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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Can we solve integral equations for a multivariable function?

Integral equations are equations in which an unknown function appears under an integral sign. Can we solve integral equations when the function is multivariable and the integral is a multiple integral?...
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Uniqueness of the solution to a certain set of integral equations (from moment conditions)

Let $\underline{\phi},\overline{\phi}\in(1,\infty)$ with $\underline{\phi}<\overline{\phi}$. I am looking for functions $f:[0,1]\times[1,\infty)\rightarrow (0,\infty)$ and $g:[0,1]\times [\...
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Is Adomian Decomposition the same as Neumann Iteration? (Integral Equations)

I've come across two different integral equation techniques to approximate the solution of an integral equation: Neumann iteration and adomian decomposition. However, they appear very similar, are ...
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How to solve integral equation of convolution using Fourier transform

I'm having trouble solving the following exercise: Use the Fourier transform to solve the integral equation $$f(x) = \int_{-\infty}^{\infty} e^{-|x-\xi|}u(\xi)\,d\xi$$ Then verify your solution when $...
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Unique solution to a specific Volterra's integral equation of the third kind

Consider an integral equation (Volterra's integral equation of the third kind) $$(d-cx) u(x) = \int_x^b u(y) dy, \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function ...
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On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1) $$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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Solution of Integral equation via Fourier Transform

Solve for f(t) using Fourier transform: $$ \int_{-\infty}^\infty f(s)f(t-s)\,ds - 2\sqrt{2} \int_{-\infty}^\infty e^{-s^2/\pi}f(t-s)\,ds = -\sqrt{2}\pi e^{-\frac{t^{2}}{2\pi}} $$ Now, I get the ...
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Convergence of a self-consistent multivariable integral equation

I want to solve the following integral equation $$\mathbf v(\mathbf r)=\int_V \mathbf v(\mathbf r^\prime)\cdot \mathbf F(\mathbf r, \mathbf r^\prime) \mathrm ~d^3r^\prime +\mathbf g(\mathbf r)$$ using ...
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Integral Equation and inverse laplace

When we get the equation in (5.81), we get complex expressions while taking the inverse laplace transform after moving to the next step. Is there any way to get rid of this complex expression?
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Verify solution to linear Fredholm integral equation of the second kind

Let $\int_a^b C_X(t, s)\psi_k(s)ds = \lambda_k\psi_k(t)$, which corresponds to a homogeneous linear (Fredholm) integral equation of the second kind. Where $C_X(t, s)$ is the covariance function and is ...
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An integral equation with zero initial condition implies unique solution

Suppose $f,g:[0,\infty)\to [0,\infty)$ are two continuous functions satisfying $$f(x)-g(x) = \int_0^x\frac{g(y)-f(y)}{g(y)f(y)}\;dy$$ for every $x\in[0,\infty)$. How to show that $f(x)=g(x)$ for all $...
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Discretize the nonlinear Hammerstein operator

I would like to know please how to discretize the following special instance of nonlinear Hammerstein operator (in matlab if it is possible): $F : H^{1}[0,1] \rightarrow L^{2}[0,1],$ $ F(x)(s):=\int_{...
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Consider a continuous function $f(x)$ with $f(1)=0$ [duplicate]

Given that $$I_1=\int_0^1x^2f(x)dx=\frac {1}{3}$$ and $$I_2=\int_0^1(f'(x))^2dx=7$$ Find the value of $$J=\int_0^1f(x)dx$$ In the first equation using a bit of symmetry I substituted $x=1-t$ and after ...
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How to solve the integral equation $f(x) \int_a^x \int_a^{t} f(\omega) d\omega dt - \left[ \int_a^x f(t) dt \right]^2 = 0$

I tried to solve the integral equation $$f(x) \int_a^x \int_a^{t} f(\omega) d\omega dt - \left[ \int_a^x f(t) dt \right]^2 = 0$$ by taking the first, second, third, and fourth order derivatives but I ...
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Converting an integral equation to a differential equation

I was recently working on a problem and ended up with an integral equation that I was hoping can be solved or at least be converted to a differential equation. I have no experience in integral ...
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Resolvent Kernel Uniform Convergence

In order to solve nonlinear Fredholm integral equation of the second kind I need to make interchange of integration and summation as shown below $$ y(x) = f(x) + \lambda \sum_{k=1}^{\infty} \frac{...
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Integral equation Objective question.

Given the eigen values of the integral equation $$y(x)=\lambda\int_0^{2\pi}\sin(x+t)y(t)dt $$ are $\frac{1}{\pi}$ and $-\frac{1}{\pi}$ with respectively eigen functions $\sin(x)+\cos(x)$ and $\sin(x)-...
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Solution of non homogeneous Fredholm integral equation.

Given that eigen values of the integral equation $$y(x)=\lambda\int_0^{2\pi}cos(x+t)y(t)dt$$ are $\frac{1}{\pi}$ and $-\frac{1}{\pi}$ with respectively eigen functions $cos(x)$ and $sin(x)$ . Then ...
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Integral equation corresponding to Initial Value Problem.

The Initial Value problem $$y’’+y=0,y(0)=1,y’(0)=0$$ is equivalent to integral equation $(A)$. $y(x)=1+\int_0^x(t-x)y(t)dt$ $( B)$. $y(x)=1+\int_0^x(t+x)y(t)dt$ $ (C)$. $y(x)=1+\int_0^x(tx)y(t)dt$ $(D)...
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Best way to solve the inverse problem of an integral equation with uncertainties?

I wish to estimate the unknown functions $f$ and $g$ and their uncertainties according to $$ y_i = \int_0^1 F_i(x)\,f(x) \,\text{d}x \; + \int_0^1 G_i(x)\,g(x) \,\text{d}x, \quad i=1\ldots N $$ where ...
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Fredholm integral equation of the second kind - no homogeneous cases anywhere?

I'm trying to solve this homogeneous Fredholm integral equation of the second kind for $y(x)$. $y(x) = \int^{\infty}_{-\infty} \ln \left( \frac{1}{x-t} \right) y(t) dt$ I'm aware of Liouville-Neumann ...
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PDE and Volterra Equation: the role of boundary conditions

I have to deal with the following PDE which describes the evolution of the generating function $\mathcal{Z}(h,t)$ for a population of cells dividing with rate $\gamma$: $$ \partial_t\mathcal{Z}(h,t)=\...
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Show that $y$ satisfies Integral equation.

Let $y$ satisfies the boundary value problem $$y''(x) + \lambda y(x) = 0, 0 < x < 1,$$ $$ay(0) = by(1), by'(0) = ay'(1),$$ where $a, b$ are constants. Show that $y$ satisfies the integral ...
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Convert the bessel equation $x^{2}*y''+xy'+(\lambda x^{2}-\alpha^{2})y=0$ to integral equation.

Convert the bessel equation $x^{2}y''+xy'+(\lambda x^{2}-\alpha^{2})y=0$ to integral equation, where $0<x<1,$ $y (x)$ is bounded as $x\to 0$, $y(1)=0$, where $\alpha>0. $ Please give some ...
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Solve for the kernel in a system of Volterra equations

I have a system of $n = 1, \dots, N$ Volterra equations with the same kernel $K$: $$f_n(t) = \int_{0}^{t} K(t, s) g_n(s) \mathrm{d}s\,, \quad t \in [0, 1] \,,$$ where $f_n$ and $g_n$ are known and $K$ ...
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linear Volterra integral equation of the first kind

Solve for $f'(x)$ in $$ e^{z}-1 =-\int_{0}^{z} f'(x)\ln(1-\frac{x}{z})dx.$$ I'm fairly certain that the growth rate of $f'(x)$ is greater than polynomial. Therefore, I tried $f'(x)=e^x$ and thus $$\...
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Contraction mapping theorem applied to integral equation

I am facing the problem of showing the existence of a solution to the following integral equation: \begin{equation}\label{Eq: IE} z(t) = e^{\lambda(T-t)-\beta\int_{t}^{T}z(s)^{\frac{1}{\beta-1}}ds}\...
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Integral in recursive function

I would like to formulate a continuous and differentiable function $x_{t}=x(t)$ where its value at time t depends on the (kind of) accumulating mean of its past values. First, the idea in discrete ...
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Structure of the Inverse of a Fredholm integral operator of the second kind

NOTE: Cross-posted on MathOverflow I am trying to solve an equation of the form $$ (\mathbb{I} + K)\phi = f $$ where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\mathbb{R}) $ is a ...
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How can I solve for $f(x)$ and $g(x)$?

How can I solve for $f(x)$ and $g(x)$: $$e^{-x^2}=\int_{-\infty}^\infty g(t)e^{-f(t)(x-t)^2}\text{d}t\;\;\;\;?$$ Not necessarily an elementary solution, a numerical one would also suffice. I am trying ...
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Proof of Salem's 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 $\sigma\in(\frac{1}{2},1)$ In https://arxiv.org/abs/2003.00581 there is written that this ...
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Weighted $L^2$ norm/homogeneous Fredholm equation of the first kind

I'm looking to find functions $K$ that satisfy $$\int_{- \infty}^{\infty} f(x)K(x)dx = 0$$ given any $L^2$ function $f$. I would also ask these functions $K$ satisfy certain symmetry conditions, but ...
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Integral Equation confusion.

I have confusion on this problem I am working on, $u(x) = \frac{3x}{4} + \frac{1}{5} + \ \int_{0}^{1} (x-t)^3u(t) \,dt$ I know this Fredholm second kind integral equations because $\phi(x) = 1$ and $f(...
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Is this integro-differential equation well posed?

Consider the integro-differential equation $$ \begin{align} K(t)\cdot\frac{\mathrm{d}}{\mathrm{d}t}\exp\left(2\int_0^t\mathrm{d}s\:\sin2\chi(s)\right)=\frac{\mathrm{d}^2}{\mathrm{d}t^2}\left[\chi(t)-\...
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1 vote
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Show that the following integral equation admits a unique solution except for equivalences using Fourier transforms

Let $f,g \in \mathcal{L}_1(\mathbb{R^n},\mathbb{K})$ if $\mathcal{N}_1 (f)<\frac{1}{|\lambda|}$ for some $\lambda \in\mathbb{K}-\{0\}$, then using fourier transform show that the integral equation $...
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How to show the solution so this Fredholm integral is unique?

Define the operator $K: L^2[0,1] \to L^2[0,1]$ by $$K f(x) = \int _0 ^1 k(x,y) f(y) \, dy .$$ where $k(x,y)$ is a continuous complex valued function on the unit square. $[0,1]\times[0,1]$ Also assume ...
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How to solve this equation:$\Phi(x,y,z)=x+\int_{\Omega(x,y,z)}\Phi(u,v,w)dudvdw$?

Assume that $\Omega(x,y,z)=[0,x]\times[0,y]\times[0,z]\subset \mathbb R^3 ,0\leq x,y,z\leq 1$. Consider equation $$\Phi(x,y,z)=x+\int_{\Omega(x,y,z)}\Phi(u,v,w)dudvdw$$The question is: is there any ...
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2 answers
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Solving for endpoints given by two integral equations

I have been trying to work out an example related to hypothesis testing for the scale parameter in an exponential distribution. By following the statistic theory I have been led to the following ...
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eigenvalues and eigenfunctions for a nondegenerate $L^2$ kernel

Consider the symmetric $L^2$ kernel $K(x,t)=\log(1-\cos(x-t))$ for $0\leq x,t\leq 2\pi$ . Find the eigenvalues and corresponding eigenfunctions of $K(x,t)$ By general procedure we consider a Fredholm ...
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solution of a Fredholm equation of first kind

I was recently studying about solution of the following homogeneous Fredholm equation of first kind $$x=\lambda\int_0^1e^{x-t}y(t)dt$$ if there's any . But the method of regularization (since the ...
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verifying eigenvalues ad eigenfunctions of a symmetric kernel

Compute the eigenvalues and the eigenfunctions of the symmetric kernel $K(x,t)=\min(x,t)$ in the basic interval $0\leq x\leq1$ and $0\leq t\leq 1$ . The standard way is to follow along lines of ...
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2 votes
2 answers
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Solving a Simple Integral Equation

I'm looking to solve the integral equation $$f(x)=a\int_x^{x+b}f(t)\,dt,$$ for positive $a,b$. However, I've gotten stuck. My current approach is to apply a Laplace Transform to obtain (denoting $F=\...
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How to solve the integral equation $ \int_{0}^y \frac{f(v)}{\sqrt{y-v}}dv = 4 \sqrt{y} $

This is a question from a mathematical contest. The curve $y=y(x)$, passing through the point $(\sqrt{3},1)$ and defined by the following property $$ \int_{0}^y \frac{f(v)}{\sqrt{y-v}}dv = 4 \sqrt{y} ...
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  • 700
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Derivative of a definite integral with respect to a function inside the integral.

Suppose I want to optimise the function $$f\mapsto \int_\mathbb{R}e^{f(x)}\mathrm{d}x$$ with respect to $f$, where $f$ is concave and twice continuously differentiable. Is there some result that ...
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Have you ever seen this kind of "Logarithmic" Volterra integral equations?

I am currently trying to solve a set of integral equations of the form \begin{equation} \ln \int^1_0 f(s, t) \ \text{d}s = b + \int_0^1 w(t, s) \ \ln f(t, s) \ \text{d}s, \end{equation} where $b \in \...
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2 votes
1 answer
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Existence of point with prescribed value from given differential equation

I came up with this problem while I was trying to prove the following geometric problem : Let $A, B$ be the distinct points in $\mathbb{R}^2$. If $r(t)$ is the non self-intersecting trajectory of ...
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Solution of the Integral equation $ y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt $

This question is from a mathematics competition question paper. We are given the integral equation $$ y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt $$ Then $y$ is given by: $y(x) = f(x) + \int_{0}^x (x-t)...
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How to handle an integral or derivative as a collinear variable in regression?

I would like to regress multiple unknowns relating a dependent variable to an second variable both of which are functions of third independent variable. I can relate the integral of the first ...
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Dealing with a (partial) integro-differential equation

I am dealing with this (partial) integro-differential equation: \begin{align} \frac{\partial v}{\partial t}(t,x) &= c_{3}v(t,x) + \int_{x}^{1} K(x,\xi) v(t,\xi)d\xi,\\ v(0,x)&= v_{0}(x) \end{...
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Regular perturbation on Fredholm equation

I want to find the three first terms in an expansion of $u(y)$ \begin{equation} u(y)=u_0(y) +\epsilon u_1(y)+\epsilon^2u_2(y)+\mathcal{O}(\epsilon^3) \space \space \space \space \space \space \space \...
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