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lec17_ODE4 - expansions but they take...

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Fixed Points Example: Lotka-Volterra Predator-Prey model Runge-Kutta method for integrating ODEs Example: Apply 4 th order R-K method to pendulum problem Midterm project due date extended until next Thursday (4/1) as per my pre-spring break comment! No HW this week – finish your projects! Reading for Differential Equations in Appendix B. Reading for Runge-Kutta on hand out.
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System of ODEs describing the change in population of predators (y) and prey (x). Interpretation of parameters: a - prey birth rate, b – prey death rate (other than being eaten), c – rate of consumption, d – predator death rate, e – predator birth rate Find fixed points by setting both derivatives to 0 and solving for x and y: There are others, see them? dx dt = x ( a cy bx ) dy dt = y ( d ex ) x 0 = d e y 0 = a c bd ce
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Direct use of the Taylor series methods for better accuracy becomes quite tedious and error prone beyond the methods we have already seen. Runge-Kutta methods are also based to Taylor series
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Unformatted text preview: expansions, but they take intermediate (extrapolated) steps in which the function is evaluated appropriately chosen times BETWEEN the time time steps (dt). The derivation is rather long and not too instructive, we’ll just quote results and use them. The 4 th order Runge-Kutta is the standard method for solving ODEs – not necessarily the fastest or most accurate, but it is quite stable. 2 nd order RK is basically just a type of Euler method and not very commonly used. 4 th order RK is the standard Assume we have an ODE of the form Truncation error for 4 th order RK is dx dt = f ( x ( t ) , t ) O ( τ 5 ) Requires 4 intermediate evaluations of the function f(x): F 1 = f ( x, t ) F 2 = f ( x + 1 2 dtF 1 , t + 1 2 dt ) F 3 = f ( x + 1 2 dtF 2 , t + 1 2 dt ) F 4 = f ( x + dtF 3 , t + dt ) x ( t + dt ) = x ( t ) + dt 6 [ F 1 + 2 F 2 + 2 F 3 + F 4 ] Equation to solve: dx dt = − x...
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