BE5B01DEN - Differential equations & numerical methods

DEN: coronaversion

Exams: To prepare for the exams, make good use of materials offered below on these pages. I definitely recommend the document Information about exam and the description of a typical exam test. You will find that we covered almost everything in our worksheets, and the remaining advanced questions cannot be memorize by heart anyway, one simply has to understand notions and how things work. So refresh the topics, for isntance using the Solved problems and Practice problems below, and then you can try the trial test. It is up to you whether you approach it as a dress rehearsal and try it just with what you have in your head, or use it as a guide for study. You will find the solution on the second page as well as the grading guide, so you can grade yourself to see how well you did.

The preferred way to take an exam (the test) is on-site in Prague. I will contact you with some dates, or you can also contact me. A special arrangement for those abroad will be made.
The optional oral exam will be taken on MS Teams for all, including those who write the written test in Prague.
To be able to participate in on-line parts of the exam, you will need a reasonably reliable Internet connection, and MS Teams with a functional camera.

Here is a direct link to the Youtube playlist for this course. Videos will be added as they are completed.

Week 13 (May 18–22):
The end is near. This week there are the last two videos to watch. We will finish off the matrices in video 12, where we learn how eigenvalues and eigenvectors are found in applications. We do this to widen our horizons, it will not be on the test, and the first 38 minutes are the most important. The last video to watch is number 11, where differential equations and numerical methods meet in applications. I especially recommend the first topic (you will understand when you see it).
This concludes the lectures for the semester, and it is time to get ready for exams. We already covered most types of problems, just two minor types are left and we will look at them this week.
• Here is the worksheet for this week.
Solution.

Week 12 (May 11 – 15):
We are almost there. Watch video 8c on iterative methods. You should understand the Jacobi and Gauss-Seidel methods well enough to be able to apply them by hand, explain the idea behind them and point out the difference between them. You should also know something about convergence of these methods.
• Here is the worksheet for this week.
Solution.
• Here is the Maple worksheet for this week.

Week 11 (May 4 – 8):
This week we will mostly explore rather than study for exams. In the worksheet we will return to two topics that will not be on the exam, but you should be aware of them. First, a homogeneous system with eigenvalues of higher multiplicity motivates us to recall the special procedure we saw in the lecture last week. Second, a non-homogeneous system will show us that the methods of variation and undetermined coefficients work also in this context. Both methods can be found in video 9b (if you want, you can jump directly to row variation at 20:39). We saw both methods before, so it is just a question of adopting them to systems. This topic is also covered in chapter 27 of the special edition of the lecture notes for the course. You are not expected to totally master this topic, just understand it enough to solve the worksheet by following the outlined procedure.
• Here is the worksheet for this week.
Solution.
We will continue taking it easy and conclude this week by watching some captivating movies about solving systems of equations through elimination (use these slides). In video 8a you should see a somewhat different point of view compared to linear algebra. Which brings us to what you should actually learn actively this week: You should be able to solve systems using the Gauss elimination and backward substitution, and be aware of practical aspects (n3 complexity, problems with numerical errors). As a bonus you can learn about the LUP decomposition, but you can skip it if you want and watch just the first hour of the video.
Finally, there is the video 8b on topics that you are not expected to master actively, but when you hear the name "matrix norm" the next week, it should ring some bells. You can see it as a sightseeing trip to a more advanced mathematics. You can leave this video for the following week, then you will have 2+2 videos instead of 3+1.
You can use these extended lecture notes on numerical methods for matrices.

Week 10 (April 27 – May 1):
We start the last chapter about differential equations, this time we address systems of linear ODEs. Look at the video 9a (slides are here) and you should easily manage the worksheet. This should leave some time for completing your semestral project. You can use these extended lecture notes, this week we cover chapter 26.
• Here is the worksheet for this week.
Solution.

Week 9 (April 20 – 24):
This time we will look at solving euqations through fixed points. Look at video 7c, the slides are the same as in the previous week. The questions in the worksheet suggest what one should learn this week.
• Here is the worksheet for this week.
Solution.
• Here is the Maple worksheet for this week.
Semestral project: This year's semestral project looks at root finding. All the necessary information and tools can be found in this Maple worksheet. If you get stuck somewhere, I will be happy to help. The deadline for completing this project is May 8 (the Victory day).
However, I recommend that you start already this week after completing the homework and try to hand the project report in the next week, so that we have time to discuss it and perhaps make some corrections or additions.

Week 8 (April 13 – 17)
This and the next week we will focus on solving equations numerically. For this week look at video 7a from the playlist. You can use these extended lecture notes, this week we cover chapter 18. You are expected to understand the method of bisection and the Newton method well enough so that you are both able to use them in actual problems and to desrcibe them and deduce necessary formulas. You should understand stopping conditions and the notion of the order of method, so that you can work with it as we did for the order of ODE-solving methods. Regarding the secant method, you should have some idea about its principle, but memorizing the formulas will not be required.
• Here is the worksheet for this week.
Solution.
• Here is the Maple worksheet. While Maple worksheets are not obligatory, this week I strongly recommend that you go through it, because it will help you with completing the semestral project.

Week 7 (April 6 – 10)
Now for the work. First, start with the worksheet and solve the linear ODE problems. Hopefully, after the feedback from the previous week this should be easier.
Next, look at the video 2c about the method of variation for solving first-order linear ODEs. Then complete the last problems in the worksheet.
• Here is the worksheet for this week.
Solution.

Week 6 (March 30 – April 3)
We will cover the linear differential equations chapter. This week we focus on basics, next week we will practice some more. I prepared these extended lecture notes on this topic.
Read chapter 15 on homogeneous linear equations. It is not difficult, you should have no trouble working out the homogeneous problems in the following worksheet.
After you work out the problems with homogeneous equations, read chapter 16 or watch video 5b from our playlist on Youtube and complete the worksheet. Video 5c may help you with working out the table.
• Here is the worksheet for this week.
Solution.

Week 5 (March 23–27)
In the last lecture that we had we covered the basic methods for solving initial value problems for 1st order ODEs numerically (Euler method, Heun, midpoint). In particular the former is important, as there can be questions related to it on exam, one should also understand the notion of order of error and how one can use it.
Review the Euler method and the two second-order methods, recall the notion of global error and order, for instance using these extended lecture notes, and read about the main idea of the RK 4 method.
• Solve the following worksheet and send it in.
Solution.
• Go through the following Maple worksheet.
Note: There are some videos on Runge-Kutta methods on Youtube. I did not find any that I would find particularly endearing, but videos 7.1.1. through 7.1.6. and 7.1.8 on this playlist seemed quite good, I think that also this one and this one explain the geometric idea quite well.

 

Goodies:
syllabus - the most important information about the course.
Schedule of classes - weekly outlines of lectures and a special printable version of slides. We strongly recommend that you print them and take them with you to lectures.
• Information on midterms and project. Sample midterm.
Information on the final.
Sample final test.
• And if you survive three years here, you will want to look at outline of knowledge that is expected from you when you go for the final state examination.

Important information for the start: The classes take place at room 459, where every student has a terminal connected to the kepler server. A basic manual is here. You can do all the work for the course without having an account, just log-in as a guest.

Lecture notes for the course: Ordinary Differential Equations and Numerical Mathematics.

Solved problems: Here you will find problems of key types with detailed solutions.
 • 1. analysis of solutions.
 • 2. separable equations.
 • 3 & 4. linear equations.
 • 5. variation.
 • 6. & 7. systems.
 • numerical mathematics.

Practice problems: Most of them are of the right difficulty for exams.
 • 1. analysis of solutions.
 • 2. separable equations.
 • 3. homogeneous linear equations.
 • 4. linear equations.
 • 5. variation.
 • 6. homogeneous systems.
 • 7. systems.
 • numerical mathematics.

Maple:
If you want to use the library NumericalMethods on your computer, download the files
NumericalMethods.mla (library of procedures)
NumericalMethods.hdb (Help library) or NumericalMethods.help (Help library for version 18 or later)
and put them into the library folder of your Maple, the traditional place is .../Maple/lib/. Additional info can be found on the page Resources on Maple