Review Part 4

Review Part 4 - Review of Part IV Methods and Formulas...

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Review of Part IV Methods and Formulas Initial Value Problems Reduction to First order system: For an n -th order equation that can be solved for the n -th derivative: x ( n ) = f p t,x, ˙ x, ¨ x,. .., d n 1 x dt n 1 P . (42.6) Then use the standard change of variables: y 1 = x y 2 = ˙ x . . . y n = x ( n 1) = d n 1 x dt n 1 . (42.7) Di±erentiating results in a ²rst-order system: ˙ y 1 = ˙ x = y 2 ˙ y 2 = ¨ x = y 3 . . . ˙ y n = x ( n ) = f ( t,y 1 ,y 2 ,...,y n ) . (42.8) Euler’s method: y i +1 = y i + hf ( t i , y i ) . Modifed (or Improved) Euler method: k 1 = hf ( t i , y i ) k 2 = hf ( t i + h, y i + k 1 ) y i +1 = y i + 1 2 ( k 1 + k 2 ) Boundary Value Probems Finite Di±erences: Replace the Di±erential Equation by Di±erence Equations on a grid. Review the lecture on Numerical Di±erentiation. 146
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147 Explicit Method Finite Diferences ±or Parabolic PDE (heat): u t m→ u i,j +1 u ij k u xx m→ u i 1 ,j 2
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This note was uploaded on 01/15/2011 for the course MATH 345 taught by Professor Staff during the Spring '08 term at Ohio State.

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Review Part 4 - Review of Part IV Methods and Formulas...

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