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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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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(...
Ollie's user avatar
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0 votes
2 answers
67 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}{\...
CBBAM's user avatar
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0 votes
0 answers
27 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 ...
Sela Fried's user avatar
-1 votes
0 answers
29 views

Linear Integral equation (proof) [closed]

So I was just going through linear Integral equation chapter there is a point which is mentioned as: If $\lambda \in Q(A)$ and $|\lambda-\lambda_o||R_{\lambda_o}|<1(\lambda\in C)\implies\lambda \in ...
πααρτθ Σαρθι's user avatar
1 vote
0 answers
103 views

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 ...
1m1's user avatar
  • 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 ...
Ricci Ten's user avatar
  • 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(...
Lyle's user avatar
  • 138
6 votes
1 answer
286 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|^{\...
MathLearner's user avatar
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, ...
AlexH's user avatar
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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, ...
Jacen Bridger's user avatar
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\...
Martin Modrák's user avatar
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') \,...
quixedjetr's user avatar
2 votes
1 answer
99 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 ...
Jie Zhu's user avatar
  • 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 \...
metamathics's user avatar
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^...
tsnao's user avatar
  • 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 $p(w|a)$ ...
叶心萤's user avatar
0 votes
0 answers
43 views

Solution to convolution integral equation

I have two positive definite kernels $k_1,k_2 : \mathbb R^d \to \mathbb R$ and a probability measure $\rho$ on $\mathbb R^d$. The kernels are of the form $k_i(x,y)=K_i(x-y)$ for some function $K_i:\...
Anson's user avatar
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1 vote
0 answers
67 views

An analytical solution of the integral equation $ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz $

How to solve for the f(x) in the integral equation? $$ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz \\ \alpha \in (1,2), C_2\text{ is a positive constant.} \\ \...
YuanLan's user avatar
  • 11
1 vote
0 answers
97 views

Convolution equation equals a constant

How would you solve this? I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, $f(t)=3+t^2/2$, does not ...
mathboyexpert1010's user avatar
1 vote
0 answers
39 views

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
110 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
48 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 ...
creillyucla's user avatar
58 votes
3 answers
5k 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
  • 3,881
0 votes
0 answers
38 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 ...
maplemaple's user avatar
  • 1,231
0 votes
1 answer
91 views

Is there a non-trivial function satisfying $\int_0^1 (f(x)+x)^a \, dx=0$ for $a$ a positive real (or integer)? [closed]

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
  • 1
0 votes
0 answers
70 views

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
106 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
211 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
0 votes
0 answers
21 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
0 votes
0 answers
57 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
0 votes
0 answers
74 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
  • 1
0 votes
1 answer
143 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
148 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
  • 373
3 votes
1 answer
95 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
70 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
81 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
  • 27
1 vote
1 answer
85 views

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
  • 157
0 votes
1 answer
37 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
175 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
  • 608
0 votes
1 answer
97 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
  • 608
0 votes
1 answer
93 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
  • 608
1 vote
0 answers
59 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
74 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
  • 337
1 vote
1 answer
112 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
2 votes
0 answers
75 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
  • 311
1 vote
0 answers
58 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
76 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
263 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
56 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
52 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

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