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A question on integral equations

  1. Sep 24, 2003 #1
    Let be the integral equation:

    f(x)-g(x)=Kf where K is the integral operator having the kernel

    my question : is there an operator R satisfying:

    g(x)-f(x)=Rg where f satisfy the original integral equation so it can be solved by mean of R operator.


    Another question is can any integral equation be transformed into a differential equation?..if so how it is made?..thanks.
     
  2. jcsd
  3. Sep 24, 2003 #2

    Tom Mattson

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    Staff Emeritus
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    Gold Member

    OK

    Just using algebra, I get:

    1. f(x)-g(x)=Kf(x)
    2. f(x)-Kf(x)=g(x)
    3. (1-K)f(x)=g(x)
    4. f(x)=(1-K)-1g(x)
    5. g(x)-f(x)=[1-(1-K)-1]g(x)

    Defining R as [1-(1-K)-1] seems to fit the bill. The only thing I am not sure of is whether (1-K) is invertible. I assumed that it was invertible in Step 4.

    edit: fixed bracket
     
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