Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Solving an integration equation with unknown kernel

  1. Jul 13, 2013 #1
    I am pondering over how to solve the following (seemingly nonstandard) integral equation.

    Let h(t) be a known function which is non-negative, strictly increasing and satisfies that h(t) → 0 as t→-∞ and h(t)→1 as t→∞. Indeed, h(t) can be viewed as a cumulative distribution function for a continuous random variable.

    Let K be a given positive integer. The problem is to find the function f(x) satisfying that

    h(t) = ∫(F(t+x)^K)f(x)dx

    where the integral is taken from -∞ to ∞, F is a functional of f and satisfies F(y)=∫f(x)dx where the integral is taken from -∞ to y. The function f to be solved is positive with support on the whole real line and is integrated to 1 over its support. Indeed, f is viewed as a density of a continuous random variable whose cumulative distribution function is F. There is a location constraint on the solution f that should satisfy that ∫xf(x)dx=0. Namely, the expectation of the random variable with density f should be zero.

    The integral equation seems nonstandard since the kernel F(t+x) is unknown though it depends only on f.

    Could anyone suggest how to deal with this problem ?

    Thank you very much !
     
  2. jcsd
  3. Jul 13, 2013 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    What does the corresponding differential equation say? (I'm not implying that this is a good hint, I'm just curious.)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Solving an integration equation with unknown kernel
Loading...