Testing Weak Singularity of Integral Equations

[sara_87]

Homework Statement

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

Homework 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.

The Attempt at a Solution

a) I don't 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

 

[mighty2000]

.

Hello!

In order to determine whether a kernel is weakly singular or not, we need to check if it meets the criteria for weak singularity. This includes being continuous when x is not equal to s and having a certain decay rate when x is equal to s.

For part a), you are correct in saying that the kernel is not weakly singular because it is continuous at x=s. However, we also need to check the decay rate. In this case, the kernel has a polynomial decay rate of 1, which is not enough to be considered weakly singular (since v must be less than 1). Therefore, we can conclude that the kernel is not weakly singular.

For part b), you are correct in saying that the kernel is discontinuous at x=s. However, we need to determine the decay rate when x=s. In this case, the kernel has a decay rate of 1/2, which is less than 1. Therefore, we can say that this kernel is weakly singular with v=1/2 and c=1.

In order to provide a more thorough explanation, we can also mention that the decay rate of a kernel at x=s determines the strength of its singularity. A decay rate of 0 means the kernel is not singular, a decay rate between 0 and 1 means it is weakly singular, and a decay rate of 1 or higher means it is strongly singular. This can help us understand why the kernel in part b) is considered weakly singular with a decay rate of 1/2.

I hope this helps clarify the concept of weak singularity in kernels. Let me know if you have any further questions. Good luck with your studies!