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Math Help - Piecewise Integral

  1. #1
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    Piecewise Integral

    I posted a related problem in Differential Equations, but I have realized that really all I need is to understand this integral and so I would more likely get help here (I know I only look at threads in areas that I know).

    Let f(t,x) = \begin{cases} 0 & x\le 0 \\ 1 & x > 0\end{cases}\,\quad f(0,x) = 0

    u(x,t) = \int_0^t f(t,x-a(t-s))\, ds,\qquad a > 0

    So if x \le a(t-s),\; \int_0^t f(t,x-a(t-s))\, ds = \int_0^t\! 0\,ds
    x > a(t-s),\; \int_0^t f(t,x-a(t-s))\, ds = \int_0^t\! 1\,ds

    The solution is suppose to be
     u(x,t) = \begin{cases} 0 & x\le 0 \\ x/a & 0<x<at \\ t & x > at\end{cases}

    EDIT: Refer to other thread for solution.
    Last edited by lvleph; October 13th 2010 at 05:21 PM.
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