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.
969
questions
1
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0
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26
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Crystallization Process Integral Equation with Arrhenius Functions
I am working on a mathematical model describing the crystallization process, represented by an integral equation involving the crystalline fraction $\xi_V(T, t)$, where $T$ is the temperature and $t$ ...
0
votes
2
answers
103
views
How do I solve this differential-integral equation? [closed]
The following equation has come up in my research and I am lost at where to start. I have tried guessing forms of the solution and Mathematica is not helpful. Any help pointing me in the right ...
1
vote
1
answer
41
views
ring meniscus at cylinder
I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry ...
54
votes
3
answers
4k
views
A very odd resolution to an integral equation
Here is something I've found on the internet
$$\begin{aligned}
f-\int f&=1\\
\left(1-\int\right)f&=1\\
f&=\left(\frac1{1-\int}\right)1\\
&=\left(1+\int+\int\int+\dots\right)1\\
&=1+...
0
votes
0
answers
37
views
Existence of a Spatial Curve with normal vector constant angle to the $z$-Axis?
Is it possible to find a spatial curve $\alpha(s) = (x(s), y(s), z(s))$ such that the lines containing the normal vector $\hat{n}(s)$ at each point on the curve $\alpha(s)$ intersect the $z$-axis at a ...
1
vote
1
answer
81
views
Is there a non-trivial function satisfying $\int_0^1 (f(x)+x)^a \, dx=0$ for $a$ a positive real (or integer)?
Here is the functional equation:
$$\int_0^1 (f(x)+x)^a \, dx=0$$
Is there a function $f(x)$ that satisfies it for any $a>0$ and $f(x)\not\equiv-x$?
What about the case where $a$ is an integer?
0
votes
0
answers
41
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Continuous dependence of spectrum of Fredholm integral operator on parameters
Let $(X,\mathcal B(X),\lambda)$ be some Euclidean space $X\subset \mathbb R^n$ equipped with the Lebesgue measure $\lambda$. Suppose we have a Fredholm integral operator $$T:L^1(X)\to L^1(X):f(\cdot)\...
0
votes
1
answer
92
views
Lower bound of a trigonometric integral
Let $\alpha$, $\beta$, and $\gamma$ be non-zero real numbers. Further, suppose that $f$ and $g$ are probability density functions defined on $\mathbb{R}$. I'm interested in computing a lower bound of ...
4
votes
0
answers
204
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What are the conditions for solution of nonlinear Fredholm equations with Banach fixed point theorem?
Consider the nonlinear Fredholm integral equation of the second kind:
$$
\varphi(x) = f(x) + \lambda \int_a^b F(x, t, \varphi(t)) \, dt
$$
where
$(f)$ and $(K)$ are given functions,
$(a, b)$ are ...
0
votes
0
answers
19
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Non-local boundary condition and integral equations
I'm solving initial value problem with non-local boundary condition $u(0,t)=\int_{0}^{l}\beta(s)u(s,t)ds = \gamma(t)$. I have already found function u for to cases $x<t$ and $x>t$. But I have ...
0
votes
0
answers
52
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Initial value problem with non-local boundary condition
I am solving that problem:
$$u_t+u_x+\sigma(x)u=g(x), x\in[0,l], t\ge0$$
$$u(x,0)=\phi(x)$$
$$u(0,t)=\gamma(t)=\int_{0}^{l}\beta(s)u(s,t)ds$$
I have already solved initial value problem with $\gamma(t)...
0
votes
0
answers
55
views
How to solve this coupled PDE eigenvalue problem numericallly?
I'm solving a system of two coupled partial integro-differential equations for two functions $ {\phi _0^\alpha (R,r')} $ and $ {\phi _0^\beta (R,r)} $:
$$ \frac{{{\partial ^2}\phi _0^\alpha }}{{\...
0
votes
1
answer
133
views
How to find the solution for this integral equation?
I'm trying to solve this integral equation:
$$\int_0^\infty \frac{t^{zi}+t^{-zi}}{e^t}dt=0$$
For the least value of $z$. This is, if $S=\{z_0,...,z_n\}$ is a set of solutions, and $|\lambda|=\mathrm{...
1
vote
1
answer
140
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Question about Salem's integral equation reformulation of Riemann hypothesis
Consider an integral equation:
$$\int_{-\infty}^{+\infty}\frac{e^{-\sigma y}f(y)}{e^{e^{x-y}}+1}dy=0$$, where $\sigma\in(\frac{1}{2},1)$
Salem proved that this equation has no bounded solution other ...
3
votes
1
answer
87
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Can you find the the function given the value of its definite integral
While solving a physics problem , I came across a equation that looks like this, $$\int_0^\pi \sigma(\theta)\,\sin(2\theta)\,d\theta=\frac{q}{\pi a^2} $$
Is there a way to solve for $\sigma(\theta)$. ...
1
vote
1
answer
54
views
Solution of a homogeneous first kind integral equation with linear kernel
I have a very simple integral equation I wish to solve, but I cannot put my finger on the appropriate method which isn't overkill for such a simple problem. I feel like I am missing something very ...
0
votes
1
answer
75
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How to numerically solve a non-Markovian integral equation?
In the book Breuer, Heinz-Peter, and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford: Clarendon Press, 2009., there is equation(10.17)
$$
\frac{\mathrm{d}}{\mathrm{d}t}c_1(t) = - \...
1
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1
answer
52
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Understanding Method of Successive Approximation for finding the unique solution to Volterra Integral Equation
Consider this non-homogeneous Volterra Integral Equation of second kind,
$$u(x)=f(x)+\int_a^{x}K(x,\xi)\,u(\xi)\,d\xi$$
where, $f(x)$ and $K(x,\xi)$ are non-zero real valued continuous functions ...
0
votes
1
answer
34
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Inverse Integral problem: substituting infinite upper bound with constant
In the handbook of integral equations, there are provided solutions for two integrals involving functions $f(x)$ and $y(t)$:
Equation 1: Given $$f(x) = \int_{a}^{x} \frac{y(t) \ dt}{\sqrt{x^2 - t^2}},$...
2
votes
2
answers
151
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Solve the integral equation $f(x)-\lambda\int^{1}_{0}\min(x,t)f(t)dt=\sin(\frac{\pi N x}{2})$
In my studies of fractional analysis I have been solving problems to familiarize with the concepts of fredholm theory, and I found the following problem which I have been having problems solving:
For ...
0
votes
1
answer
83
views
Find the spectrum of the following operator: $Af(x)=\int^{\pi}_{0}\sum^{\infty}_{n=1}3^{-n}\cos(nx)\cos(nt)f(t)\,dt$
In a book of functional analysis I encountered this problem
Find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 3^{-n} \cos(nx)\cos(nt) f(t)\,dt$ in $L_2[0,\pi]$.
To do this I just ...
0
votes
1
answer
65
views
How to solve this Fredholm integral equation of the second kind: $\ f(x) - \lambda\int\limits_{0}^{1} 5x^{2}t^{2}f(t) dt = 4x +bx^{2}\ $
Recently I have been studying Fredholm theory to solve integral equations, and as I am trying to familiarize myself with the theory I have been trying to solve certain problems.
This one is one of ...
1
vote
0
answers
54
views
Constrained solution to Fredholm equation with Lagrange multipliers
I am numerically solving a Fredholm equation of the second kind:
\begin{align}
f(x) &= g(x) + \int_0^\infty K(x,t)f(t)\,dt,
\end{align}
by using a Gaussian quadrature rule to convert it into a ...
1
vote
1
answer
59
views
What kernels are unitary
The Fourier transform is a integral transform with kernel $e^
{−2πiξx}$. The Fourier transform is unitary in that it preserves the $L2
$ norm.
Is there a general way to show or guess that a kernel is ...
1
vote
1
answer
78
views
Existence of solution to Fredholm integral equation of the second kind under a condition on the spectral radius
Consider the Fredholm integral equation of the second kind given by
$$
f(x)=g(x)+\int_a^bk(x,y)f(y)\ \mathrm dy.
$$
In any source I could find online, including some more advanced ones, existence of ...
1
vote
2
answers
189
views
How to solve system of integral equations? [closed]
Suppose we have the following 'system' of integral equations:
$$\int_0^1 e^x f(x) dx=0$$
and
$$\int_0^1 e^{2x} f(x)=0.$$
Is there a non-zero bounded integrable solution $f(x)=0$ satisfying both ...
2
votes
0
answers
65
views
Solve and integral equation with symmetric kernel [closed]
I have the following integral equation with symmetric kernel
$$
x(t)=\sin(\pi t)+\pi \cos (\pi t) +\lambda \int_{0}^{1} k(t,s)x(s)\,ds $$
where $k(x,t)$ is a symmetric kernel given by
$$k(t,s)=
\...
1
vote
0
answers
56
views
Statistical inference for the integral equation
Consider a integral equation
$$
\begin{aligned}
\mathbb{E} \left[ Y|A \right] &=\mathbb{E} \left[ g\left( W \right) |A \right]\\
\int{yp\left( y|a \right) dy}&=\int{g\left( w \right) p\left( ...
0
votes
0
answers
40
views
Solution of the Volterra integral equation of the 2nd kind
Please tell me where I made a mistake? Or maybe I used the wrong method to solve it?
Link to my attempted solution: https://ru.overleaf.com/read/xcxsthdjpnmx#17beab
For the successive approximation ...
5
votes
2
answers
261
views
System of integral equations describing probability
During my work on my thesis, I've stumbled upon the following problem:
Let $f_1$ and $f_2$ be some arbitrary PDFs with support $\mathbb{R}$. Does there exist a joint bivariate distribution $f_r(x, y)$,...
0
votes
0
answers
47
views
Continuous dependence on initial conditions of Fredholm integral equation of the second kind
In several papers and other sources, I have seen statements about it being `well-known' that the Fredholm integral equation of the second kind is well-posed, in contrast to a Fredholm integral ...
0
votes
0
answers
49
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Fredholm integral equation, exercise 12 Functional analysis Kreyszig
I'm trying to do a exercise of Kreyszig book of functional analysis but I'm stuck, I'm trying to solve the integral equation
\begin{equation}
x(s)-\mu \int_{0}^{2\pi}sin(s)cos(t)x(t)dt =\hat{y}(s)
\...
0
votes
0
answers
37
views
Seeking Detailed Explanation for Transforming an Integral Equation Using Euler's Formula and Error Function
I am working on understanding the transformation of a specific integral equation into a simpler form using Euler's Formula and the Error Function. The original equation is:
$$
u(x, t) = u_0\left\{1 - ...
1
vote
3
answers
182
views
Solve $f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau$
The question:
$$\begin{equation*} f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau \end{equation*}.$$
I believe we need to take the Laplace transform of all terms. I am getting stuck with this part:
...
1
vote
2
answers
54
views
Dimension of null space of an operator $T$
Let, $K(x,y)$ be a kernal in $[0,1]\times [0,1]$ defined as $K(x,y)=\sin(2\pi x)\sin(2\pi y)$. Consider the integral operator
$$T(u)(x)=\int_0^1 u(y)K(x,y)\,dy$$
where, $u\in C[0,1]$. Which of the ...
1
vote
0
answers
50
views
Is there a general procedure to solve numerical integral equations with non-elementary integrals?
I'm feeling a little lost in trying to solve equations of the form:
$$f(x)+\int_a^b\phi(x,t)\mathrm{d}t=0$$
Where the integral in the LHS is non-elementary, and the variable $x$ is the unknown.
If the ...
0
votes
0
answers
140
views
Fredholm equation of the second kind with a quotient kernel
I'm trying to find a solution to a Fredholm equation of the second kind of the form
$$f\left(x\right)=g\left(x\right)+\lambda\intop_{a}^{b}\mathcal{K}\left(\frac{t}{x}\right)f\left(t\right)\mathrm{d}t....
0
votes
0
answers
22
views
Solving for generalized eigenfunction of weakly divergent integral operator
I'm interested in solutions to the generalised integral eigenfunction equation
$$
f(z^2,d-1)=2 \int_{z}^\infty \frac{f(r^2,d)}{\sqrt{1-\frac{z^2}{r^2}}}dr = \int_{z^2}^\infty \frac{f(y,d)}{\sqrt{y-z^2}...
1
vote
0
answers
69
views
Is it possible to write any arbitrary partial differential equation as an integral equation?
Note that I am not a mathematician; I am simply deducing using the very fallible means of deduction via intuition, which by no means is rigorous.
My question concerns the possibility for any PDE to be ...
3
votes
1
answer
125
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Fredholm Integral Equation of the Second Kind in $L_2[0,1]$
Given space $L_2[0,1]$ and the equation
$$\displaystyle x(t) + \lambda \int_{0}^{1}(\frac{1}{2} - |t-s|)x(s) ds = \cos(\pi*t)$$
And I want to find a solution to the equation for all values $\lambda \...
2
votes
0
answers
147
views
Eigenfunctions of the integral kernel 1/(x^2 + x'^2)
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
2
votes
0
answers
66
views
Cauchy integral equation with derivative
Does anybody know the solution of this singular Cauchy-like integral equation:
$$
y(x) = \int_{-\infty}^{\infty} \frac{y'(x')}{x-x'}dx'\\
y(0) = 1, \lim_{ \lvert x \rvert \to \infty } y(x)= 0
$$
The ...
1
vote
0
answers
51
views
How to calculate the kernel of an integral given the original function and its product [closed]
I am trying to solve for the kernel of the following integral.
$\int_{-\infty}^{\infty}K(x,t)f(t)dt = g(x)$
I know g(x) and I know f(x) but I am unsure of how I may solve for the kernal. I am trying ...
0
votes
0
answers
72
views
Solve the integral equation for $f$. [duplicate]
Find all functions $f$ that satisfy:
$$\int f(x)dx \cdot \int (1/f(x))dx = -1$$ So far, I have tried a handful of methods. I have substituted a variable $u$ for $f(x)$, I have substituted an ...
0
votes
0
answers
50
views
Existence of solution to "weird" integral equation
in my current work I come across an integral of the form
\begin{align}
x(a) = \int_{\Omega} f(a, u) x(u) du
\end{align}
where $\Omega$ is $\Omega \subseteq \mathbb{R}$, e.g. $\Omega = (0,1)$. The ...
1
vote
1
answer
85
views
About a counterexample for an integral-functional equation in number theory.
I was reading
http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf
How did the counterexample for the equation on page 8 look like ??
Specificly :
(quote)
“Tur´an’s lecture (probably a quite ...
1
vote
1
answer
69
views
Asymptotic solution of an integral equation
Consider an integral equation of the form
$$\sigma_B(\Lambda)+\int_{-B}^{B} K\left(\Lambda-\Lambda^{\prime}\right) \sigma_B\left(\Lambda^{\prime}\right) d \Lambda^{\prime}=f(\Lambda)$$
where the ...
0
votes
2
answers
165
views
Fredholm integral equation of the second kind with constant kernel
I'm trying to read Kress' Linear integral equations, and I'm stuck at the first example. There must be something obvious I'm missing, and to that end, should I read something before this text?
$f(x)=\...
0
votes
1
answer
114
views
Solution of a simple integral equation [closed]
I have the following integral equation:
$$
f(x)=\exp{\left(-\int_{-x}^{\infty}f(y)\,dy\right)}\,\,,
$$
where a condition on $f(x)$ holds:
$$
f(x=0)=\frac{1}{2}\,.
$$
I know that the solution is:
$$
f(...
4
votes
1
answer
80
views
An analytical solution of the integral equation $ \int_0^\rho \left( \frac{s}{\rho} \right)^3 f(s) \, \mathrm{d}s +\int_\rho^1 f(s)\,\mathrm{d}s=1$
While elaborating on the solution for the Green's function of a mechanics problem involving disks moving on an interface, I came across the following integral equation for the unknown function $f(s)$:
...