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 | |
|
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 |