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Math Help - Convolution and Leibniz's Law

  1. #1
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    Convolution and Leibniz's Law

    I'm reviewing this document:

    http://ocw.mit.edu/courses/mathemati...18_03S10_i.pdf

    The short story is that w(t) is the weight function (solution to the associated homogeneous equation) of some generic equation my'' + cy' + ky = f(t). We want to break the interval between 0 < t < x into smaller intervals, [t_{i}, t_{i+1}], i = 0, 1, 2, 3, ..., n and define f_{i}(t) \begin{cases}f(t_{i}) \text{ if } t_{i} < t < t_{i+1}\\0 \text{ elsewhere}\end{cases} so that over the interval it's a constant function. This means that f(t) \approx f_{0}(t) + f_{1}(t) + ... f_{n-1}(t).



    I get everything until page 3 equation (8). If w(t) is the weight function for f(t), then I would think that for f_{i}(t), the weight function would just be w(t-t_{i}) (restricted to the interval). I don't understand why we need to amplify this by f(t_{i})\Delta t.

    (I forgot to include a part about Leibniz's Formula but I'm too tired.)
    Last edited by ragnar; June 2nd 2011 at 10:38 PM.
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  2. #2
    Super Member Aryth's Avatar
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    The response of the system to an impulse of area A delayed by time t_i is Aw(t - t_{i}).

    The Area of an impulse at time t_i is F(t_{i})\Delta t
    Last edited by Aryth; June 4th 2011 at 01:08 PM.
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  3. #3
    A Plied Mathematician
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    @ Aryth: use [tex] tags instead of [tex] tags for now. It'll work better.
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