The MIT course insists about the usefulness of drawing the isoclines of the function "f" before drawing the "line elements", which correspond in fact to the tangent field.

From a practical point of view, it is indeed more convenient to draw the isoclines first, and then use them to draw the tangent field, rather than drawing directly the tangent field.

Some exercises in a pdf file below will help you to practise...The corrected version will be available next week.

### Week 4: Resolution of first order differential equations

Watch the on-line course on differential equations:

Then solve the exercises on the exercise sheet hereafter. (A corrected version will be available next week)

If you have questions about an exercice or a specific point in the course

**do not hesitate**to ask me by email.### Week 5: Qualitative study of differential equations: Kepler's laws

This week we study differential equations coming from mechanics, leading to the famous Kepler's laws.

To get a flavour about Kepler's laws, before going to the mathematical analysis, you can watch the following clips:

NASA clip about Kepler's laws:

http://www.youtube.com/watch?v=wjOOrr2uPuU

Another small course from the Cosmic Classroom:

http://www.youtube.com/watch?v=QsopVzjd2sU

Hereafter is a course on Kepler's laws that you can consult

This week's home work consists in writing a one page summary about Kepler's laws and their resolution (exact? to what extent?) and sending it to me.### 6th Week: Qualitative study of differential equations: Epidemiology

This week we will learn a little about another possible application of differential equations: the epidemiology of infectious diseases.

First, what is epidemiology? Let's have a look at the video:

http://www.youtube.com/watch?v=jCK2mflwESM

Hereafter, you will find a basic course (from Fred Brauer) on the modelling of infectious diseases with differential equations: read it from page 20 to page 32.Some imaginative people can make funny epidemiological studies with almost the same mathematical tools:

http://mysite.science.uottawa.ca/rsmith43/Zombies.pdfhttp://mysite.science.uottawa.ca/rsmith43/BieberFever.pdf

This week's homework is to make a research on the internet and find another research article about epidemiology with differential equations. Send me by email the references of this research article, and a summary with mathematical formulas of the epidemiological model that is described inside.

### 8th week: Maximal solutions

This week we keep on studying geometrical properties of ODE's, through the notions of explosion and barriers.

The notion of explosion of the solution of an ODE will lead us to study more precisely on which intervals are defined the solutions of ODE's, and what is a maximal solution.

Below you will find a course and some exercises. One of them is about the subject of the last week: the modelling of an epidemy.

You can also consult the online course of the MIT:

You can find here illustrations of questions 1 and 2:

1/: the zones with V_xy positive, negative and zero

2/ the tangent fields associated, and the trajectories of 3 solutions (in green), starting from points inside the zone V_xy <0, and following the local tangent fields. You can see that the trajectories decrease, but are stopped by the bottom line V_xy = 0 that they cannot cross, since at this line the tangent is horizontal.

### 9th week: Autonomous equations

This week we will study with further details autonomous differential equations, and what we can learn about their stability properties.

The online-course presented last week is still topical. Watch it carefully, especially the examples that are presented by the teacher. Consult also the small guide to phase lines also available on the web site:

https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/e56d0d4835f94e96d61d5182e4efc8d9_MIT18_03SCF11_s10_2text.pdfSomes exercises about autonomous equations are proposed in the exercise sheet below.

### 11th week: Numerical schemes for the integration of differential equations (1)

This week we will begin the study how to solve numerically differential equations.

Hereafter is a very nice course in pdf file presenting briefly the most common numerical methods for ordinary differential equations (read only the first 10 pages).

You can also consult the following course:

http://www.youtube.com/watch?v=LbKKzMag5Rc&list=EC64BDFBDA2AF24F7E

and the associated exercices:

http://www.youtube.com/watch?v=X5-ucBtneVM&list=EC64BDFBDA2AF24F7E

This week there is one assignment: some exercises in a pdf file. Their corrected version will be available next week.

### 12th week: Numerical schemes for the integration of differential equations (2)

This week we will study on the example of chemical kinetics the convergence of numerical integration methods.

You will find hereafter the historical proof of convergence of the Euler method and an exercise. The correction version will be available next week.

The homework to prepare is also hereafter.

### 13th week: Practical application of previous courses with Python

This week, you will have to prepare some programs about the numerical resolution of ODE"s. The exercise sheet below contains an exercise about the numerical resolution of ODE's.

You can make the exercise with python, or another similar language, such as matlab for instance.

Here you will find some explanations about Newton's method to find zeros of functions:

https://math.dartmouth.edu/opencalc2/dcsbook/c3pdf/sec36.pdf

### 14th week: Practising numerical methods

This week, we will still work with Python (or Matlab, Scilab, C...).

In the following exercise sheet, you will find another exercise about the programming of numerical integration.

### Projects

For the project, you have to choose a document, such as a research article, containing a modelling of a physical/biological/chemical/epidemiological phenomenon using ODE's, and its numerical simulation. Then you have to study the modelling presented in the document and reproduce the numerical simulations with a program written in Python (or another language), with at least 3 different integration methods that you have to program yourselves.

Three articles are proposed hereafter, but you are free to choose another one. In particular, you can choose the article about zombies in the epidemiology course. In this case, send me the document that you have chosen, so that I validate that it meets the requirements of the course. You can prepare the projects alone, or in groups or 2 or 3.