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Euler’s method question

  1. Apr 3, 2012 #1
    1. The problem statement, all variables and given/known data

    I need some help with the last part of the following problem:

    [​IMG]


    3. The attempt at a solution

    My approximation to the solution to the IVP at t=-0.8 using 1 step of the Euler's method was:

    x(-0.8)=0.8

    Whereas the approximation with 1 step of 4th order Runge-Kutta method was:

    x(-0.8)=0.8214

    And since the exact solution is

    [itex]x(-0.8) = e^{-0.8 +1} -2 \times (-0.8) -2 = 0.8214027582[/itex]

    the error in Euler's method would be

    [itex]|0.8214027582-0.8| =0.0214027582[/itex]

    And the error for Runge-Kutta is

    [itex]|0.8214027582-0.8214| =2.7582 \times 10^{-6}[/itex]

    I'm stuck here. So how many steps does Euler's method take to produce an answer with an error no larger than 2.7582 x 10-6 (the error of Runge-Kutta)?

    I tried to use the following equation:

    [itex]e_n \leq \frac{k}{n}[/itex]

    Where k is a constant and n is the number of steps and en is the error. I then tried to solve for the constant bu substituting in the values from Euler's method:

    [itex]0.021402758 = \frac{k}{1} \ \implies k =0.021402758[/itex]

    Then substituting in the new error

    [itex]2.7582 \times 10^{-6}=\frac{0.021402758}{n} \ \implies n = 7760[/itex]

    But doesn't 7760 steps seem too much? Where did I go wrong? I appreciate it if anyone could help me with this problem.
     
  2. jcsd
  3. Apr 3, 2012 #2

    rcgldr

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    7760 seems to be OK. You could confirm this using a program or a spreadsheet with Δt = (0.2 / 7760) to see if it corresponds with your answer.
     
    Last edited: Apr 3, 2012
  4. Apr 3, 2012 #3
    But I am wondering if I've even used the correct method for finding the number of steps?
     
  5. Apr 3, 2012 #4

    rcgldr

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    What was the source of the error equation you used, class notes, a textbook, ... ?
     
  6. Apr 3, 2012 #5
    It's from a textbook...

    Edit: page 635 of the textbook called "Differential Equations" by Blanchard, Devany and Hall.
     
    Last edited: Apr 3, 2012
  7. Apr 3, 2012 #6

    rcgldr

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    It appears that you have. As mentioned before if you want to check this, you could confirm this using a program or a spreadsheet using Euler method with Δt = (0.2 / 7760) to see if it corresponds with your answer (for the spread sheet you would need to use 7761 rows, the initial state and 7760 steps).
     
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