Topic |
Hrs |
Description |
Review of Series |
5 |
- Convergence concepts
- Geometric series and the ratio test
- Error control and Shanks Transformation
- Power series
- Analytic functions, radius of convergence
|
Linear Ordinary Differential Equations |
9 |
- Power Series Solutions near an ordinary point
- General Solutions
- Initial-Value Problems
- Non-homogeneous problems
- Homogeneous ODE's with Constant Coefficients
- Non-Homogeneous ODE with Constant Coefficients
- ODE's of Euler type
- Change of independent variable
- Singular points: regular vs irregular
- Series solutions near a regular singular point (Frobenius method)
|
Linear Operators and Eigenvalue Problems |
6 |
- The matrix case
- Operators and eigenvalues in function space
- Sample eigenvalue problems
- Orthogonality and eigenfunction series
|
Eigenfunction-series solutions for Boundary-Value Problems |
6 |
- Separation of variables: eigenvalue problems, formal series
- The Big Four standard eigenvalue problems
- Propagating coefficients
- Initialization
- Special modes; superposition
- Nonhomogeneous PDE
- Nonhomogeneous BC
|
More General Eigenfunction Series |
5 |
- Eigenvalue Problems
- Orthogonality and Eigenfunction Series
- Sturm-Liouville Theory
- Full Fourier Series
|
Heat Equation in Depth |
3 |
- Derivation; BC's and their meanings
- Transient versus long-run behaviour
- Splitting methods for nonhomogeneous boundary conditions
|
Laplace's Equation in Depth |
4 |
- Derivation and Interpretation
- Superposition and splitting
- Neumann problems and consistency conditions
- Physical solutions are bounded functions
- Problems in polar coordinates (Euler equations reappear)
|
Wave Equation in Depth |
4 |
- Derivation; BC's and their meanings
- Travelling wave solutions
- D'Alambert's solution
- Signal speed; reflections
- Forcing and resonance
|
Numerical Methods |
2 |
- Finite Differences
- Spreadsheet Implementations
|
Total Hours on Outline |
44 |
- (44 lecture-hours + 2 test-hours = 46 meetings this term)
|