SM222 Lecture 4 Spring 2006

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SM222 Lecture 4Spring 2006

• 1 First-order ODEs
• 1.4 Numerical solutions

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Assignments

• Due
• 1.3 Problems, p. 22, 1, 5, 6
• Due Tues. 17 Jan. 2006
• Pending
• 1.4 Problems, p. 28, 2, 5 (Maple or Excel)
• Due Tues. 17 Jan. 2006
• 3.6 Problems, p. 132, 7, 8 (Maple, Excel or
  TI)
• Due Wed., 18 Jan. 2006
• Assign
• 2.2 Problems, p. 46, 4, 5, 7, 12, 23
• Due Friday, 20 Jan. 2006

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Points to Control

• Whats the idea behind a numerical method?
• Linearization link between
• Tangent at a point
• Solution in the vicinity of the point
• Eulers method
• The method
• How you assess the method relative error

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Basis for a numerical method

• Why a numerical method? Example
• Basis behind method direction field solution
  curve
• See F05_sl01_4_append.mws

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Basis for a numerical method

• IVP direction field

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Linearization tangent and local behavior

• Observe use the tangent line to propagate the
  solution beyond a point. How?
• Linearization (Taylor to 1st-order)
• The trick (Euler) For

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Eulers Method (cont)

• Example
• Solution

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Flow Eulers method

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Eulers Methodexecution (Excel)

• Example
• EulerEx.xls

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Eulers Method (results)

• Example
• See S05sl01_3_append.mws
• Analytic solution (2.2)
• How assess accuracy?

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Eulers method assessing accuracy (p. 126)

• Example
• Observation anticipate/estimate propagation of
  error linearization
• Can we improve?

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Summary Eulers method
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Appendix Eulers method (Maple)

• Procedure (see F05sl01_3_append.mws)
• gt Euler0 proc( F, x0, y0, h, N )
• gt local i, L, X
• gt X evalf( x0, y0 )
• gt L X
• gt for i from 1 to N do
• gt X X h, hF(op(X))
• gt L L, X
• gt end do
• gt return matrix( N1, 2, L )
• gt end proc

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Appendix Eulers method (Voyage 200)

• Procedure (see Class Demonstration)
• euler1(x0, y0, h, n )
• Prgm
• Local u, v, w
• Disp 0,u,v
• x0 -gt u
• y0 -gt v
• For m, 1, n, 1
• rhs(u,v) -gt w
• uh -gt u
• vhw -gt v
• Disp m,u,v
• EndFor
• EndPrgm