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ODE-Explicit - Integration of Ordinary Integration...

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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
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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 0 0 ( ) t = y y
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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 0 0 ( ) t = y y 0 0 '( ) ' t = y y ( 1) ( 1) 0 0 ( ) m m t - - = y y 1 = y y 2 ' = y y ( 1) m m - = y y 2 ( 0) 2 2 ( ) ( 0) 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 = = = = ′′ = = - + +
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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 = = = =
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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 r t e dt + (Basel, July 27, 1667 - January 1, 1748) ( 29 2 1 ( ) 3 2 1 4 t y t e t = + + 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 t y Numerical solution?
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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 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 1 1 2 n n n n n n n n dy y y t t o t dt y y h o h y h f t + + + = + - + = + + = + + 1 ( , ) n n n n y y hf t y + = + 0.2 0.8 1 1 2 3 4 5 6 7 y 1 0.2 1.4 h y = = 0 0 0 0 1 ( , ) 2 0 y f t y t = = = Leonhard Euler (Basel, April 15, 1707 – September 18, 1783) y = f(x) ( ) 0 u t ∂ρ + ∇ ρ =
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Marco Lattuada – Statistical and Numerical Methods for Chemical Engineers Explicit ODE Solvers – Page # 7 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 = + + 1 ( , ) n n n n y y hf t y + = + 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 t y 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t In the next slides, the following notation will be used: y(t n ) = exact values of the variables y in t n y n = approximate values of the derivatives y’ in t
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