Math 440 Fall 2008 Problem Set


Week 1:
Read section 1.1, and do 1.1 #2,4,5,6,7
                         1.2 #3

Mon: review ODE concepts, independent/dependent, IVPs,
        separable, 1st order linear, 2nd order constant coeff.
Wed: ODE IVPs vs BVPs, Calc review.  Defn of derivative and partial
     derivative.  PDE concepts.  Introduction to Wave, Heat,
     Laplace, Black-Scholes.
Fri: IBVPs, BCs, Well-posedness, Linear ops, Superposition.

Week 2:  Read sections 1.2, 1.3.  
	Do 1.2.1, 1.2.4, 2.3.5, 1.3.1, 1.3.2, 1.3.3
        HANDOUT PROBLEMS

Mon:   Finish linearity, operators. 1-D Wave Equation: Derivation.   

Wed: Classification of 2nd order PDEs: Hyperbolic, Parabolic, Elliptic

Fri:  ODE review: Homogeneous and particular solutions. Integrating factor.
                  Undetermined coefficient.  Variation of paramter.  

Week 3:  Read the derivation of the heat equation in another text. Worksheet
	(available Friday) 

Mon:   Matrix presentation of Classification.  Introduction to heat 	
	equation.  

Wed: Derivation of the heat equation in 3d. 

Fri: Continue derivation.  Reduce to 1d constant coefficient.  

Week 4: Read sections 1.4 and 2.1.
	Do 1.4.1, 1.4.2, 2.1.1, 2.1.3, 2.1.5, 2.1.6, 2.1.8

Mon:  Method of Characteristics.  Geometery of method.

Wed: Characteristics 2d wave.  Derive d'Alembert's solution. 

Fri:  Maple.

Week 5: Read section 2.3.
        Do 2.1.8, 2.1.10, 2.3.2, 2.3.4, 2.3.7 (and verify your solution), 2.3.12

Mon:  Domain of Dependence/Infuence. Reflected characteristics. i
      Prove well-posedness.

Wed:  Separation of Variables (part 1).

Fri:  Separation of Variables (part 2). 

Week 6: Play around with Maple and/or Mathematica. Plot 
        solutions to problems in section 2.3.

Mon:   Separation of variables (part 3). Linearity.

Wed:   Non-homogeneous problems.

Fri:   Maple.

Week 7: Read 3.1 and 3.2
        Do 3.1.2 (a,d,e), 3.1.3 (a,d,e), 3.1.4 (a,d,e) 3.1.5, 3.1.8
        ( I had a typo before - 3.1.4)

Mon:  Fourier (sine) series expansions

Wed:  Fourier series, convergence of Fourier series
      (HOMEWORK DUE -for Weeks 4, 5 and 6)

Fri:  Convergence

Week 8: 

Mon:   Review

Wed:   TEST.  In class + Takehome.

Fri:   No class.

Week 9:

Mon: Fourier Series 

Wed:  Test Review

Fri:  Computer - Fourier Series

Week 10:  Read 3.3, 4.1
          Do 3.3.4, 4.1.6, 4.1.8 Solve HE with Neumann BC's

Mon:  Recap of semester. 

Wed: Heat Equation and Fourier Series

Fri: Max/Min Principle

Week 11:  Read 4.2, 5.1, 5.2
          Do 5.1.7

Mon:  Characterstics, Max/Min and uniqueness of soln to Heat Eqn

Wed:  Laplace's Eqn on a rectangle

Fri:  Maple and/or Laplaces's Eqn (on disk?)

Week 12:  Read 5.2, 5.3
          Do 5.2.1, 5.2.2
          Generate Maple/Mathematica worksheet to visualize solution 
          of Laplaces eqn on both rectangle and disk.

Mon:  Laplace's eqn on disk

Wed:  Finish Laplace's eqn on disk

Fri:  Project discussion.  Maple/Mathematica solutions.