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lecture_19_ODEs_2p

# lecture_19_ODEs_2p - Lecture 19 Numerical solutions of ODEs...

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4/17/2011 1 Example canonical ODEs Falling object Pendulum Lecture 19: Numerical solutions of ODEs The Lorenz system Numerical solution of an ODE: general method 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):

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4/17/2011 2 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 Other canonical example of an ODE: pendulum equation reads:
4/17/2011 3 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

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4/17/2011 4 Example 3: the Lorenz system
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lecture_19_ODEs_2p - Lecture 19 Numerical solutions of ODEs...

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