CAAM 336 Syllabus

Week 1
 Mon 21 Aug Classification of ODEs 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
 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

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