AE202 Lecture 7 Ordinary Differential Equations

AE202 Lecture 7 Ordinary Differential Equations - Numerical...

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Numerical Integration in MATLAB Aditya Paranjape February 16, 2011
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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 )).
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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 0 f ( t ) dt (1) Caveat : Don’t know x(t)!
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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 0 f ( t ) dt (1) Caveat : Don’t know x(t)! Approximation: x ( t ) = x (0) + Z T 0 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.
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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 0 f ( t ) dt (1) Caveat : Don’t know x(t)! Approximation: x ( t ) = x (0) + Z T 0 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.
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