Week 2.
Lecture 3 on ODEs.
Solution of linear constant-coefficient ODES. Homogeneous systems.
Week 2.
Lecture 4 on ODEs.
Solution of linear constant-coefficient ODES. Inhomogeneous systems 1.
Week 3.
Lecture 5 on ODEs.
Solution of linear constant-coefficient ODES. Inhomogeneous systems 2.
Week 3.
Lecture 6 on Matrices.
Terminology, notation, classification. Multiplication.
Week 4.
Lecture 7 on Matrices.
Determinants. Cramer's rule. Geometric interpretation. Evaluation. Dirty tricks: row and column manipulations.
Week 4.
Lecture 8 on Matrices.
Gaussian Elimination. Use with multiple right hand sides. Finding the inverse matrix.
Week 5.
Lecture 9 on Matrices.
From a pair of first order linear constant coefficient ODEs to eigenvalues and eigenvectors. General method.
Week 5.
Lecture 10 on Matrices.
Eigenvalues and eigenvectors for a 3x3 matrix. Mass/spring systems as motivation.
Week 6.
Lecture 11 on Laplace Transforms.
Definition. Examples. LT of derivatives. Solutions of some example ODEs
Week 6.
Lecture 12 on Laplace Transforms.
The unit impulse. Solution of ODEs with the unit impulse as the forcing function.
Week 7.
Lecture 13 on Laplace Transforms.
The shift theorem in s. The unit step function. The shift theorem in t.
Week 7.
Lecture 14 on Laplace Transforms.
Convolution. Examples. Use in ODE solutions.
Week 8.
Lecture 15 on Numerical Analysis.
Definition of iteration schemes. Ad hoc methods. Numerical examples. Perturbation analysis.
Week 8.
Lecture 16 on Numerical Analysis.
Derivation of the Newton-Raphson method. Numerical examples. Perturbation analysis. Performance for double zeros.
Week 9.
Lecture 17 on Fourier Series.
Definition of Fourier Series. Symmetry considerations for integration. A couple of examples.
Week 9.
Lecture 18 on Fourier Series.
Further examples. Rate of convergence. Discontinuities.
Week 10.
Lecture 19 on Fourier Series.
Use of Fourier Series as a periodic forcing term for ODEs.
Week 10.
Lecture 20 on Least Squares.
Definition of residuals. Minimisation of the sum of the squares of the residuals. Fitting of lines, quadratics
and general curves.
Week 11.
No lecture.
Week 11.
No lecture.