Dr. Neil Balmforth

Applied PDEs

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


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: Tu-Th, 11-12, immediately following class

Recommended text:
R. Haberman, ``Applied PDEs''

The TA: Fanze Kong (fzkong@math)
Additional TA office hour: Wednesdays, LSK 303A, 1:30pm-3:00pm

Lecture notes: I, II, III, IV, V,
And by popular demand, I have written some more: VI, VII
(Beware! These are a work in progress and may contain typos
- please let me know if you spot any)

Synopses from a previous year: 1, 2, 3, 4, 5, 6a, 6b, 7a, 7b, 8, 9, 10a, 10b, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,

More lecture notes:
Laplace transforms,
Method of characteristics

Laplace transform question

Assi 1, with solution ( pde23a.m, pde23ax.m)
Assi 2, with solution ( pde23b.m, pde23bx.m)
Assi 3, with solution
Assi 4, due March 30

Sample 80min exams: 1, 2, 3
Older 50min midterm
Actual exam, with solution
Previous finals: 2018, 2019
Additional problems on traffic flow and more sample final exam problems

Video lectures for the final few weeks of the course, should I become mysteriously waylaid:
Laplace transforms I (beware of a slip in this lecture - the Laplace transform of f(t)=t exp(-t) is 1/(s+1)^2)
Laplace transforms II
Method of characteristics I
Method of characteristics II
Method of characteristics III
Method of characteristics IV
Method of characteristics V

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), 7.9.4(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

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