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 parenleftbigg t, x, ˙ x, ¨ x, . . . , d n 1 x dt n 1 parenrightbigg . (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) Differentiating results in a first-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 ) . Modified (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 Differences: Replace the Differential Equation by Difference Equations on a grid. Review the lecture on Numerical Differentiation. 146
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147 Explicit Method Finite Differences for Parabolic PDE (heat): u t mapsto→ u i,j +1 u ij k u xx mapsto→ u i 1 ,j 2 u ij + u i +1 ,j h 2 (42.9) leads to: u i,j +1 = ru i 1 ,j + (1 2 r ) u i,j + ru i +1 ,j , where h = L/m , k = T/n , and r = ck/h 2 . The stability condition is r < 1 / 2.
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