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



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

    (I forgot to include a part about Leibniz's Formula but I'm too tired.)
    Last edited by ragnar; Jun 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 $\displaystyle t_i$ is $\displaystyle Aw(t - t_{i})$.

    The Area of an impulse at time $\displaystyle t_i$ is $\displaystyle F(t_{i})\Delta t$
    Last edited by Aryth; Jun 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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