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Volterra equations

  1. Oct 4, 2009 #1
    I am reading an example from a book on Volterra equations but there's one point i dont understand. the book says:
    for certain types of linear integrodifferential equations, the reduction can be made directly by integration. consider for instance, the linear equation:

    [tex]f'(t) - \int^{t}_{0}k(t,s)f(s) ds=g(t)[/tex],
    with f(0)=f0. Integrating this we get:
    [tex]f(t)- \int^{t}_{0}[/tex][tex]\int^{T}_{0}k(T,s)f(s) dsdT=G(t)[/tex]

    I don't understand how that can be derived by integration.
     
  2. jcsd
  3. Oct 4, 2009 #2

    tiny-tim

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    Hi sara_87! smile:

    It's the "fundamental theorem of calculus" …

    the derivative of ∫at f(x) dx = f(t) :wink:
     
  4. Oct 4, 2009 #3
    but where did the T and G come from?
     
  5. Oct 4, 2009 #4

    tiny-tim

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    If that's g'(t) in the top line, then G is simply g.

    T is just a "dummy" value of t.

    Try differentiating the bottom equation, and you'll see that it works!
     
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