# Integral equation

1. Jan 18, 2010

### sara_87

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) $$\int_0^x sin(x-s)y(s)ds$$

b) $$\int_{-3}^3 \frac{y(s)}{x-s}ds$$

2. Relevant equations

A discontinuous kernel k(x; s) is weakly singular (at x = s) if k is continu-
ous when x $$\neq$$ s and if $$\exists$$ constants v$$\in$$ (0; 1) and c > 0 such that $$\left|k(x,s)\right|\leq c \left|x-s\right|^{-v}$$ for x $$\neq$$ 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