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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 - ...

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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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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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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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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
46 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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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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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 ...
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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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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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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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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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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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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} &...
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Nonlinear, homogeneous Volterra Equation with Positive Kernel

Suppose we have a homogeneous Volterra equation of the second kind: $$ x(t) = \int_0^t K(t,s,x(s))\,\mathrm{d}s, \qquad 0\leq t\leq T, $$ and the sign of $K$ is determined by only $x$ for all $t$ and $...
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Numerical solution of integral equation involving unknown random variable transformation.

Let Y be a random variable normally distributed, and let $X=g(Y)$ be a transformation of Y such that g solves a certain integral equation $$I[g,F_X]=0$$ where $F_X$ is the distribution function of ...
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Solve this integral equation.

Let $h(x)$ be a known well-behaved function, I have to solve for $\sigma(t)$: $$ \phi(x) = \int_a^b\log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right]\sigma(t)dt $$ Where, $b>a>0$, ...
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Solutions of Fredholm integral equation of first kind

In the context of Fredholm integral equation of the first kind: $$ f(x) = \int_0^\infty K(x,y)\phi(y) dy $$ where $f, K$ are given and $\phi$ is unknown, I read somewhere that if the homogeneous ...
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Problem on Definite Integration

Q. Find all the values of $ 'a' $ which satisfy the equation: $$ \int_{0}^{\frac{π}{2}} (sin(x) + acos(x))^3 dx - \frac{4a}{π-2} \int_{0}^{\frac{π}{2}} xcos(x) dx =2 $$ My Attempt: I tried to ...
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1answer
127 views

Can the following functional equation be reduced to a standard ODE?

Consider the functional equation $$ f\left(x\right) := \dfrac{x^3}{2}\left\{G''\left(x\right) k + \int_{\underline{x}}^{x}\dfrac{\partial^2D_H\left(x,y\right)}{\partial x^2}G'\left(y\right)d y + \...
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Fredholm integral equation of the 2nd kind with a weakly singular convolution kernel

I've reviewed the literature but could not find a solution of the integral equation $$\int_{0}^{1} dx\frac{\phi(x)}{|x-y|^{\alpha}}+a\phi(y)=b.$$ for the function $\phi$ where $\alpha\in(0,1)$ and ...
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72 views

Resolvent kernel of Fredholm integral equation.

For the linear integral equation $ y(x)=x+\int_{0}^{1/2} y(t) dt$. Find Resolvent kernel $R(x,t,1)$. I tried to find resolvent kernel of Volterra integral equation by taking kernel as 1....
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$\int^1_0 f (f(x)t) \,\mathrm{d}t =\frac{1}{2}f(x)$ for every $x$

$$\int^1_0 f (f(x)t) \,\mathrm{d}t =\frac{1}{2}f(x)$$ for every $x$. I have to find all linear functions that looks like: $f(x)=Ax+B$ I thought maybe to differentiate it but I get nothing ... ...
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Eigenvalue Problem for Fredholm (Generalised?) Integro-Differential Equations

Consider the following problem: For $\Omega\subset\mathbb{R}^2$ a bounded domain, find $(\lambda, f(x))\in\mathbb{R}\times L^2(\Omega)$ such that \begin{align*} Lf(x) & = \lambda Kf(x) && ...
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1answer
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Minimum norm solutions: Integral Equation

I wish to find the matrix that minimises the following Loss function $$\mathbf{A}^* = \text{argmin}_\mathbf{A} \int_{\mathbf{x}\in\mathbb{X}} \|\mathbf{A}\mathbf{x}-\mathbf{y}(\mathbf{x})\|^2d\mathbf{...
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A system of nonlinear generalized Abel's integral equation

How can solve the following system of nonlinear generalized Abel's integral equation: $\begin{cases} u(x)-2v(x)+\int_0^x ‎\frac{u^2 (t)+v^2 (t)}{(x-t)^{‎\frac{1}{5}‎}}\; ‎dt=g_1 ‎(x), ‎\\‎ v(x)-u(x)...
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Integral equation to determine curve shapes

In the Cartesian plane, $\mathbb{R}^2$, we have a rectangular domain $\Omega \subset \mathbb{R}^2$, with $x\in[0,a], y\in[0,b]$, where $a,b\in\mathbb{R}^{+}$. Over a subinterval $x\in[x_{1},x_{2}]\...
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1answer
94 views

An special example of an integral equation

How can solve the following special integral equation: $$\int_0^x \frac{x^2 t^3 + t^4 +1}{(x-t)^{\frac{1}{4}}}u(t)\; dt=\frac{128}{231}x^{\frac{11}{4}}+\frac{32768}{100947}x^{\frac{31}{4}}+\frac{...
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Converting Differential Operator to Integral Equation

The Green's function of following differential operator $$(\mathcal{L}y)(x)=\frac{d}{dx}\left(x\,\frac{dy}{dx}\right)-\frac{n^2}{x}\,y(x), \:0<x<1$$ with boundary conditions $$y(0)=y(1)=0$$ ...
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1answer
49 views

Applying Fourier transform to $f(x)-\int_\limits{-\infty}^{\infty}K(x-t)f(t)dt=g(x)$

Notes: $f(x)-\int_\limits{-\infty}^{\infty}K(x-t)f(t)dt=g(x)$ where $f,g,k\in L_1{\mathbb{R}}$ therefore the equation is solvable with a unique solution iff $1-\hat{K(\xi)}\neq 0\:\:,\forall\xi\in\...
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Unique solution to an integral equation? (Moment generating function)

I have a solution to the following integral equation: \begin{equation} \gamma(t) = \dfrac{2}{t}\int_{0}^{1} \gamma(t x)(e^{(1-x)t}-1)dx, \, \gamma(0)=1. \end{equation} Can one show that the ...
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Explain Nyström Method

To solve an integral equation numerically, we can apply Nyström's method. Therefor we approximate the integral operator $(A\varphi)(x) := \int \limits_G K(x,y)\varphi(y)dy,\; x \in G$ by a sequence ...
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1answer
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Finding a function satisfying the integral relation

Let $u$ be an absolutely continuous function on $(a,b) \subset \mathbb{R}$. Find the unique absolutely continuous function $v$ satisfying, $$ \int_a^bu(x)v(x)dx + \int_a^bu^\prime(x)v^\prime(x)dx = ...
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1answer
53 views

Integral equation with fixed boundaries

How to solve integral equation $$ f(x) = 1 + \int_0^1 f(x-t) dt $$ for $x\geq 1$ and we also know that $f(x) = e^x-1$ for $x \in (0,1) $? I would like to obtain solution for $x \in [2,3]$ without ...
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Differentiating the single-layer potential

Suppose $f\in L^2[-1,1]$ and consider the single layer potential with moment $f$ on $[-1,1]$ $$ Kf(x,y) = -\frac{1}{2\pi}\int_{-1}^1 \ln|(x,y) - (\xi,0)|f(\xi)\, d\xi $$ Formally I shown that for $x\...
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1answer
179 views

Use of a Laplace Transform to solve Abel's Integral Equation

I am following an exercise where I have to solve Abel's integral equation: $$f(t)=\int_{0}^{t} \frac {\phi(\tau)}{(t-\tau)^{\alpha}} d\tau$$ I have taken the first step and shown that: $$\bar{\phi }...
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Definition of fundamental eigenvalue and sign of fundamental eigenfunction.

Consider the following integral eigenvalue equation $u = \lambda Ku$, where $K\colon L^2(F)\longrightarrow L^2(F)$ is a symmetric, compact, self-adjoint operator with positive, continuous kernel of ...
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Question on notation of 3rd degree Taylor expansion

Let $\;f \in C^3(\mathbb R^m;\mathbb R)\;$ and $\;g,h:\mathbb R^n \to \mathbb R^m\;$ be two smooth functions. If $\;I(h)=f(g+h)-f(g)-Df(g)\cdot h-\frac{1}{2}D^2(f(g))h \cdot h\;$ then it follows $...
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Solution of Volterra integral equation of first kind in which the kernel is not regular

I encounter a very hard question about integral equation. I have the following first kind Volterra equation $\int_{0}^{t} K(s,t)f(s)ds=g(t)$ We have some special features on the kernel: 1. $K$ is ...
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37 views

Any algorithm to solve the following integral equation?

I would like to ask if there exists numerical algorithm to solve the following integral equation $-\frac{1}{2}e^{-|\theta(u)-a|}=\int_{0}^{u} C(v) \frac{\theta(u)}{2\sqrt{\pi}(u-v)^{\frac{3}{2}}}e^{-\...
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1answer
105 views

Solving a Fredholm integral equation with a logarithmic kernel

I'm trying to solve this integral equation to find $y(x)$ but am struggling. Note, $a$ and $c$ are just two parameters. $$\int_0^{\infty}y(t)\,\text{ln}\left|{\frac{t-x}{t+x}}\right|dt=\pi\left[\pi+2\...
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2answers
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Solving an equation of the form $f = T(f)$, Alternative to fixed point iteration

Let $T$ be some integral operator and suppose $f:[0,\infty) \rightarrow [0,\infty)$ is a function which satisfies: $$f = T(f).$$ A popular method for finding $f$ is to take some arbitrary $f_0$ and ...
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1answer
147 views

Eigenvalues of Fredholm Integral problem

I came across this exercise and I am stuck on how to solve it: Consider a Fredholm Integral of the second kind: $u(x)=f(x)+\lambda\int_{0}^{L}K(x,t)u(t)dt$. Define the 2nd iterate Kernel $K_2(x,y)=...
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Cast the homogeneous Volterra equation into a first-order differential equation - Solutions are not the same

Consider the homogeneous Volterra equation $f(x) = \int_0^x dx' K(x,x')f(x')$ and assume that the kernel is separable, $K(x,x')=m(x)n(x')$ leading to $f(x) = m(x)\int_0^x dx' n(x')f(x')$. Assume ...
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Fredholm Integral Equation of First Kind with finite limits, 1-D

$$ \int_{-1}^{1} \frac{f(\xi)}{h^2 + (x-\xi)^2} d\xi = 1 $$ for $x \in [1,1]$ and $h < 1$ I went through all the relevant examples in Andrei D. Polyanin Alexander V. Manzhirov's 'Handbook of ...
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4answers
51 views

Show that the given map F is a contraction map & find a solution of F, if any.

Given a space of all real-valued continuous function $C[0,1]$ with $sup$ norm, define $$F:C[0,1]\rightarrow C[0,1]\text{ by } F(x)(t)=x(0)+\lambda\int_{0}^{t}x(s)ds$$ $\lambda\in\mathbb{R}$ with $|\...
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2answers
60 views

Solution of Differential equation as an integral equation

I was looking for the solution of the following problem. Prove that if $\phi$ is a solution of the integral equation $$y(t) = e^{it} + \alpha \int\limits_{t}^\infty \sin(t-\xi)\frac{y(\xi)}{\xi^2}d\...