Lecture_25_E7_L25_ODE2

# Lecture_25_E7_L25_ODE2 - 1 E7: INTRODUCTION TO COMPUTER E7:...

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E7 L25 1 E7: INTRODUCTION TO COMPUTER E7: INTRODUCTION TO COMPUTER PROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS ENGINEERS Lecture Outline 1. Review of ordinary differential equations 2. Numerical integration algorithms: Euler Modified Euler (predictor-corrector) 3. Examples Copyright 2007, Horowitz, Packard. This work is licensed under the Creative Commons Attribution-Share Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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E7 L25 2 Review of Ordinary Differential Equations (ODEs) Review of Ordinary Differential Equations (ODEs) 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 f(t,y)  is a known function of its arguments
E7 L25 3 Solution of ODEs (Initial Value Problem) Solution of ODEs (Initial Value Problem) Given the ODE ( ) ( , ) dy t f t y dt = and an initial condition 0 0 ( ) y t y = find the function ( ) y t which satisfies: 0 0 ( ) y t y = ( ) ( , ( )) dy t f t y t dt = (slight abuse of notation) 0 t t for

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E7 L25 4 What is the difference between ode45 and quad? What is the difference between ode45 and quad? Quad is used to solve integrals. For example: ( ) ( ) dy t f t dt = Only a function of time t This can be solved by quadratures: ( ) ( ) ( ) o t o t y t f d y t τ = +
E7 L25 5 What is the difference between ode45 and quad? What is the difference between ode45 and quad? Quad is used to obtain definite integrals numerically (e.g. quadratures): ( ) b a q f t dt = Quad syntax: q = quad(@ f ,a,b) where f is only a function of time.

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E7 L25 6 What is the difference between ode45 and quad? What is the difference between ode45 and quad? ode45 is used to solve ODEs: ( ) ( , ) dy t f t y dt = a function of time AND  y This cannot cannot solved by quad , unless we can use separation of variables, e.g. ( ) ( ) ( ) f t g t h y =
E7 L24 7 ODE integration VS integration by quadratures ODE integration VS integration by quadratures Most ODEs cannot be integrated by quadratures dependent variable appears on the RHS unless we can use separation of variables: numerical integration, e.g. quad ( ) ( , ) dy t f t y dt = ( , ) ( ) ( ) f t y g t h y = ( ) ( ) dy g t dt h y = 1 1 0 0 ( ) ( ) y t y t dy g d h y τ = 0 0 ( ) y y t = 1 1 ( ) y y t =

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E7 L24 8 Higher order ODEs Higher order ODEs Remember the basic ODE: and can be vectors! where: t is the independent scalar variable (often time) y is the dependent variable f(t,y)  is a known function of its arguments then will also be a vector ( , ) y f t y = & y & y
E7 L24 9 Higher order ODEs Higher order ODEs System of n first-order coupled ODEs : ( , ) y f t y = & 1 n × 1 n × 1 n ×

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E7 L24 10 Higher order ODEs Higher order ODEs System of n first-order coupled ODEs : ( , ) y f t y = & ( 29 ( 29 ( 29 1 1 1 2 2 2 1 2 1 2 , , ,..., , , , , n n n n n y f t y y y y f t y y y y f t y y y = = = & & M &
E7 L24 11 Higher order ODEs Higher order ODEs System of n first-order coupled ODEs : ( , ) y f t y = & ( 29 ( 29 ( 29 1 1 1 2 2 2 1 2 1 2 , , ,..., , , , , n n n n n y f t y y y y f t y y y y f t y y y = = = & & M &

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Euler’s disk (ME170) – courtesy Prof. O’Reilly
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## This note was uploaded on 09/17/2011 for the course ENGINEERIN 7 taught by Professor Patzek during the Spring '08 term at Berkeley.

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Lecture_25_E7_L25_ODE2 - 1 E7: INTRODUCTION TO COMPUTER E7:...

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