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Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs). Differential equations are the ubiquitous, the lingua franca of the sciences; many different fields are linked by having similar differential equations ODEs have one independent variable; PDEs have more Examples: electrical circuits

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Ordinary Differential Equations (ODEs)

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  1. Ordinary Differential Equations (ODEs) • Differential equations are the ubiquitous, the lingua franca of the sciences; many different fields are linked by having similar differential equations • ODEs have one independent variable; PDEs have more • Examples: electrical circuits Newtonian mechanics chemical reactions population dynamics economics… and so on, ad infinitum

  2. Example: RLC circuit

  3. To illustrate: Population dynamics • 1798 Malthusian catastrophe • 1838 Verhulst, logistic growth • Predator-prey systems, Volterra-Lotka

  4. Population dynamics • Malthus: • Verhulst: Logistic growth  

  5. Population dynamics Hudson Bay Company

  6. Population dynamics V .Volterra, commercial fishing in the Adriatic

  7. In the x1-x2 plane

  8. State space Integrate analytically! Produces a family of concentric closed curves as shown… How to compute?

  9. Population dynamics self-limiting term  stable focus Delay  limit cycle

  10. As functions of time

  11. Do you believe this? • Do hares eat lynx, Gilpin 1973 Do Hares Eat Lynx? Michael E. Gilpin The American Naturalist, Vol. 107, No. 957 (Sep. - Oct., 1973), pp. 727-730 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2459670

  12. Putting equations in state-space form

  13. Traditional state space: Example: the (nonlinear) pendulum McMaster

  14. Linear pendulum: small θ For simplicity, let g/l = 1 Circles!

  15. Pendulum in the phase plane

  16. Varieties of Behavior • Stable focus • Periodic • Limit cycle

  17. Varieties of Behavior • Stable focus • Periodic • Limit cycle • Chaos …Assignment

  18. Numerical integration of ODEs • Euler’s Method simple-minded, basis of many others • Predictor-corrector methods  can be useful • Runge-Kutta (usually 4th-order) workhorse, good enough for our work, but not state-of-the-art

  19. Criteria for evaluating • Accuracy use Taylor series, big-Oh, classical numerical analysis • Efficiency  running time may be hard to predict, sometimes step size is adaptive • Stability  some methods diverge on some problems

  20. Euler • Local error = O(h2) • Global accumulated) error = O(h) (Roughly: multiply by T/h )

  21. Euler • Local error = O(h2) • Global (accumulated) error = O(h) Euler step

  22. Euler • Local error = O(h2) • Global (accumulated) error = O(h) Taylor’s series with remainder Euler step

  23. Second-order Runge-Kutta (midpoint method) • Local error = O(h3) • Global (accumulated) error = O(h2)

  24. Fourth-order Runge-Kutta • Local error = O(h5) • Global (accumulated) error = O(h4)

  25. Additional topics • Stability, stiff systems • Implicit methods • Two-point boundary-value problems shooting methods relaxation methods

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