lecture_19_ODEs_6p - 4/17/2011 Lecture 19: Numerical...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
4/17/2011 1 Example canonical ODEs Falling object Pendulum The Lorenz system Numerical solution of an ODE: general method Lecture 19: Numerical solutions of ODEs Euler explicit scheme Euler implicit scheme Trapezoidal method Mid point method Built in Matlab functions Numerical error and instability Example 1: falling object Canonical example of an ODE: falling object ODE system associated with it (Newton’s law): Example 1: falling object ODE system associated with it (Newton’s law): This system can be integrated, which provides us with velocity The velocity system can be integrated, which provides us with trajectory information This trajectory information solves the problem. Example 2: pendulum equation The pendulum equation for the system reads: Other canonical example of an ODE: pendulum equation Example 2: pendulum equation Pendulum equation: With the standard approximation of small angles: Leads to the standard harmonic oscillator: With oscillation frequency: Example 2: pendulum equation In general, it is useful for numerical integration to write the ODE in the form of a system of first order ODEs: start with the following: Now with the following change of variables: One obtains the following system (which is equivalent): With the solution derived in the supplement lecture
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/07/2011 for the course ENGIN 7 taught by Professor Horowitz during the Spring '08 term at University of California, Berkeley.

Page1 / 5

lecture_19_ODEs_6p - 4/17/2011 Lecture 19: Numerical...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online