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.
1,018 questions
13
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4
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A very weird integral equation
I tried to come up with an integral equation for fun, and made this creature:$\def\d{\mathrm d}$
$$
f(x)-\int_x^{2x}f(t)\,\d t=0,
$$
so I followed these steps:
$$
\begin{aligned}
&f(x)+\int_x^{2x}...
- 131
1
vote
2
answers
126
views
Volterra Integral Equation
I have found a Volterra integral equation of the first kind of the following type:
$$y(r)=\int_0^r\,K(t,r)\,f(t,r)\,\mathrm dt,$$
in which my unknown function is $f(t,r)$. I have tried to find ...
- 91
0
votes
1
answer
62
views
Proving a limit subject to an integral equation
Given a specific integral equation \begin{equation}
f(x) = 2- \frac{1}{\pi}\lim_{C\rightarrow \infty} \int^C_{-C}\frac{f(y)}{(x-y)^2+1} d y,
\end{equation}
I want to show that $$\lim_{C\rightarrow ...
- 855
1
vote
0
answers
107
views
Does multi-rate update depth bound the Volterra degree?
Fix a discrete-time system with input sequence $(x_t)_{t\ge 0}\subset\mathbb{R}^d$, output $(y_t)_{t\ge 0}\subset\mathbb{R}^p$, and $L$ internal levels. For each level $\ell\in\{1,\dots,L\}$ choose a ...
- 123
0
votes
0
answers
51
views
Find a function defined for all inner points of a unit $3D$ ball such that its integral over nonempty intersection of unit ball with any plane is $1$
We are looking for a continuous function defined for all inner points of a unit $3D$ ball (sphere) such that its integral over nonempty intersection of unit ball with any plane is $1$. This question ...
- 824
3
votes
1
answer
130
views
Find continuous function of two variables defined on unit disc that has integral $1$ over any chord of unit circle
If we assume that our function depends only on distance from origin we can come up with differential equation for new function of one variable (distance from origin), I was doing it once but could not ...
- 824
1
vote
0
answers
97
views
Abel type integral equation
I am trying to solve the following Abel type integral equation, in which $f\left( s\right)$ is not a symmetric function:
$$\int_{-r}^{r}\frac{f\left( s\right) }{\sqrt{r^{2}-s^{2}}} \ ds=g\left(
r\...
- 91
0
votes
0
answers
33
views
Applications of matrix-valued differential / integral / integro-differential equations
I am considering the following type of equation
$$ A(t) = F(t) + \int_0^t \mu(t,s) A(s) \, {\rm d} s, \qquad 0\le t\le T, $$
where $F : [0,T] \to M_n, \mu: \Delta \to M_n$ are given continuous ...
- 41
1
vote
0
answers
91
views
Hint to solve integral equation
I have found the following integral equation:
$$\int_{a}^{a+r}\frac{f\left( s\right) ds}{\sqrt{s-c}\sqrt{a+r-s}}=-\frac
{\mu\pi}{2R}\left( a+c-r\right) \text{, }a>0 \text{, }r>0 \text{ and }...
- 91
5
votes
1
answer
116
views
How to Derive the Closed-Form Kernel of an Integral Equation Without Relying on Pattern Matching?
I'm studying the book Integral Equations by M. Rahman.
To solve the integral equation
$$
u(x) = f(x) + \lambda \int_0^x e^{x - t} u(t) \, dt,
$$
the resolvent kernel method gives the solution:
$$
u(x) ...
- 2,894
2
votes
0
answers
108
views
The spectrum of a rank one update to a compact, self adjoint operator
Consider the Hilbert-Schmidt operator $K$ defined on $L^2([0,T])$: \begin{align}
Kf(\tau) = \int_0^T \left(\mathrm{e}^{-\frac{|\tau-s|}{a}}-\mathrm{e}^{\frac{\tau + s - 2T_0}{a}}\right)f(s)\,\mathrm{d}...
- 82
4
votes
1
answer
166
views
Find $f$ such that $\int_0^1 f(x) \mathrm{d}x = \sum_{k=1}^{\infty} f(k)$
The Question
I was just watching this video that proves the result
$$
\int_0^1 x^{-x} \mathrm{d} x = \sum_{k=1}^{\infty} k^{-k}
$$
(So we already know one possibility: $f(x) = x^{-x}$)
But I was ...
- 314
6
votes
2
answers
226
views
Does $f \in C(0,1]$ and $f(x) + \int_0^x \frac{f(y)}{\sqrt{x-y}} dy=0$ implies $f = 0$?
Suppose that $f$ is continuous function on $(0,1]$ and satisfies
$$
f(x) + \int_0^x \frac{f(y)}{\sqrt{x-y}} dy = 0, \qquad x \in (0,1].
$$
Does it follow that $f(x) = 0$ for $x \in (0,1]$?
The ...
- 4,826
6
votes
1
answer
119
views
Surface charge distribution on a conductor - does it exist mathematically?
Physics told us that when a conductor reaches the electrostatic state
all charges reside on its surface, and
the electric potential is constant within the conductor
Viewing this statement from a ...
- 12k
0
votes
1
answer
77
views
Non-local Schrodinger-like equation and its spectrum
Consider the following definition of non-local Laplacian operator,
$$\Delta^K u(x) = \int_{\Omega} dy\,K(x,y)\left(u(y)-u(x)\right),\tag{1}$$
where $K(x,y)=K(y,x)$ is measurable and "good" ...
- 506