Questions tagged [integral-equations]
This tag is about questions regarding the integral equations. An integral equation is an equation in which the unknown function appears under the integral sign. There is no universal method for solving integral equations. Solution methods and even the existence of a solution depend on the particular form of the integral equation.
984
questions
3
votes
0
answers
62
views
Solution of equation with unknown under the integral
I have a problem which I have reduced to solving the following equation for the unknown $r_0$:
$$
1/2 = \int_0^D f(r)p(r,r_0)dr
$$
where $D \in \mathbb{R}$, and $f$ is continuous density function.
$p(...
- 103
0
votes
1
answer
79
views
Why are kernels often singular on the diagonal?
Many kernels/integral operators are given in terms of a function that is singular near the origin:
For example, the heat kernel on $\mathbb{R}^d$:
$$
\operatorname{K}\left(t,x,y\right) =
\frac{1}{\...
- 6,275
0
votes
0
answers
29
views
Linear integral equations of functions of two variables
This article by Peterson seems to provide a solution to an integral equation in two variables that I am interested in. However, I find the article hard to understand. Does somebody know of a more ...
- 57
1
vote
0
answers
103
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Can we say anything about this recursive integral equation? [closed]
Let $0 \le B(r)$ with $0 < r$ be truncated function, all real.
The truncation is \forall r_${max} \le r: B(r) =0; \forall r \le r_{min}: B(r) =0$
For $0<x$, define $I(x) = \int^x_1{B(r)dr}$ and ...
- 19
3
votes
4
answers
194
views
problem on double integral
Let $G:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be defined as
$$
G(t, x)=\begin{cases}
t(1-x), & \text { if } t \leq x \leq 1 \\
x(1-t), & \text { if } x \leq t \leq 1
\end{cases}.
$$
For ...
- 520
2
votes
0
answers
50
views
General solution for linear Volterra-like integral equation?
A linear Volterra integral equation looks like this (see the wiki)
\begin{align}
x(t) = f(t) + \int_0^t K(t, s)x(s)~\mathrm{d}s.
\end{align}
If the Kernel function $K$ is of the form $K(t, s) = K(...
- 138
6
votes
1
answer
287
views
Finding solutions to a complex integral equation
I would like to determine whether there exists some (holomorphic) function $f:\mathbb{C}\to
\mathbb{C}$ such that the following integral equation
$$
f\left( z \right) =\frac{C}{\left| z-z_0 \right|^{\...
- 569
4
votes
1
answer
107
views
Solve an integral equation using functional analysis
I'm trying to solve the following equation:
Is there a continuous function $f: [0,1] \rightarrow \mathbb{R}$ that satisfies
$$f(x) + \int_0^x e^{x \cos(t)}f(t) \ dt = x^2 + 1, x \in [0,1]$$ if so, ...
- 55
1
vote
0
answers
22
views
Confusion between Separable (degenerate) kernel and Convolution (difference) kernel [closed]
Let $k(x,t)$ denote kernel in Integral Equation.
Take $k(x,t)=e^{x-t}$. At first look, it seems convolution kernel. But It can be written as $e^{x}×e^{-t}$. Then, Is it separable kernel ?
Similarly, ...
1
vote
2
answers
103
views
An integral equation over two CDFs on the unit interval
I have $F_1,F_2$ two CDFs of random variables over $[0,1]$ and a number $0 < m < 1$. I'd like to somehow characterize the solutions to the constraint:
$$
\forall x, 0 < x < 1: m(1 -x) - m\...
- 217
2
votes
0
answers
13
views
Relationship between integration rule and numerical solution to Fredholm equation with non-unique solutions
In my research (computational physics) I need to solve a Fredholm integral equation of the form
$$ f(x) = g(x,x_i) + \mathcal P \int_0^\infty \frac{g(x,x')}{\tfrac{1}{2}\left(x_0^2-x'^2\right)}f(x') \,...
- 43
2
votes
1
answer
100
views
How to solve this integral equation $ \int_{-\infty}^\infty f(z)x^z dz = F(x)$ for f(x)?
My question is: solving $f(x)$ with known $F(x)$ and equation
$$ \int_{-\infty}^\infty f(z)x^z dz = F(x).$$
I met this problem when I tried to extend the idea of generating functions for discrete ...
- 239
0
votes
1
answer
26
views
IVP equal to integral equation
I have just recently started getting into differential equations and their solutions. Now I have discovered this theorem:
Let $m \in \mathbb{N}, I=[a,b] \subset \mathbb{R}, f: I \times \mathbb{R}^m \...
- 79
0
votes
0
answers
63
views
An integral equation involving bivariate Fourier transform
I was recently trying to solve one PDE, and in doing so I stumbled upon the following integral equation which I cannot aproach: find $\varphi$ such that
$$
\int_0^\infty \int_\mathbb{R} \sin(k a) \, e^...
- 320
1
vote
0
answers
29
views
Question about existence of solutions to integral equations of the first kind
We have three random variables $U, W, A$ and consider the integral operator. The integral operator $T$ is defined as $$Tf= \int f(w,u)p(w|a)dw = p(u|a). $$
for any fixed variable $u$, where $p(w|a)$ ...
- 31