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.

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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 $t$ ...
Josef Resl's user avatar
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 ...
user1297645's user avatar
1 vote
1 answer
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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 ...
creillyucla's user avatar
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+...
Alma Arjuna's user avatar
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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 ...
maplemaple's user avatar
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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?
Anixx's user avatar
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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)\...
Václav Mordvinov's user avatar
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 ...
user775349's user avatar
4 votes
0 answers
204 views

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 ...
Olga Gonzalez's user avatar
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0 answers
19 views

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 ...
Kyle Crane's user avatar
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0 answers
52 views

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)...
Kyle Crane's user avatar
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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 }}{{\...
HERMIT's user avatar
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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{...
Simón Flavio Ibañez's user avatar
1 vote
1 answer
140 views

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 ...
stephan's user avatar
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3 votes
1 answer
87 views

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 $\sigma(\theta)$. ...
Al-Ahsan Abhro's user avatar
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 ...
Silver Pages's user avatar
0 votes
1 answer
75 views

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) = - \...
ZQW's user avatar
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1 answer
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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 ...
mat09's user avatar
  • 101
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1 answer
34 views

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}},$...
David khoder's user avatar
2 votes
2 answers
151 views

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 ...
AdrinMI49's user avatar
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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 ...
AdrinMI49's user avatar
  • 584
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 ...
AdrinMI49's user avatar
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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 ...
quixedjetr's user avatar
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 ...
jrudd's user avatar
  • 327
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 ...
Václav Mordvinov's user avatar
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 ...
stephan's user avatar
  • 327
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)= \...
C L 's user avatar
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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( ...
叶心萤's user avatar
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 ...
Mark's user avatar
  • 1
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 $f_r(x, y)$,...
NikoWielopolski's user avatar
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 ...
Václav Mordvinov's user avatar
0 votes
0 answers
49 views

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) \...
scottish's user avatar
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 - ...
Rob_'s user avatar
  • 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: ...
user5587's user avatar
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 ...
Empty's user avatar
  • 13k
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 ...
Simón Flavio Ibañez's user avatar
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....
Roy's user avatar
  • 1
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}...
AnotherShruggingPhysicist's user avatar
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 ...
JS4137's user avatar
  • 123
3 votes
1 answer
125 views

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 \...
Margaret's user avatar
  • 107
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} \...
Yuli Nazarov's user avatar
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 ...
Dimitry Chuprakov's user avatar
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 ...
JustAnotherGuyOnline's user avatar
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 ...
Brody Cates's user avatar
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 ...
Red's user avatar
  • 36
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 ...
mick's user avatar
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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 ...
user824530's user avatar
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)=\...
LiquidMikerrs's user avatar
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(...
Ruth Murphy's user avatar
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 $f(s)$: ...
keynes's user avatar
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