BE5B01DEN: Schedule of classes

I will try to follow the following schedule, but I cannot rule out changes caused by higher causes (illness, natural disasters etc.) or by a flash of inspiration getting grip of me in the middle of a lecture.

Week 1:
a) Introduction to ODEs. Method of separation.
b) Some popular applications.
Labs: Solving separable ODEs.

Week 2:
a) Linear ODEs of order 1 (method of variation).
b) Analyzing solutions (slope fields, stability of equilibria). Theory: Peano's and Picard's theorem.
Labs: Solving ODEs by variation. Slope fields, stability.

Week 3:
a) Introduction to numerical analysis. Errors and their propagation in calculations. Representation of numbers. Taylor approximation, big-O notation.
b) Numerical derivation. Numerical integration.
Labs: Numerical integration, error estimation using order.

Week 4:
a) Numerical solition of ODE: the Euler method. Error estimate, order of method.
b) Second order methods. Runge-Kutta methods for ODE. Estimating error via control method. RKF45.
Labs: Euler method, predicting error using order.

Week 5:
a) ODEs numerically: the Taylor method. Brief intro to interpolation.
b) Homogeneous linear ODEs. Theorems on solutions, characteristic numbers.
Labs: Homogeneous linear ODE's.

Week 6:
a) Structural theorem for non-homogeneous linear ODEs. The method of undetermined coefficients for special RHS.
b) Linear ODEs: method of variation. Numerical methods for higher order LODE: Taylor method, FDM.
Labs: Solving general linear ODEs (undetermined coefficients).

Week 7:
a) Solving equations numerically: roots. The bisection and Newton methods. Stopping conditions, order of method.
b) The secand method, more on root finding.
Labs: The bisection and Newton methods.

Week 8:
a) fixed point approach. Iteration. Relaxation.
b) Refresher on eigenvalues and eigenvectors.
Labs: Fixed point iteration.

Week 9:
a) Homogeneous systems of linear ODEs.
b) Non-homogeneous systems of equations. Numerical methods for systems.
Labs: Solving homogeneous systems of ODE.

Week 10:
a) Orbits. Stability for systems of linear ODEs.
b) Applications: Population dynamics, pendulum and spring, RLC circuits.
Labs: Midterm. Solving systems of linear ODEs using variation.

Week 11:
a) Systems of linear equations: Gaussian elimination, LU(P) decomposition.
b) Residual. Errors in elimination. Matrix norms, condition number.
Labs: Solving systems using GEM and back substitution.

Week 12:
a) Iteration for systems of linear equations. The Gauss-Seidel method. Relaxation.
b) Numerical approach to eigenvalues and eigenvectors. Power iteration.
Labs: The Gauss-Seidel iteration.

Week 13:
a) A brief overview of Laplace transform.
b) Review.
Labs: Review.