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Homework Help: Integral equation

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data

    If we want to show whether a kernel is weakly singular or not, what do we do?

    eg. consider:

    a) [tex]\int_0^x sin(x-s)y(s)ds[/tex]

    b) [tex]\int_{-3}^3 \frac{y(s)}{x-s}ds[/tex]

    2. Relevant equations

    A discontinuous kernel k(x; s) is weakly singular (at x = s) if k is continu-
    ous when x [tex]\neq[/tex] s and if [tex]\exists[/tex] constants v[tex]\in[/tex] (0; 1) and c > 0 such that [tex]\left|k(x,s)\right|\leq c \left|x-s\right|^{-v}[/tex] for x [tex]\neq[/tex] s on its set of defnition.

    3. The attempt at a solution

    a) I dont think this is weakly singular because when x=s, the kernel is continuous.

    b) when x=s, the kernel is discontinuous because the equation tends to infinity. and we can say that v=1/2 and c=1.

    I feel like this is wrong. but even if it's right, i think it lacks explanation
     
  2. jcsd
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