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Reverse Runge Kutta?

  1. Nov 5, 2009 #1
    1. The problem statement, all variables and given/known data

    e^(x) + y = dy/dx, [-1,1], y(0) = 1, N = 4.


    2. Relevant equations



    3. The attempt at a solution
    h = b-a / N = 0.5
    x0= 0, y0 =1
    x1= 0.5, y1 = 2.472
    x2= 1, y2 = 5.433
    x3= 1.5, y3= 11.195
    x4=2, y4= 22.146
    =====================
    These were the values I got for the x's and y's. However, I would like to know if it is possible to do a reverse (backwards) runge kutta to [-1,1] range. If so, how do I compute this? (what would be my starting x0, y0).
     
  2. jcsd
  3. Nov 5, 2009 #2

    HallsofIvy

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    Certainly- just take [/itex]\Delta x[/itex] to be negative. To use 4 steps from 0 down to -1, take [itex]\Delta x[/itex]= -1/4.

    But if your problem was to solve on [-1, 1] with N= 4, you should really be taking just two steps from 0 to 1, with [itex]\Delta x= 1/2[/itex] and two steps from 0 to -1 with [itex]\Delta x= -1/2[/itex].
     
  4. Nov 10, 2009 #3
    [bump]
    Question revised:
    1) Apply Runge-Kutta of order 4 to solve the ODE on [-1, 1] with N=4. At the very least solve for the points in the interval that are in [0,1]. Is it possible with the initial condition y(0)=1 to obtain numerical solutions in [-1,0) using Runge-Kutta? If so then how can it be done?
    ODE = e^(x) + y = dy/dx, [-1,1], y(0) = 1, N = 4.
    Compare with using the initial condition for the interval [-1,1]

    =============================================
    Basically, solving for y(0)=1 over [-1,1], N=4, I got the same answers as above which are again:
    h = b-a / N = 0.5
    x0= 0, y0 =1
    x1= 0.5, y1 = 2.472
    x2= 1, y2 = 5.433
    x3= 1.5, y3= 11.195
    x4=2, y4= 22.146
    *Can anybody confirm these are the correct #'s, or if its necessary to go to x4, since it's 4th order runge kutta.

    The second part of this question is whether it is possible with the initial condition y(0)=1 to obtain numerical solutions in [-1,0).
    For this part, I am kind of confused.
    I basically used x0=0 and y0=1, over [-1,0) with h = 0-(-1)/4 = .25
    From this, I calculated and my answers were:
    x0=0, y0=1
    x1=.25, y1=1.605
    x2=.5, y2=2.473
    x3=.75, y3=3.705
    x4=1, y4=5.437
    Upon calculating these answers, I do not see any significance....or maybe there is? I do not know.

    Assistance would be helpful
     
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