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Unformatted text preview: Numerical Integration in MATLAB Aditya Paranjape February 16, 2011 Differential Equations A differential equation describes the variation of a variable, called the state variable ( x ), as a function of an independent variable ( t ). ˙ x ( t ) = dx dt = f ( x ( t ) , t , p ( x , t )) , f ( t ) , where x ( t ) is the instantaneous value of the state variable, and p ( x , t ) is an external parameter or an event . If we plot x ( t ) versus t , then f ( x ( t ) , t , p ( t , x )) is the slope of this curve at a given t . Figure: Red arrow denotes the slope for each time in this time history plot. The slope equals f ( x ( t ) , t , p ( x , t )). Euler’s Method ˙ x ( t ) = f ( t ) , We are interested in calculating x ( t ) for t ∈ [0 , T ] Can write x ( t ) = x (0) + Z T f ( t ) dt (1) Caveat : Don’t know x(t)! Euler’s Method ˙ x ( t ) = f ( t ) , We are interested in calculating x ( t ) for t ∈ [0 , T ] Can write x ( t ) = x (0) + Z T f ( t ) dt (1) Caveat : Don’t know x(t)! Approximation: x ( t ) = x (0) + Z T f (0) dt = x (0) + f (0) T (2) Caveat : If T is large and f varies rapidly, the error in x will be large. Euler’s Method ˙ x ( t ) = f ( t ) , We are interested in calculating x ( t ) for t ∈ [0 , T ] Can write x ( t ) = x (0) + Z T f ( t ) dt (1) Caveat : Don’t know x(t)!...
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This note was uploaded on 01/26/2012 for the course AE AE 202 taught by Professor Martin during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 Martin

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