Grader: Chenchen Mou

The course will discuss the solution of Boundary Value Problems for classical Partial Differential Equations.

The text that will be used is:

David L. Powers, **Boundary Value Problem** 6th
Edition, Harcourt Academic Press.

If you have the fifth edition you can use it. There are only minor differences with the newest edition. Some of the exercises have changed number. In case one such exercise is assigned as a HW the number relative to the fifth edition will appear in square brakets (e.g. [12]).

There will be two midterm. The midterms will test all the material covered up to the Friday before the test date.

The exercise listed are for HW collection. I will collect them every two weeks and grade 2 or 3 exercises among the one assigned.

The final greade will be based on the following rules: 50% final, 40% midterms, 10% HW. Curving will be done on the final result.

**Assignments for the Long Distance students are due 1 weeks
after the normal students due date.**

Arguments Covered

- Ordinary Differential Equation (review material)
- Fourier Series and Integrals
- The Heat Equation
- The Wave Equation
- Problem in Several Dimension (time permitting)
- Numerical Methods (time permitting)

**First week**

Material covered:

- 0.1 (Homogeneous Linear Equations)
- 0.2 (Nonhomogeneous Linear Equations)
- 0.3 (Boundary Value Problems)
- 2.1 (Derivation and Boundary Conditions)

Exercises:

- (0.1) 3, 11, 12, 18, 22
- (0.2) 3, 7, 11, 16
- (0.3) 3, 7, 16 [14]
- (2.1) 3, 5

**Second week**

Material covered:

- 1.1 (Periodic Functions and Fourier Series)
- 1.2 (Arbitrary Period and Half Range Expansions)

Exercises:

- (1.1) 1, 6,7
- (1.2) 1, 7, 8, 9, 15

**Third week**

Material covered:

- 1.4 (Uniform convergence)
- 1.5 (Operation on Fourier Series)

Exercises:

- (1.4) 1, 2, 4
- (1.5) 6, 8, 9

**Fourth week**

Material covered:

- 1.6 (Mean, Error and Convergence in Mean)
- 1.8 (Numerical Determination of Fourier Coefficients)

Exercises:

- (1.6) 2, 3, 4
- (1.8) 3

Second HW due on Wednesday 21 September

Material covered:

- 1.10 (Complex Methods)
- 1.9 (Fourier Integral)
- 1.11 (Applications of Fourier Series and Integrals)

- (1.10) 1, 3, 6
- (1.9) 3, 5
- (2.1) 3, 5

Sixth week

Material covered:

- 2.2 (Steady States Temperatures)
- 2.3 (Example: Fixed Ends Temperatures)

Exercises:

- (2.2) 3, 7, 9
- (2.3) Project 2.2 [9]

The first midterm will be on Wednesday 10/5. It will cover all the material up to section 2.2. You may use a scientific calculator but no laptop or calculator able to do symbolic differentiation or integration. No cheat sheet will be allowed.

Here is some preparation material: fall 2005 miterm and solution set and a practice test with solution set.

Third HW due on Wednesday 12 October

Eigth week

Material covered:

- 2.4 (Example: Insulated bar)
- 2.5 (Example: Different Boundary Conditions)
- 2.6 (Example: Convection)

Exercises:

- (2.4) 1, 5, 8
- (2.5) 5 , 10 , 11 , 12
- (2.6) 7, 10

Nineth week

Material covered:

- 2.7 (Sturm-Liouville Problems)
- 2.8 (Expansion in Series of Eigenfunctions)
- 2.9 (Generalities on the Heat Conduction Problem)

Exercises:

- (2.7) 1, 5, 7 [9], 11 [7]
- (2.8) 1, 2, 3
- (2.9) 1

Fourth HW due on Wednesday 2 November

The second midterm will be on Monday 11/9. It will cover all the material up to Chapter 2 included. You may use a scientific calculator but no laptop or calculator able to do symbolic differentiation or integration. No cheat sheet will be allowed.

Here is some preparation material: fall 2004 miterm and fall 05 midterm with solution set.

Tenth week

Material covered:

- 3.1 (The Vibrating String)
- 3.2 (Solution of the Vibrating String Problem)
- 3.3 (D'Alembert Solution)
- 3.4 (One Dimensional Wave Equation: Generalities)

Exercises:

- (3.1) 4
- (3.2) 7, 8, 12 [14]
- (3.3) 1, 4 [9], 12
- (3.4) 8

Fifth HW due on Wednesday 30 November

Projects for extra credit. Please, if you decide to turn in extra credit, choose one of this problems or let me know one of you choice.

Eleventh week

Material covered:

- 4.1 (The Potential Equation)
- 4.2 (Potential in a Rectangle)
- 4.3 (Further Example for a Rectangle)
- 4.5 (Potential in a Disk)

Exercises (suggested):

- (4.1) 2, 6, 8
- (4.2) 5, 6
- (4.3) 1, 10, 11
- (4.5) 1, 7, 8

Material covered:

- 5.3 (Two Dimensional Heat Equation: Double Series Solution)
- 5.4 (Problem in Polar Coordinates)
- 5.5 (Bessel's Equation)
- 5.7 (Vibration of a Circular Membrane)

Exercises (suggested):

- (5.3) 2, 6, 7
- (5.4) 2, 5, 7
- (5.5) 3, 4, 5
- (5.7)

Solution set for the midterms: midterm 1 and midterm 2.

Preparation material: Final from fall 04, fall 05 and fall 07.