The course Differential Equations deals with analytical and numerical methods to study the properties of solutions to ordinary differential equations (ODE's) and partial differential equations (PDE's), when these solutions are not known explicitly. 

More information on the aim and learning outcomes of the course can be found in the course plan (Links to an external site.).

Course Responsible

Simone Calogero

Student representatives

The following students have been appointed: 

Erik Redmo Axelsson  
Lisa Vind                          

Literature

[1] Simone Calogero: Lecture notes on ODE's (pdf)   

[2] Stig Larsson, Anders Logg, Axel Målqvist: Matematisk analys och linjär algebra (del II)

Remark: Chapter 3 of [2] has already been explained in the course MVE620 (en variabel analys) and its content is therefore assumed to be known. Be sure to read again this chapter before the course starts. IMP: Reference [2] here refers to the first version of the book (you should have a PDF of it). If you bought the book, then the number of the chapters is different. In particular, Chapter 5 become Chapter 4 in the final version of the book.

[3] Simone Calogero: Lecture notes on PDE's (pdf)

Little help with the numerical assignment on the 2-body problem: pdf

Online resources (optional):

Several well written notes on ODE's and PDE's can be found here

Schedule

The lectures take place in several different rooms, don't get lost!

The course schedule can also be found on TimeEdit.

Remark: The exercise session is on Wednesdays afternoon, however many exercises are also discussed in other lectures. The solutions of some of these exercises will be reviewed in the class, and then later the solutions will be posted on the program below (if they are not already in the lecture notes).

PART	1:	Ordinary differential equations
When	Where	Content
31/10 10-11.45	HA2	Strong solutions of ODE's. The projectile problem. [1] Sections 1, 2.
2/11 10-11.45	KC	Initial value problem for first order systems of ODE's. [1] Sections 3.1, 3.2, 3.3. [2] Sections 5.2, 5.3, 5.4.
2/11 13.15-15	KC	[1] Exercises 1, 2, 4, 5, 6, 7, 8, 9, 10. (Solutions) [2] Exercises 5.13, 5.14, 5.15, 5.16.
7/11 10-11.45	HA2	Weak solutions of ODE's. [1] Sections 3.4. Exercise 11. (Solution)
9/11 10-11.45	KB	Aunomous dynamical systems: Fixed points, linearization, stability.  [1] Sections 4.1, 4.2, 4.3, 4.4.
9/11 13.15-15	KB	[1] Exercises 13, 14, 15, 16. (Solutions)
11/11 10-11.45	KC	Phase portrait of dynamical systems. Hamiltonian systems. [1] Section 4.5. Exercises 17, 18, 19. (Solutions)
14/11 10-11.45	HA2	2-dimensional linear dynamical systems: Jordan form, solutions. [1] Section 4.6.
16/11 10-11.45	HC2	Phase portrait of 2-dimensional linear dynamical systems. Node, focus, saddle, center. [1] Section 4.6.
16/11 13.15-15	KB	[1] Exercises 20, 22, 23, 24, 25, 26, 27, 28, 29, 30. (Solutions)
18/11 10-11.45	KA	Stability analysis of non-linear dynamical systems.  [1] Sections 4.7 Exercises 31, 32, 33. (Solutions) REM: we skip Section 4.8 (Ljapunov functions).
21/11 10-11.45	EC	Numerical solutions of ODE's. Finite difference method. Convergence and stability. [1] Section 5; [2] Chapter 6.
23/11 10-11.45	KC	ODE models: Lotka-Volterra system. The 2-body problem. [1] Section 6
23/11 13.15-15	KC	Implementation of the finite difference method.  Jupiter notebook with all Python codes in ref. [1].
25/11 10-11.45	KA	Boundary value problems for ODE's. Introduction to the finite element method. [1] Section 7 (pages 58-59). Exercises 48, 49, 50, 50, 51, 52.
PART	2:	Partial differential equations
28/11 10-11.45	HA2	Introduction to PDE's. The transport equation. Conservation laws.  [3] Sections 1, 2.1, 2.2.
30/11 10-11.45	KB	Burger equation. Weak solutions of conservation laws. Shock waves.  [3] Sections 2.2, 2.3.
30/11 13.15-15	HC2	[3] Exercises 1, 2, 3, 4, 5, 6, 7. (Solutions)
2/12 10-11.45	HC2	Applications of transport equations and conservation laws. The Euler equation.  [3] Section 2.3. Exercises 8, 9.
5/12 10-11.45	HA2	Wave equation. Initial value problem in one dimension on the whole line and on the half line. Spherical waves in three dimensions.  [3] Section 3.1, 3.2.
7/12 10-11.45	HC2	Applications of the wave equation: Sound and electromagnetic waves. [3] Section 3.3. Exercises 25, 26, 27, 28.
7/12 13.15-15	KB	[3] Exercises 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23. (Solutions)
9/12 10-11.45	HC2	Heat equation. General solution in the whole space. Applications to Brownian motion and heat flow. [3] Section 4.1, 4.4. Exercises 29, 30.
12/12 10-11.45	HA2	Initial boundary value problem for the heat equation. Separation of variables method. Fourier series. [3] Section 4.2. Exercise 31, 32, 33.
14/12 10-11.45	KB	Boundary value problem for the Poisson equation. Finite element method in 2 dimensions. [3] Section 5.1, 5.2, 5.3 (pages 37-38). Exercises 38, 39, 40, 41, 42.
14/12 13.15-15	KB	Review + Exercises (ODE's).
16/12 10-11.45	HC2	Review + Exercises (PDE's).

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Assignments

The assignments consist of the numerical exercises 45 (Lotka-Volterra system) and 47 (2-body problem) in reference [1]. Each assignment gives max 3 points.

Remarks:

• The assignments are not compulsary, but strongly recommended.
• The assignments have to be submitted through Canvas at this link: Submit.  
• The assignments have to be submitted in a single Jupiter notebook. 
• The assignments can be worked on groups of max 3 students. However each member of the group has to submit the assignments. Write the name of the students you worked with in the comment box of the submission page . 
• The deadline for submission is December 31, 2022 at 23.59. The grade on the project (max 3+3 points) will be communicated in the first week of January. The grade points of the assignments are valid only for the exams in 2023. 
• One of the three grade points of each assignment is based on the "aesthetical'' quality of the report, e.g., on having nice and informative plots. 

Examination

The final written exam gives max 30 points. The following rule will be used to transform the exam + assignments points to the finale grade:

Exam + assignments points	Final grade
0 - 17	Not passed
18 - 22	3
23 - 27	4
28 - 36	5

The exam consists of two parts. In the first part you will be asked to give and explain one of the definitions in the following list from ref. [1] (max 3 points):

Definition 3.1, Definition 4.1, Definition 4.2, Definition 4.3, Definition 4.4, Definition 4.5, Definition 5.1, Definition 7.1

and one in the following list from ref. [3] (max 3 points)

Definition 2.1, Definition 2.2, Definition 2.3, Definition 3.1+3.2, Definition 4.1, Definition 5.2

The explanation of each definition is expected to be about 1 page. You should briefly explain the motivation behind the concept being defined and why it is useful (e.g., how did we use it in the course?). Use pictures if you like. 

The second part of the exam consists of 6 exercises, each giving max 4 points. 1 of the exercises will be  theoretical (proving or disproving some simple property of ODE's or PDE's), while the other 5 will be computational (e.g., study stability of fixed points, computing solutions of PDE's, etc.) 

Remarks: 

• The exam can be written in English or Swedish.  According to Chalmers rules, you can bring an English-Swedish dictionary at the exam. Digital dictionaries are not allowed.
• No aids are permitted for the exam (not even a calculator)
• The text of the exam will contain a table of standard integrals and some other formulas, which you may or may not need to solve some of the exercises. Other non-standard formulas, e.g., trigonometric or vector identities, will be provided if you need them. 
• As a general rule, if a formula appears in a definition or in the statement of a theorem (not in the proof!) in the lecture notes, then you have to remember it for the exam.

Solutions to this year exams

14 January 2023: pdf

3 April 2023: pdf

24 August 2023: pdf

Old Exams

As this course is given for the first time, there are no old exams.

Here are some sample exams to practice:  

Sample Exam 1 (pdf) Solution (pdf)

Sample Exam 2 (pdf) Solution (pdf)

Sample Exam 3 (pdf) Solution (pdf)

Sample Exam 4 with answers (pdf) 

 

Examination procedures

In Chalmers Student Portal (Links to an external site.) you can read about when exams are given and what rules apply to exams at Chalmers. In addition to that, there is a schedule (Links to an external site.) when exams are given for courses at the University of Gothenburg.

Before the exam, it is important that you sign up for the examination. You sign up through Ladok (Links to an external site.). 

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok.

At the annual (regular) examination: 
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office (Links to an external site.). Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination: 
Exams are reviewed and retrieved at the Mathematical Sciences Student office (Links to an external site.). Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

Course evaluation

At the beginning of the course, at least two student representatives should have been appointed to carry out the course evaluation together with the teachers. The evaluation takes place through conversations between teachers and student representatives during the course and at a meeting after the end of the course when the survey result is discussed and a report is written. 

Guidelines for Course evaluation (Links to an external site.) in Chalmers student portal.

 

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