The integral-equations tag has no wiki summary.
4
votes
2answers
886 views
Spectrum of Indefinite Integral Operators
I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities.
For the first, suppose $T:L^{2}[0,1]\rightarrow ...
7
votes
2answers
320 views
Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$
I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$.
Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$.
1st derivative: ...
2
votes
1answer
402 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 ...
1
vote
1answer
621 views
How to find eigenfunctions of a linear operator (follow-up question)
This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely.
I am interested in calculating characteristic ...
1
vote
4answers
271 views
Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$
I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$.
This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
3
votes
1answer
257 views
Is there a solution to this integral equation?
The problem is related to this question: How to find eigenfunctions
of a linear operator (follow-up
question) I posted earlier.
Suppose I want to solve the following integral equation:
$$\int_0^1 ...
3
votes
1answer
298 views
How to find eigenfunctions of a linear operator
I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral.
As a special case suppose I am interested in this function:
...
3
votes
2answers
163 views
Can we bound from above sub-solutions of Volterra integral equations?
Let us review Gronwall's lemma. If $v \in C^0([\tau, T])$ is such that
$$v(t) \le c + \int_{\tau}^t u(s)v(s)\, ds,\qquad t \in [\tau, T]$$
where $c$ is a real constant and $u \in C^0([\tau, T])$ is ...
2
votes
2answers
82 views
Solution of an integral equation $\phi(x)+\int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $
Solve the following integral equation:
$\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $
I need to solve the integral equation above. Can anyone help me please?
1
vote
2answers
203 views
Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$
Integral equation
$$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has:
a unique solution for $\lambda \neq \frac{4}{\pi +2}$;
a unique solution for $\lambda \neq ...
0
votes
1answer
99 views
Solving integral equation $a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$
I've got this nasty-looking integral equation involving taking two minimums:
$$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$
where $\delta(\cdot)$ is the ...
