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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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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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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
27 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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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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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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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
50 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
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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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1answer
38 views

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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144 views

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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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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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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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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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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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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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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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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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 ...
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Integral linear equation of Fredholm.

How can I prove?. Prove that if $[(b-a)|\lambda|sup_{t,s \in [a,b]}|\kappa (t,s)|]<1$, then a integral linear equation of Fredholm has a unique solution in $C[a,b]$. We know by definition that; ...
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Converting nonlinear ODE with neumann BC into Fredholm integral equation

I have a nonlinear ODE I wish to convert into a Fredholm integral equation. This site shows how to convert an ODE with Dirichlet BC into an integral equation, however I cannot figure out how to do it ...
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1answer
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Is there anything that we can say about $f$ in terms of $\theta$ if $\int_0^tf(\tau)\text{d}\tau = \theta(t)\int_0^t\theta(\tau)\text{d}\tau$ is true?

I have an integral equation in which I have $\theta(t)\int_0^t\theta(\tau)\text{d}\tau$ as a term, which I would like to transform to a single integral, so that I end up with a Volterra integral ...
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1answer
103 views

How To Solve An Integral Equation

Our professor posted an integral equation for us to solve. It is $$f(x) = a - \int^x_b (x-t)f(t)dt$$ Where $a$ and $b$ are constants. This was in the context of using Leibnitz's rule, so I attempted ...
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43 views

Integral equation with two variables

I have an integral equation that has the following form: $$f(x,t)=g(x,t)+\int_0^t h(u)f(x+u,t-u)du$$ for all $x,t\in \mathbb{R_+}$ and $f(x,0)=1, \forall x\in\mathbb{R_+}$ with $g$ and $h$ are given ...
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1answer
94 views

Conversion of second order ode into integral equation

The second order differential equation $$-\phi''(x,\psi)+g(x)\phi(x,\psi)=\lambda^2\phi(x,\psi) $$ where $\lambda\in\mathbb{C}$ and $x>0$. with conditions $$\phi(0,\psi)=0,\phi'(0,\psi)=1$$ can be ...
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69 views

How to obtain the explicit solution of the following integral equation

I'm considering the following integral equation $$ f(x,y,z)=x+\int_0^x\int_0^y\int_0^z f(u,v,w) dudvdw $$ It seems that $$ f(x,y,z)=\sum_{n\geq 0}\frac{x^{n+1} y^n z^n}{(n+1)!n!n!} $$ is the ...
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2answers
79 views

Showing existence and uniqueness for a solution to a homogeneous Fredholm type integral equation of the second kind

I'm studying for an exam in real analysis. Thus, only such techniques should be considered. I'm looking at old exams, and repeatedly see questions similar to the one below. Show that there exists a ...
3
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37 views

Self-Dual Irregular Gears

I recently was reading about nautilus gears, and I was wondering what other irregular gears one can make. In particular, I'm interested in systems in which both gears are congruent (or at least ...
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1answer
57 views

Integration of Bessel function multiplied with an algebraic and trigonometric functions.

I tried my best to solve the following definite integration. Struggling a lot, ended with no luck. I know similar but a bit little simplified identity. The question is how to change this identity ...
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2answers
42 views

Can there exists a solution to the following integral equations? [closed]

I have come across the following two integral equations: \begin{align} a&=f(1)+f(2)+\int_{0}^{3}t f(t)\mathrm{d}t\\ b&=\int_{0}^{3}t^2f(t)\mathrm{d}t. \end{align} Now, my question is can there ...
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42 views

Non-linear differential-integral equation which becomes second degree polynomial.

How can I solve a differential equation like this one? $$k \to y(k), \forall k\in[a,b]\subset \mathbb R$$ $$y(k)^2=\int_{k}^{k+\Delta} y'(t)dt$$ where $\Delta$ independent of $k$ has primitive ...
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0answers
44 views

Numerically solving a system of linear integro-differential equations in Matlab

Given the following system of linear integro-differential equations $$ \frac{d}{d t}B(t)+\int_{0}^{+\infty}C(x,t)dx+A(t)=0,\\ \left[\frac{\partial}{\partial t}+V(x)\right]C(x,t)+B(t)=0,\\ \frac{d}{d ...
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0answers
43 views

Show that $y(x)$ is a solution of the given integral equation

Show that $y(x)=e^x$ is solution of the integral equation $y(x)+\lambda \int\limits_0^1 \sin{xt} \text{ }y(t) dt=1$. To solve this I proceeded as follows: \begin{equation*} \begin{split} \text{LHS} &...