## 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