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Questions tagged [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. (Handbook of Mathematics - I.N. Bronshtein · K.A. Semendyayev · G.Musiol · H.Muehlig)

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How to solve $F(t) = a\sin{t} - 2 \int^1_0 F(u) \cos(t - u) du$

How to solve $F(t) = a\sin{t} - 2 \int^1_0 F(u) \cos(t - u) du$ The answer is $F(t) = at e^{-t}$ I am clueless about these types of integrals where we have variables on both sides of equation. ...
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Differentiating an Integral Equation

We're trying to solve the following problem by converting to differential equations: $$\phi(x) = x - \int_0^x(x-s)\phi(s)\,ds$$ We can differentiate both sides and use the product rule and the ...
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How many solutions does equation $\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$ have on the segment [0, 3]?

The task i'm trying to solve is: How many solutions (roots) does equation have: $$\int\limits_x^{x+\frac{1}{2}} \cos \left( \frac{t^2}{3} \right) dt = 0$$ on the segment [0, 3] ? By the moment i'...
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A Homogeneous Fredholm Equation of Second Kind

in my probability research I encounter the following integral equation for continuous non-negative $f: (0,\pi/4] \to \mathbb R$: $$ f(\varphi) = \int_0^{\pi/4} \frac {4} {\pi} \sin \varphi_0 \cos \...
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Efficient method to find $H$ given by $H(x)=\int_0^x f(x-u) f(x-au) e^u \, du$

Question Let $f:[0,\infty) \rightarrow [0,\infty)$ be some continuously differentiable function and $a \in (0,1)$ then we define the function $H:[0,\infty) \rightarrow [0,\infty)$ by letting: $$H(x)=\...
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Prove the inverse of integral operator exists and bounded by using eigenvalue

This theorem is taken from Linear Integral Equation by Rainer Kress. $\textbf{Theorem:}$ bounded integral operator $T_0:C^{1,\alpha}[0,2\pi]\to C^{0,\alpha}[0,2\pi]$ defined by $(T_0\psi):=\frac{1}{...
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Solving Integral Equation by Converting to Differential Equations

Consider the problem $$\phi(x) = x - \int_0^x(x-s)\phi(s)\,ds$$ How can we solve this by converting to a differential equation?
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Resolvent Kernel of a general Fredholm Equation of second kind

I'm trying to solve the following problem: Find the resolvent kernel, and solve the integral equation: \begin{equation} u(t) = p(t) + \int_{0}^{x}a(t)b(s)u(s) ds. \end{equation} Here we have an ...
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What extra condition must be satisfied by the kernel to find the unknown function in this system of equations? How?

There are two integral equations of $$\alpha(x,y)=\int_x^y K(x,y,t)U(t)dt$$ $$\beta(x,y)=\int_x^y K(x,y,t)U(t)tdt$$ where $K(x,y,t)$ is a non-negative known function which $t$ lives in $[y,x]$; $U(t)$...
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Integral equation with convolution kernel and non-stationary kernel

I have encountered this kind of integral equation: $$\int f(t')g(t-t')dt'=\int f(t')h(t,t')dt'\quad ,$$ where $f,g,h$ are smooth functions and $f$ is arbitrary. Can $h$ be related directly to $g$ ...
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How to solve differential equation that contains an integral term

I'm trying to find an analytical solution for the following differential equation that contains an integral term: \begin{align} \frac{d{v}_p(t)}{dt} &= \alpha + \beta\frac{d{v}_f(t)}{dt} + \...
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Integral equation equal to a constant from aerodynamics application

I am stuck on part of my aerodynamics textbook which involves an integral equation. I would like some help understanding how to solve the equation. Problem Statement Find $\gamma(\theta)$ such that $...
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70 views

Homogeneous Fredholm integral equation with asymmetric logarithmic kernel

In the study of few-particle quantum systems, I have come upon the following integral equation: \begin{equation} f(x) = \lim_{\Lambda\rightarrow \infty}\frac{3\sqrt{3}}{\ln(4x^2+6\lambda) + 2\gamma} \...
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1answer
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What is the final value when moving $x$ by an infinitesimal percentage of $f(x)$ until $100$%?

Let $f : \mathbb{R}\to\mathbb{R}_+$ be a smooth function. Given some $x_0\in\mathbb{R}$ and a direction $s\in\{-1,1\}$, I'm interested in image of $x_0$ under $T_n$ composed with itself $n$ times, i.e....
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Find the solution for the Integral equation

I have an inegral equation like this $\qquad n(\phi)=\int_0^\sqrt{\phi} f(w)\sqrt{2w+\phi}dw$. I need to find $f(w)$ analytically. Here $n(\phi)$ is known. Here $\phi$ is a constant.
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Abel Integral Equation

I've got the following question: We've got an Abel integral equation of the first kind $$\int\limits_0^t {\frac{{f(\tau )}}{{\sqrt {t - \tau } }}d\tau } = \varphi (t)$$ and the solution has got the ...
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36 views

how to alter $x'' + 3x = 6t$ to $x(t) = (t^3-t) + 3\int_0^1 t(1 - \tau)x(\tau)d\tau$ at $x(0) = x(1) = 0$

differential eq.$$x'' + 3x = 6t$$ alternative expression of differential eq.(Fredholm eq.)$$x(t) = (t^3-t) + 3\int_0^1 t(1 - \tau)x(\tau)d\tau$$ boundary value $$x(0) = x(1) = 0$$ What I did ...
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Fredholm Equation with Exponential Sum Kernel

I'm trying to solve the following integral equation to find the function $f(x)$ \begin{equation} f(x) = K(x) - \int_0^\infty K(x-t)f(t)dt \end{equation} where \begin{equation} K(x) = \sum_{i=1}^N ...
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Kernel evaluations of and order approximations of 2nd order Volterra integral equation

The integral equation $u:[a,b]\to \mathbb{R}$ $$u(t) = f(t) + \int\limits_a^t K(t,s)u(s)ds$$ defined on the interval $[a,b]$, with $f:[a,b]\to \mathbb{R}$ and $K: [a,b]^2 \to \mathbb{R}$ some known ...
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1answer
30 views

Distribution of a random convergent sequence of nested intervals

Starting from the interval $[0,1]$, generate two uniform random numbers $x_1,y_1$ and sort them so $x_1<y_1$. This yields an interval $[x_1,y_1]$. Generate two numbers uniformly from this interval, ...
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Show that this second type Fredholm equation doesn't admit a solution using fredholm theorems

Given the following fredholm integral equation $g(s) =f(s)+\lambda\int_{0}^{2 \pi}sin(t+s) g(t) dt$ defined from $C[0,2\pi]$ over itself. Show that if $f(s) =s$ then the equation doesn't have ...
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1answer
41 views

Applying FTC to Integral Equation (Spivak)

The following is an exercise from Spivak's Calculus: Find all continuous functions $f$ satisfying $$\int_0^xf(t)dt=(f(x))^2+C$$ for $C\neq 0$, assuming that $f$ has at most one zero. I ...
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57 views

Solving $f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $ with Iteration Method

I have problem solving following integral equation $$f(x) = \frac{x^2}2 +x - \int_0^x f(t)dt, x\in[0,1] $$ using iteration method with initial approach $f_0(x)= \frac{x^2}2 +x$ I applied Picard ...
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55 views

Fixed point of unusual integral equation

I am a little rusty in this area so please forgive the slowness. I am trying to prove or disprove the existence of fixed points for the following integral equation. Throughout I am interested in the ...
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1answer
53 views

Weird integral equation with non convolution kernel

Let $f$ and $g$ two rugular functions. My question is the following: Under what condition can we say that for given $g$, there exists $f$ such that we have: $$\int\limits_0^1 {f(x - s,s)ds = g(x)} $$ ...
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A question about linear integral equation

What is the Motivation for study Linear Integral equations What I know is : Many physical problems which are usually solved by differential equation methods can be solved more effectively by ...
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Compact Integral Operators induced by positive Kernels

Let $K$ be a compact operator induced by the kernel $k(s,t)\in L^2([0,1])^2$ with $k(s,t)>0$. Prove that $\|K\|<1$ if and only if $(I-K)$ has a bounded inverse $(I-K)^{-1}$ which is induced by a ...
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1answer
62 views

Non trivial solution of Fredholm integral equation of second kind with constant kernel

Let us consider the following integral equation$$f(x) + \lambda \int_0^1 {K(s,x)f(s)ds = 0,{\text{ x}} \in {\text{(0}}{\text{,1)}}{\text{.}}} $$ I'm looking of the values of $\lambda$ so that the ...
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Non existence of solution of a special first kind Fredholm integral equation

Let $k \in {L^2}((0,4) \times (0,1))$, $g \in {L^2}(0,1)$. We consider the following first kind Fredholm equation $$\int\limits_0^4 {k(s,x)f(s)ds=g(x), x\in(0,1).} $$ Where $f$ is the unknown. How ...
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Weird solution to an integral equation

So according to this wiki link https://en.m.wikipedia.org/wiki/Integral_equation The solution to the Fredholm equation of the first type of the form $$g(s)=s\int_{0}^{\infty}dtK(st)f(t)$$ For a ...
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Integral equation in a rectanglar domain

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$ Observe that the kernel is not defined on a square. My question: Can I apply the classical theory in ...
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Third kind Fredholm integral equation

Let us consider the following integral equation $$a(x)u(x) + \int\limits_0^1 {K(s,x)u(s)ds} = f(x)$$ Let f in $L^p(0,1)$ for some $p \in [1,\infty] and let $ $K \in L^q((0,1) \times (0,1))$. Assume ...
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First kind Volterra integral equation regularity

Let $K(x,y) \in {L^2}({(0,1)^2})$ and $g \in {L^2}(0,1)$. We consider the following integral equation $$\int\limits_0^x {K(x,t)f(t)dt = g(x)} $$ My question: what can we say about the regularity of $f$...
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Connection between the eigenfunctions of the compact operators $T[f](x\in H_1)=\int_{H_1}k(x,y)f(y)dy$ and $R[f](x\in H_2)=\int_{H_1}k(x,y)f(y)dy$?

Let $H_1$ and $H_2$ be Hilbert spaces. Suppose we have a compact integral operator $T:H_1 \to H_1$ given by $$ T[f](x) = \int_{H_1} k(x,y)f(y)dy, \quad \quad x \in H_1. $$ Suppose we also have a ...
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Solution $y(x)$ for $-\cos(x)= \int_0^{2\pi}\max(y(t), y(x+t))dt$

The application here is to design a value which will produce a sine wave like pressure or flow rate as a function of time. Pressure, or flow rate is a function of the open area inside of a valve. ...
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Solving for integrand from integrated quantities.

Given equations of the form: $A(r) = \int_{t_{1}}^{t_{2}}F(r,t)dt$ $B(t) = \int_a^b F(r,t)r^2dr$ where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information ...
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Solution $y(x)$ for $\sin(x) = \int_0^{2\pi} \max(y(t), y(x+t)) dt$

I've been banging my head against a wall for a few weeks to find a feasible solution for $y(x)$. $$\sin(x) = \int_0^{2\pi} \max(y(t), y(x+t)) dt$$ I don't think there is an unique solution, but I ...
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2answers
98 views

Degenerate kernel method to solve Fredholm integral equation of the second kind

$$ f(x) + \int_0^1 (xy+x^2y^2) f(y) dy = x^3 +\frac16x^2+ \frac15x $$ I have this fredholm integral equation of the second kind and am not sure how to answer this equation. I know that is has to be ...
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integral equation and Fourier transform of “almost” the convolution

I am facing an integral equation where one of the terms looks like this: $$ V(t) = \int_t^{+\infty} K(x-t) \cdot V(x) \, dx $$ where $$K(x) = N(\frac{-b-a\sigma^2}{\sigma \sqrt{x}}) - d^{-2a}N(\frac{...
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Solve for f in the integral equation $f(t) = \sin(t) + \int f(s)ds $

Solve for f in the integral equation $$f(t) = \sin t + \int_{0}^{t} f(s)ds$$ using $ (V^nx)(t) =\int_{0}^{t} \frac{(t-s)^{n-1}}{(n-1)!}x(s)ds $ to to where V is the Volterra operator $V$ on $L^2(0,1)$...
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1answer
41 views

Solving second order ordinary differential equation with variable constants

I'm having trouble solving a differential equation I found: $$ u''(x) + x\int_0^xu(t)dt = f(x) $$ where: $ x\in[0,1], \quad u(0) = 1, \quad u(1) = -1 $, and $f(x)$ any given function. One of my ...
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Homogeneous Fredholm integral equation of the first kind with positive symmetric kernel

Given the equation $$\int^{1}_{-1}K(|x-t|)\varphi(t)dt=0,$$ where the kernel is positive: $K(x)>0$; equation is satisfied for $x\in [-1,1]$. $K(x)$ and $\varphi(x)$ are real and continuous ...
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I need some to explain to me collocation method and points

I have been researching the whole internet for hours just to get a good insight about collocation method and points. I would appreciate if someone could explain these terms in details and give ...
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1answer
132 views

Solving an integral equation (possibly Fredholm, 1st kind) containing quartic exponentials with Fourier Transforms

I've been reading an economics paper regarding rational inattention by Sims (link: https://www.sciencedirect.com/science/article/abs/pii/S0304393203000291) and have been trying to follow his steps in ...
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42 views

Integral equation in polar coodinate system

I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...
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81 views

Multivariate linear integral equations

In the univariate case, linear integral equations have the form (0): $$ f(x) = \lambda \phi(x) - \int_a^b K(x,y) \phi(y) dy $$ where $ a < x,y < b $ and $K:[a,b]\times[a,b] \to \mathbb R$ is the ...
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27 views

Singular integral equation with Volterra part

I am trying to solve an integral equation of the form \begin{align} y(x) + A(x) \int_{-1}^x y(t) d t + B(x) PV \int_{-1}^{1} \frac{y(t)}{x-t} dt &= f(x), \end{align} for $y(x)$ where $A$, $B$ ...
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49 views

System of linear Fredholm integral equations

I am looking for a reference for theorems and results talking about the existence and the uniqueness of the system of linear Fredholm integral equations. Thanks in advance
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14 views

Estimation for Scrödinger integral equation

For the equation: $${-y'' + q(x) y = k^2 y}$$ whith the initial conditions: $$y(0;k) = 0 $$ $$y'(0;k)= 1 $$ Its equivalent integral equation is: $${y(x; k)} = {\frac {sin(k x)}{k} + \int^x_0 {\...
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1answer
208 views

Integral equations and the Fredholm alternative / theory

The Fredholm alternative states that either: $$ 0 = \lambda \phi(x) - \int_a^b K(x,y) \phi(y) dy $$ has a non-trivial solution, or: $$ f(x) = \lambda \phi(x) - \int_a^b K(x,y) \phi(y) dy $$ always has ...