1
vote
0answers
27 views

eigenvalues of homogeneous integral equation of second kind, with singular kernel

There is a homogeneous integral equation of second kind with a singular kernel(non-symmetric). The equation has the form: $\int_{a}^{b} k(x,t)Γ(t)dt =λΓ(x).$ It's 2-norm is infinity, $||k(x,t)||_2 ...
0
votes
0answers
29 views

How to Solve This Special Case of Multidimensional Integral Operator?

I'm dealing with an integral equation of the following form: $1 = f(x)\int dy f(y)B(x,y)$ where $B(x,y)$ is a known function, and I want to solve for $f(x)$. If I treat $f(x)f(y)$ as one big unknown ...
3
votes
0answers
117 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
2
votes
0answers
315 views

Homogeneous Fredholm Equation of Second Kind

I'm trying to show that the eigenvalues of the following integral equation \begin{align*} \lambda \phi(t) = \int_{-T/2}^{T/2} dx \phi(x)e^{-\Gamma|t-x|} \end{align*} are given by \begin{align*} ...
0
votes
1answer
190 views

Linear versus non-linear integral equations

I'm having trouble solving an integral equation. It appears to me to be a homogeneous Fredholm equation of the second kind. However, I'm being told that this can't be a Fredholm equation, because it ...
3
votes
1answer
538 views

Eigenvalues of an operator

I think this question isn't that hard, but I am a bit confused: Define $$(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$$ Then $A$ is an operator on functions. Find the eigenvalues and the ...