Dr. Neil Balmforth

Applied PDEs

This course provides an introduction to practical analytical solution methods for PDEs.

The final exam will take place on April 23rd. It is an (open-book) take-home affair. The exam will be posted on this page, and must be submitted electronically via CANVAS (email is acceptable if there are uploading issues). It will be posted shortly before 12:45pm, due at 3pm.

The Final

I will be checking my email if there are any queries about the exam

For those who have accommodations at the Centre for Accessiblity, you will be granted extra time; your exams will be due at 4.08pm.
I have received email expressing concerns over the possibility that students will coordinate efforts in completing this examination. Please note that the exam is intended to be an individual effort; any such collaboration is unfair on the students that do not indulge in such clandestine activity.

13th April: some typos have been corrected in the solutions to Ass 5 and in the lecture notes on characteristics

In view of the measures taken given the virus, lectures are now suspended.

Lecture notes for the week of March 16-20
Lecture notes for the remainder of the course

Tuesday (March 17) afternoon, I will be preparing video lectures and will post these on YouTube for your enjoyment. The links will appear here soon. In the meantime, please look over the lecture notes above.

Videos for the week of March 16-20:

Laplace transforms I
BONUS: spot the deliberate error in the first half of this lecture ! (Introduced to enhance your learning experience.)
Answer at the bottom of the webpage.
Laplace transforms II

Videos for the week of March 23-27:
Method of characteristics I
Method of characteristics II

Videos for the week of March 30 - April 3:
Method of characteristics III
Method of characteristics IV

The finale:
Method of characteristics V (Am afraid I ran out of steam a little in this one...)

The syllabus:
I. PDEs and canonical examples
II. Separation of variables and Fourier series
III. Eigenfunction expansions
IV. Transform methods
V. Characteristics methods
Assessment will involve coursework (homework problems) and examination.

Office hours: Tuesday 11am, Thursday at noon.
Recommended text:
R. Haberman, ``Applied PDEs''

The TA: Wesley Ridgway (wridgway-at-math)

Summary notes
More summary notes

Ass 1 with solution, pde20a.m pde20ax.m (output plot)
Ass 2 with solution, pde20bx.m
Ass 3 with solution,
Ass 4, due March 26,
Ass 5, not to be graded,

Midterm date, after much agonization: 27th February

Sample midterm

Midterm from a previous year (with solution)
Last year's midterm (with solution)
The midterm (with solution)

Previous finals: 2018, 2019
Additional problems on traffic flow and more sample final exam problems

Additional relevant problems from Haberman (4th edition):
* Separation of variables and Fourier series - 2.5.3, 2.5.9, 3.4.12, 4.4.3(b)
* Halfway house (requiring Sturm-Louiville theory, but trig functions) - Worked example of section 5.7 upto eq (5.7.11), Physical examples of section 5.8
* Separation of variables and Bessel functions - 7.7.1 (assume r is less than a), 7.7.3 (the frequencies of vibration are the possible values of w in the cos(wt) and sin(wt) functions of the separation-of-variables general solution), 7.8.2(d), 7.9.1(b), 9.7.4(a)
* Separation of variables and Legendre functions - final example in section 7.10, problem 7.10.2

More relevant problems from Haberman (4th edition):
* Fourier Transforms - example in Sec 10.4.1; problems 10.4.3, 10.4.6; example at the end of Sec 10.6.3; problems 10.6.1(a), 10.6.18
* Laplace transforms - problems 13.4.3, 13.4.4, 13.5.3
* Characteristics - example starting with eq (12.2.13); problems 12.2.5(b) and (d); Sec 12.6.5; problems 12.6.3, 12.6.8, 12.6.9

Answer: The Laplace transform of f(t)=t exp(-t) is 1/(s+1)^2

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