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E7_L28_ODE3_08_BW

E7_L28_ODE3_08_BW - 1 E7 INTRODUCTION TO COMPUTER E7...

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Unformatted text preview: 1 E7: INTRODUCTION TO COMPUTER E7: INTRODUCTION TO COMPUTER ROGRAMMING FOR SCIENTISTS AND ROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS ENGINEERS Lecture Outline 1. Numerical integration of ODEs: – Euler (review) – Modified Euler – Runge-Kutta (used in ode45) 2. More ODE examples verted pendulum E7 L28 – Inverted pendulum 2 Review of Ordinary Differential Equations (ODEs) Review of Ordinary Differential Equations (ODEs) any problems in science and engineering lead to Many problems in science and engineering lead to Ordinary Differential Equations ( ODE s) of the form ( ) ( , ) dy t f t y dt where: • t is the independent scalar variable (often time) • y is the dependent variable, which can be a vector E7 L28 • f(t,y) is a known function of its arguments 3 Solution of ODEs (Initial Value Problem) Solution of ODEs (Initial Value Problem) iven the ODE ) y t • Given the ODE ( ) ( , ) dy t f t y dt • and an initial condition ( ) y t y • find the function ( ) y t (slight abuse of notation) ( ) y t y • which atisfies: ) y t E7 L28 satisfies: ( ) ( , ( )) dy t f t y t dt t t ! for 4 onsider the ODE Numerical integration of ODEs Numerical integration of ODEs ) ( ) t f t Consider the ODE ( ) ( , ) y t f t y and the initial condition ( ) y t y e want to obtain the estimates where ) E t We want to obtain the estimates where ( ) k y t 1 2 k t t k h " # 0,1,2,.... k E7 L28 and is the integration step size integration step size . h 5 Numerical integration of ODEs Numerical integration of ODEs oday we will review Today we will review • Euler’s method – Approximates a Taylor expansion up to its first order nd describe: and describe: • Modified Euler’s method (predictor-corrector) (p ) – Approximates a Taylor expansion up to its second order • Rungue-Kutta method (used in ode45, ode23) E7 L28 – Approximates a Taylor expansion up to its fourth order 6 Approximating Approximating y(to+h)- Taylor’s series expansion Taylor’s series expansion ( ) ( ) y t y t h " # 2 1 ( ) 2 y t h " # ( ) y t h " $ Euler (n=1) modified Euler (n=2) E7 L25 h h 7 4 th th order Taylor’s series expansion order Taylor’s series expansion 1 2 1 ( ) ( ) ( ) ( ) 2 y t h y t y t h y t h " $ " # " # [3] 3 [4] 4 1 1 ( ) ( ) 3! 4! y t h y t h " # " # Runge-Kutta(n=4) E7 L25 h h h 8 Euler’s method Euler’s method Given : Initial condition: ) ( ) E t y t tegration step size t a co d t o ( ) ( ) y t y t Integration step size h 1 k k t t h " " 1 ( ) ( ) ( , ( ) ) E E E k k k k y t y t f t y t h " " # E7 L27 Graphical explanation of Euler’s method Graphical explanation of Euler’s method 9 ) ( ) t f t ) t ( ) y t ( ) ( , )...
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E7_L28_ODE3_08_BW - 1 E7 INTRODUCTION TO COMPUTER E7...

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