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## M 332: Partial Differential Equations

Spring 2006, 3:10-4:00p MWF, ENGRG E205
Textbook: Applied Partial Differential Equations, Zachmann and DuChateau
Instructor: A. Malqvist
Office: Weber 204
Office Hours: MWF 4-5
Telephone: 491-4295
Email: axel@math.colostate.edu

Course Content:
Partial differential equations: elliptic, parabolic, hyperbolic; Solution techniques: Fourier series, transforms;

Assessment:
There will be one take home mid term exam and a take home final. Homework will be collected weekly.

The grades in class will be based on the take home exam (100 points), the take home final (200 points) and the homework (200 points). The grade scale will generally be 450-500, A; 400-449, B; 350-399, C; 300-349, D; below 300, F. The grades in each of the ranges will include +'s and -'s.

### Topics to be covered:

A. Introduction to partial differential equations (Roughly chapter 1)

A.1. Different kinds of PDE:s (linear, non-linear, heat equation, poisson equation, wave equation)
A.2. Initial and boundary conditions
A.3. Well-posed problems

B. Eigenfunction expansion (Parts of chapter 2)

B.1. Simple separation of variable example to motivate Fourier series
B.2. Fourier series
B.3. Convergence

C. Boundary-value and Initial-boundary-value problems on bounded domains (Chapter 3 and possibly parts of chapter 2)

C.1. Separation of Variables
C.2. Poisson equation
C.3. Heat equation
C.4. Wave equation
C.5. Sturm-Liouville problems
C.6*. Bessel functions

D. Integral transforms (Chapter 4)

D.1. Fourier transform
D.2. Laplace transform

E. Boundary-value and Initial-boundary-value problems on unbounded domains (Chapter 5)

E.1. Solution by integral tranform
E.2. Duhamel's principle

F*. Uniqueness and continuous dependence on data (Chapter 6)

F.1*. Green's identities and energy inequalities
F.2*. Maximum principles

* "if we have time".

### Homework:

Week 2: 1.3.7-9, 1.6.1(b-e) (4+4+4+4*2=20 points), hand in Monday 30 January in class. Solutions (pdf).
Week 3: 2.1.4(c-e), 2.1.6(b-d), Let x be defined on [-1,1] and let c be a number in [-1,1], further let f(x)=-1, when x is less then c, and f(x)=1, when x is greater then c. For which values c will all F-coefficients a_n=0 and for which values c will all b_n=0? (3*3+3*3+2=20 points), hand in Monday 6 February in class. Solutions (pdf).
Week 4: 2.1.1bd, 2.1.2bd, 2.1.3bd (2*4+2*3+2*3=20 points), hand in Monday 13 February in class. Solutions (pdf).
Week 5: 3.1.2, 3.1.3 (both on page 110 in the book) (10+10=20 points), hand in Monday 20 February in class. Solutions (pdf).
Week 6: 3.1.6, 3.1.7 (smallest eigenvalue meening the one with smallest absolute value) (both on page 111 in the book) (10+10=20 points), hand in Monday 27 February in class. Solutions (pdf).
Week 7: 3.1.1, 3.1.2, (both on page 123 in the book) (15+5=20 points), hand in FRIDAY 3 March in class. Solutions (pdf).
Week 9: 3.2.1 (page 147), 4.2.1abc (you just need to compute the F-transform in one way and you don't have to compute the L2-norm here), 4.2.3 (the once corresponding to 4.2.1abc) (page 190) (8+3*2+3*2=20 points), hand in FRIDAY 31 March in class. Solutions (pdf).
Week 10: 4.3.1(a-d), 4.3.3, 4.3.4, (on pages 202-203) (2*4+6+6=20 points), hand in FRIDAY 7 April in class. Solutions (pdf).
Week 12: 5.1.2 (page 220), 5.1.2 (page 226), 5.1.6 (page 226), 5.2.2 (page 233) (7+7+3+3 points), hand in FRIDAY 21 April in class. Solutions (pdf).
Week 14: 6.2.1 (not alpha*beta<0) (page 275), 6.2.3 (not k>0) (page 275), hand in FRIDAY 5 May in class. Solutions (pdf).

### Mid term take home exam:

Week 7: Mid term exam (25+25+25+25=100 points), hand in FRIDAY 10 March in class. Solutions (pdf).

### Final take home exam:

Week 13: Final exam (60+55+50+35=200 points), hand in FRIDAY 28 April in class. Solutions (pdf).

### Second Chance take home exam:

Week 15: 2.1.8a (page 53), 3.2.5 (page 147), 5.1.4 (page 211), 5.1.1. (page 220), 5.1.1 (page 226) (20*5=100 points, can only replace the five worst homeworks and can at most give a C as final grade), hand in THURSDAY 12pm in my office, this deadline in sharp meaning that assignments handed in later will not be graded). Solutions (pdf).