ODE-Explicit

ODE-Explicit - Integration of Ordinary Integration of...

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Unformatted text preview: Integration of Ordinary Integration of Ordinary Differential Equations Differential Equations d d y y ’/dt = f(t, ’/dt = f(t, y y ) ) Marco Lattuada Swiss Federal Institute of Technology - ETH Institut für Chemie und Bioingenieurwissenschaften ETH Hönggerberg/ HCI F135 – Zürich (Switzerland) E-mail: [email protected] http://www.morbidelli-group.ethz.ch/education/index Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 2 Definition of the problem: Integration of a system of ordinary differential equations (ODEs) in the explicit form: with the following initial conditions: Definition of the Problem Definition of the Problem ( 29 ( 29 ( ) ( ) , d t t f t dt ′ = = y y y ( ) t = y y Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 3 Order of the ODE: An system of ODEs of order m is defined as: with the following initial conditions: This system can be transformed into a system of ODEs of first order by introducing as new variables: Higher Order ODEs Higher Order ODEs ( 29 ( ) ( 1) , , ', , m m f t-= y y y y K ( ) t = y y '( ) ' t = y y ( 1) ( 1) ( ) m m t--= y y … 1 = y y 2 ' = y y ( 1) m m-= y y … 2 ( 0) 2 2 ( ) ( 0) ( 0) 1 y t y y y y t with y t y t = = ′′′ ′′ ′ ′ =-+ + = = ′′ = = - 1 1 2 2 2 3 2 3 3 3 2 1 2 ( ) y y y y y y y y y y y y y y t ′ = = ′ ′ = ⇒ = ′′ ′ = =-+ + Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 4 Autonomous Systems Autonomous Systems Definition: A system is autonomous when the independent variable t does not explicitly appear in the system. ( 29 ( ) t f ′ = y y It is always possible to transform a non-autonomous system of ODEs into an autonoumous one ( 29 ( ) , y t f t y ′ = 1 1 1 2 2 2 ( , ) 1 y y y f y y y t y ′ = = ⇒ ′ = = Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 5 Example: One ODE Example: One ODE ( ) 2 (0) 1 y t y t y ′ =- = ( 29 ( ) ( ) '( ) ( ) ( ) ( ) ( ) ( ) p s ds p s ds y t r t p t y t y t e k r t e dt-∫ ∫ =-= + ∫ Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) ( 29 2 1 ( ) 3 2 1 4 t y t e t = + + 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 t y Numerical solution? Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 6 Example: One ODE Example: One ODE ( ) 2 (0) 1 y t y t y ′ =- = ( 29 2 1 ( ) 3 2 1 4 t y t e t = + + 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 t y Let us use the Taylor expansion of y(t): ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 1 2 2 ( , ) n n n n n n n n n n n n dy y y t t o t t dt y y h o h y h f t y o h + + + = +-+-′ = + + = + + 1 ( , ) n n n n y y hf t y + = + 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 t 1 0.2 1.4 h y = = 1 ( , ) 2 y f t y t = ⇒ = = Leonhard Euler...
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This note was uploaded on 03/08/2010 for the course BCB 570 taught by Professor Juliedickerson during the Spring '10 term at Iowa State.

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ODE-Explicit - Integration of Ordinary Integration of...

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