Euler's method

Discussion in 'Math' started by boks, Dec 11, 2008.

  1. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
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    x'' - 2x' + x = 4t, x(0) = 1, x'(0)=1

    Introducing y_1 = x and y_2 = x'

    we have the system

    y_1 ' = y_2

    y_2 ' = 2y_2 - y_1 + 4t

    right?

    Using Euler's method with step size h=0.1, we get

    y_{1, n+1} = y_{1, n} + 0.1y_{2,n}

    y_{2,n + 1} + 0.1(2y_{2,n} - y_{1,n} + 4t)

    It's easy to see that y_{1, 1} = 1 + 0.1 \cdot 1 = 1.1, but what is y_{2,1}?
     
    Last edited: Dec 11, 2008
  2. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    What issues do you have with euler's method? What don't you understand? Why are you asking us to do simple problems for you?

    -blazed
     
  3. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
    Updated first post.
     
    Last edited: Dec 11, 2008
  4. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    I don't see what the issue is here man...

    Well, first of all, I'm going to see that by:

    You actually meant:

    y_{2,n + 1} = 0.1(2y_{2,n} - y_{1,n} + 4t)

    In which case

    y_{2,1} = 0.1(2y_{2,0} - y_{1,0} + 4t)

    You have y_{2,0} and you have y_{1,0}.

    Proceed. Good luck.

    -blazed

    Edit: Your t should be t_n where t_{n+1} = t_n + h
     
  5. steveb

    Senior Member

    Jul 3, 2008
    2,433
    469
    Actually, this is a good question and this issue can be a point of confusion when first learning. One often hears of Forward Euler and Backward Euler methods. The forward method is probably what you are using since it is more common and I think BlazedAces' answer is correct for that approach. However, in the backward Euler method the time value would be t(n+1) instead of t(n).

    Anyway, I thought I'd mention that since what appears to be a simple problem can often have little points of confusion.

    This simple Forward Euler method is generally not used for numerical analysis since the accuracy is much poorer than higher order methods. However, it is often used in real-time micorprocessor based control systems since the calculations can be perfomed quickly and with minimal delay.
     
  6. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
    Yeah, it's the forward method. Then I use t(n)
     
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