Schedule



Week 1
 Mon 21 Aug Classification of ODEs 
Matlab tutorial: getting started with Matlab
Matlab primer
Chapter 1
 Weds 23 Aug  Derivation of the heat equation, connection to the diffusion equation.
 Section 2.1
 Fri 25 Aug   Harvey
 Section 2.2
Week 2
 Mon 28 Aug
  Harvey  
 Weds 30 Aug
  Harvey  
 Fri 1 Sept
  Harvey
Week 3
 Mon 4 Sept
 LABOR DAY----NO CLASS!!!
 
 Weds 6 Sept
   Finite differences for the steady heat equation, begin order of approximation.
   Texts on order    Matlab demo code
  Section  7.5.1
 Fri 8 Sept
 Order of approximation and finite differences continues  Section 7.5.1
Week 4
Mon 11 Sept
  Vector spaces, Vector subspaces, linear operators, PDEs as operator equations.
  Begin existence and uniqueness of solutions to operator equations. Range and Null Space. Basis and Dimension.
 Sections 3.1,3.2,3.3
Weds 13 Sept
  Conclude basis and Dimension. Begin discussion of inner products and inner product spaces.     Sections 3.3, 3.4
Fri 15 Sept
  Continue with inner products and inner product spaces. Introduced generalized orthogonality of vectors.  Section 3.4
Week 5
Mon 18 Sept
  Introduce the L2 inner product and begin the projection theorem. Sections 3.4.1, 3.4.2

Weds 20 Sept
  Projection theorem, the Gram matrix and examples
   Gram matrix example
  Section 3.4.2
Fri 22 Sept
  Eigenvalues and eigenvectors for general matrices
 
Week 6
 Mon 25 Sept
Relationship between the matrix transpose and dot product. 
  Eigenvalue properties of symmetric matrices. 
Section 3.5.1, 3.5.2
 Weds 27 Sept
  Spectral method for symmetric linear systems
  Example of using null space to find eigenvectors 
  Section 3.5.3
 Thurs 28 Sept
  Exam 1:  6:30-9:30pm in Herzstein Hall 210
  Fall 2016 exam 1     Solutions
NOTE: Big O and little o are new topics. Problem 4 is material for Exam 2.
  Fall 2017 Exam 1 Solutions
 
 Fri 29 Sept
 Relationship between Boundary value problems and linear algebraic systems  Section 5.1
Week 7
 Mon 2 Oct
  Symmetric linear differential operators and eigenpairs,   Section 5.2.1, 5.2.2
 Weds 4 Oct
  Eigenfunctions with different boundary conditions
 Fourier examples code
Section 5.2.3
 Fri 6 Oct
 A step-by-step breakdown the stages of the spectral method for linear BVPs
  Section 5.3.1-5.3.3
Week 8
 Mon 9 Oct
 FALL RECESS--- NO CLASS!!!

 Weds 11 Oct
  A step-by-step breakdown the stages of the spectral method for linear BVPs
Spectral method with inhomogeneous boundary conditions
  e^x fourier series
 Section 5.3.4
 Fri 13 Oct
  Start Finite Element method for BVPs
  FEM with sine series basis
  Section 5.4
Week 9
 Mon 16 Oct
    Bilinear form and consistency of solutions for two formulations.
     Notes
 Section 5.5
 Weds 18 Oct  The energy norm, the discrete problem, and solve some examples, The piecewise polynomial space   Beginning of  Section 5.6
 Fri 20 Oct  Steady heat equation with non-constant diffusivity using finite elements   Section 5.6.1
Week 10
 Mon 23 Oct
 FEM continued
  
 Weds 25 Oct  Inhomogeneous Dirichlet boundary conditions with piecewise finite elements
  Notes 
 Section 5.6.2 
 Fri 27 Oct  FEM example
  Example
 
Week 11
 Mon 30 Oct
Solution of time-dependent equations: the heat equation.  Start solving heat via spectral  Section 4.3.1  
 Weds 1 Nov Solution of homogeneous heat equation (Dirichlet conditions) using the spectral method   Section 6.1  
Thur 2 Nov
Exam 2: 6:30PM - 9:30PM in Herzstein Hall 210
  Fall 2016 exam 2    Solutions
  Fall 2017 Exam 2     Solutions
 
 Fri 3 Nov  Continue the time-dependent spectral method
 
Week 12
 Mon 6 Nov
 Time-dependent spectral method applied to special cases of f(t,x) = C and dealing with inhomogeneous Dirichlet boundary conditions using `shifting the data'  
 Weds 8 Nov  Time-dependent spectral method applied to Neumann Boundary conditions. 
 Section 6.2
 Fri 10 Nov Finite element methods for the heat equation. 

 Section 6.4
Week 13
 Mon 13 Nov
 Solving time dependent problems using Forward and Backward Euler + stability

 Weds 15 Nov       Sample Euler time stepping code  
 Fri 17 Nov     Sample  code        (Check for bugs!)
Week 14
 Mon 20 Nov
Stability of ODE, and the CFL condition for PDE
 Weds 22 Nov Stability and CFL continued, Worked examples, open discussion
 Fri 24 Nov   Thanksgiving break!
Week 15
Mon 28 Nov

 
Weds 30 Nov
 
Thurs 30 Nov
Exam 3: 6:30PM - 9:30PM in Herzstein Hall 210
  Fall 2016 exam 3    Solutions