Question about singular integral eq.

[rugphy]

Hi,
I've questions about the Fredholm integral equation :
1. Is an following eq.
$a(x)y(x) + \int^{∞}_{-∞}\frac{\sin(x-y)}{x-y}dy = f(x)$
be defined as singular integral equation, if it is how can i get the solution?
2. What is the definition of singular kernel and is the kernel in (1)?

please help me or give me some suggestions/links to materials about these.

Thank you.

 

[mathman]

I suggest you look carefully at your post. The integral is constant (not dependent on x). I believe you need some function of y under the integral sign.

 

[rugphy]

Hi,
I've questions about the Fredholm integral equation :
1. Is an following eq.
$a(x)y(x) + \int^{∞}_{-∞}\frac{\sin(x-y)}{x-y}dy = f(x)$
be defined as singular integral equation, if it is how can i get the solution?
2. What is the definition of singular kernel and is the kernel in (1)?

please help me or give me some suggestions/links to materials about these.

Thank you.

Sorry, there are flaws in the above eq. It should be
$a(x)y(x) + \int^{∞}_{-∞}\frac{\sin(k(x-t))}{π(x-t)}y(t)dt = f(x)$
where a(x) and f(x) are known functions, and y(x) is unknown function.

 

[rugphy]

I suggest you look carefully at your post. The integral is constant (not dependent on x). I believe you need some function of y under the integral sign.

Thanks you very much for that remark. In future, I will do it more carefully.

 

[mathman]

It is called singular because the absolute value of the kernel is not integrable. Solving Fredholm equations is a major branch of analysis - there is no simple answer.