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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2 views

Degenerate/ separable kernel

Why e^tx is non- degenerate kernel, While e^(x-t) is degenerate or separable? Iam confused because in both cases after expansion we get results in powers of x and t and according to the definition of ...
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1answer
54 views

Continuity of the following integral.

Let $f(x,y):[0,1]^2 \rightarrow R$ be a continuos function, and $F(x)$ defined as $$F(x)= \int^{1}_{0} f(x,y)1_{(y\leq x)} dy.$$ where $1_{(y\leq x)}= 1$ if $y\leq x$ and $1_{(y\leq x)}=0$ if $y> x$...
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1answer
50 views

Solve PDE. Is this the right solution? $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0 $

This is the equation: $\frac{\partial u^{2}}{\partial x \partial y} + \frac{\partial u}{\partial y} + x + y + 1 = 0 $ I also have a solution but I don't know how it removed -1 in front of the arrow
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Can we find a closed form expression for the solution of $y(t)=y_0-ct+\alpha\int_0^t(t-s)y(s)^p\:{\rm d}s$?

Let $\alpha>0$, $p>1$, $T>0$, $c\ge0$ and $y_0>0$. Are we able to find an explicit form for the solution $y:[0,T]\to\mathbb R$ of $$y(t)=y_0-ct+\alpha\int_0^t(t-s)y(s)^p\:{\rm d}s$$ for ...
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1answer
30 views

Does the problem $\operatorname{div}(\vec{p}) = f$ on $\Omega$ with $\vec{p}\cdot\vec{\nu}=0$ on $\partial \Omega$ admits non-conservative solutions?

Define $V:=\{\vec{p} \in L^2(\Omega;\mathbb{R}^n) \mid \operatorname{div}(\vec{p}) \in L^2(\Omega)\}$ where $\Omega$ is a non-empty connected bounded open subset of $\mathbb{R}^n$ with $C^2$ boundary ...
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How to understand the ill-posedness of Volterra equation of first kind and well-posedness of second kind?

Consider a Volterra equation of first kind as following: $$ f(t) = \int_0^t K(t,s) x(s) ds $$ We can change it to second kind by taking derivative of both sides: $$ \tilde f(t) = x(t) + \int_0^t \...
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The existence of a conditional density function

$\newcommand{\real}{\mathbb{R}}$ I am wondering if a conditional density function $p(x_2 | x_1)$ satisfying the following conditions exists: $p(x_2 | x_1) \geq 0, \int_\real p(x_2 | x_1) dx_2 = 1$ ...
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Analog to Ramanujan Master theorem and questions regarding operational methods

A well-known and very useful result by Ramanujan is his Master theorem regarding integrals of the type $$\int_0^\infty \mathrm{d}x\,x^{s-1}f(x)=\Gamma(s)\phi(-s)$$ where $\phi(s)$ is the coefficient ...
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63 views

Find the eigenvalues and eigenfunctions of the following integral equation.

I have the integral equation $$u(x)=1+\lambda \int_0^1 K(x,t)u(t)dt $$ $x \in (0,1)$, $\lambda \in \mathbb{R} $ and $$K(x,t)=\begin{cases} x(t+1) & t \leq x \\ t(x+1) & x \leq t \end{cases} $$ ...
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31 views

Finding a density function that maximizes an integral

I have the following maximization problem originating in stochastic control theory. However, I shall present it as a general optimization problem. For a $U \subseteq \mathbb{R}$ (may as well consider $...
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1answer
45 views

Integral equation with kernel $x-y$

I am trying to solve the integral equation $$ f(x)=\int_0^x (x-y)f(y)dy, \ 0\leq x\leq 1 $$ in the space $C([0,1])$ of continuous functions on $[0,1]$. My reasoning is this: Since $u(y)=(x-y)f(y)$ is ...
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2answers
69 views

Solve this integro-differential equation.

I am given the integro-differential equation $$y'(x)=\frac{7}{6} -11x +\int_0^1 (x-t) y(t)dt $$ for $x \in (0,1)$ With $$y(0)=0$$ I want to solve this by using the Laplace transform. So I have to take ...
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1answer
45 views

How to convert an integral equation to a boundary value problem.

I have the integral equation $$f(x)=\int_0^x (x-t)f(t)dt$$ and want to turn it into an boundary value problem, so I differentiate both sides with respect to $x$ and I get $$f'(x)=\int_0^x f(t)dt$$ ...
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86 views

Solve for $\alpha$ when $f(\alpha)=2\int_{0}^{1}f(y)dy$? [closed]

I am stuck with the following problem and do not know how to tackle it. Any help would be appreciated, Let $\alpha\in[0,1]$ then, what would be the value or the bound of $\alpha$ if it satisfies the ...
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35 views

Kernel of Integral Equation

Given the IVP $$-y''(x)+xy(x)=0,\,\,\,\,\,y(0)=-2, y'(0)=-1 $$ Convert it to a Integral Equation and find the kernel $k(x,s)$. I called $y''(x)=u(x)$ and got $$y(x)=-2-x+\int_{0}^{x}(x-s)u(s)\,ds$$ ...
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21 views

Integral Equations where the Unknown Function depends on the Integration Dummy

Consider the following the integral equation: $$ f(s) = \int_a^b K(s,t) g(s,t) dt, $$ where $f$ and $K$ are known functions, and $g$ is the unknown function we want to solve for. For example, $K(s,t)$...
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43 views

Unique Solution to a Set of Bivariate Integral Equation?

Suppose we have the following two integral equations: $$ f_1(x) = \int K(x,t) \varphi(x,t) dt, \quad f_2(x) = \int K(h(x),t) \varphi(x,t) dt, $$ where $f_1,f_2,K,h$ are known functions, $h$ is ...
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1answer
37 views

Numerical solution of system of integral equations

I want to solve numerically a system of equations defined as \begin{align} \sum_{i,j} f_{ij}(r) &= 1 \\ 0 &= A_{ijkl}f_{kl}(r) + \frac{1}{r^6}B_{ijkl}f_{kl}(r) \\ &+ f_{kl}(r)\left[\int_{...
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1answer
47 views

Explicit solution to a Integral equation

I have a question about my try to solve the following equation: $y(x)=2\cos(x)+\epsilon\int_0^\infty (\frac{1}e)^\tau*y(x+\tau) d\tau $ where y is bounded. My approach was write out the improper ...
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1answer
32 views

Converting integral equation to its primary initial value problem

I converted below initial value problem to Volterra equation of second kind $$ y'(x)-2xy(x)=e^{x^2}, \hspace{3mm} y(0)=1 $$ Supposing $u(x)=y'(x)$ and integrating both sides from $0$ to $x$ yields the ...
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3answers
57 views

Solving an integral equation by converting it to a differential equation [closed]

I have to solve an integral equation given as $$ x(y)=\sin(y)+\varepsilon \int_0^\infty e^{-s} x(y+s) \mathrm{d}s$$ I know that I have to differentiate it, but I am doing hard with it. Can anyone ...
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1answer
43 views

Solving an integral function with finite limits

I am trying to solve an integral equation of the form: $$A = \int_0^1 \int_0^1 \mathrm{d}x~\mathrm{d}y~\rho(x) \rho (y) xy,$$ where $A$ is a known constant. I am trying to find the unknown function $\...
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35 views

Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \overrightarrow{x} = \overrightarrow{b} $$ with $\overrightarrow{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for ...
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261 views

Find $f$ such that $f \star f(x) = \frac{1}{1-x}$.

I'm looking for a measurable function $f$ defined on $]0,1[$ such that : $$f \star f(x) = \int_{0}^1 f(x-y) f(y) \ \mathrm{d}y = \frac{1}{1-x}$$ for (almost) any $x \in ]0,1[$. Is it possible to find ...
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35 views

Converting a Differential Equation to an Integral Equation

Consider the following equation $\phi''(x) + \lambda \phi(x) = 0$, with the condition $\phi(0) = \phi(1)$, and $\phi'(0) = \phi'(1)$ I need to convert this into an integral equation. So, now ...
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20 views

Fredholm equations, orthogonalization and local functions?

Given a Fredholm integral equation, are there any approach using orthogonalization : the integral is seen as a scalar product then if one uses Gram-Schmidt method on : $$f(x)=g(x)+\int_a^b K(x,y)f(y)...
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95 views

Find the control for this system of ODE minimizing the energy.

Suppose I have the system of ODE $$x'(t)=Ax(t)+Bu(t)$$ $$x(0)=x_0$$ $t$ is defined on an interval $I$ of $\mathbb{R}$ containing $0$, for $x(t)=(x_1(t),...x_n(t))^T$, $x_i : I \to \mathbb{R} $ ...
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0answers
26 views

Existence of solution to a system of Fredholm integral equations

Consider the following system of Fredholm integral equations with constant kernel matrix $$ f(x)=K(x)\int_{0}^{1}f(s)ds $$ where $K(x)\in C([0,1];M_{2\times 2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \...
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Proof Verification: For the heat equation $~u_t=u_{xx}~,$ which of the following option(s) is\are true for a suitable kernel $~k(x,y)~?$

Problem: Let $~u(x,t)~$ be a solution of the heat equation $~u_t=u_{xx}~$ in a rectangle $~[0,\pi]\times[0,T]~$ subject to the boundary conditions $~u(0,t)=u(\pi,t)=0,~~0\le t\le T~$ and the initial ...
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0answers
42 views

How do I solve $af(z)+c \int_{z}^{0} \int_{-h}^{z} f(t)dt^{2}=d $

How do I solve $$af(z)+c \int_{z}^{0} \int_{-h}^{z} f(t)dt^{2}=d $$ while $-h<z<0$ and $a,c,d$ are real constants. I tried to rewrite the equation as $$a{y}''+cy=d $$ where, $y(z)=\int_{z}^{...
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39 views

Solving a functional equation with inverse

The following question has bothered me for a long time. Any suggestions will be highly appreciated. I need to find a monotone positive function $f(x)>0$ and $f'(x)<0$ (whose inverse $f^{-1}$ is ...
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2answers
440 views

Probability that a Particle which moves Unit distance in a Random direction on each step will be inside the Unit Sphere after $n$ steps

The following integral equation arises while calculating the probability that, a particle which starts at the origin and moves a unit distance in a random direction on each ‘move’, will be within the ...
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0answers
22 views

Finding the characteristic number and the eigenfunctions of a homogeneous Fredholm integral equation

The kernel of the homogeneous Fredholm integral equation is given as: $$ K(x,t) = \begin{cases} x(t-1), \text{ } 0 \leq x \leq t \\ t(x-1),\text{ } t\leq x\leq 1 . \end{cases} $$ The given options ...
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232 views

Solving a pair of dual integral equations

I have the equations $$ \int\limits_0^\infty dk \ A(k)k \sinh(ka)\cos(kx)=0 \ \ ; \ \ 1<|x|<\infty \tag{1} $$ $$ \int\limits_0^\infty dk \ A(k) \cosh(ka) \cos(kx)=1 \ \ ; \ \ |x|<1 \tag{2} ...
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1answer
24 views

Determining the number of solutions of the non homogeneous integral equation.

This is a question from a competitive exam. We are given the integral equation: $$ \phi (x) = \cos(7x) + \lambda \int_{0}^{\pi} \left[ \cos(x)\cos(t) - 2\sin(x)\sin(t) \right]\phi(t) dt $$ and are ...
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0answers
50 views

Integral equation with unconventional form

I need to find a solution to this unconventional integral equation: I need to find some nonnegative function $f$ that satisfies the following for any $\phi \in[0,\pi]$: $\int^\phi_0f(\theta)f(\phi-\...
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1answer
40 views

Solution of the Integral equation $y(x) = 1 - 2x -4x^2 +\int_{0}^x \left[ 3 + 6(x-t)-4(x-t)^2 \right]y(t)dt$

I want to find the solution for the integral equation: $$y(x) = 1 - 2x -4x^2 +\int_{0}^x \left[ 3 + 6(x-t)-4(x-t)^2 \right]y(t)dt $$ I tried finding the resolvent kernel of the equation and it is ...
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0answers
25 views

An Integral Equation with a Curve

We are given a field $F:\Bbb{R}^n\to\Bbb{R}^{n×n}$ infinitely differentiable and a constant $C \in \Bbb{R}^n$. I'm looking for a curve $k:[0;\infty)\to\Bbb{R}^n$, which is infinitely differentiable ...
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3answers
98 views

Finding a function as a solution to a system of integral equations

In my research, I reached a point where I need to find a nonnegative function, that satisfies the following: For any $\phi\in[0,\pi]$: $[\int^\pi_\phi f(\theta)d\theta][ \int^{\pi-2\phi}_{-\phi} f(\...
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1answer
40 views

Solutions of a differential and its related integral equation

We are looking for solutions to the following differential equation $$ tf'(t)-\mu f(\frac{t}{\mu})+\mu f(0)=0\;\; ;\;\;t\in \mu I, $$ where $I$ is an interval containing $0$, $\mu\in (0,1)$ is a ...
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1answer
37 views

Laplace transform of solution of an integral equation for non-recursive logistic map computations

Consider the logistic map, $x_n = r \: x_{n-1} \left( 1 - x_{n-1} \right)$ If we generalize this to a complex function $f : \mathbb{C} \mapsto \mathbb{C}$, we get, $f \left( z \right) = r \: f \left( ...
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2answers
48 views

Can we solve $u(x)=-1+x+\frac{x^2}{2}+2e^x-\int_0^xu(t)dt$ with using noise terms phenomenon?

question is can $u(x)=-1+x+\frac{x^2}{2}+2e^x-\int_0^xu(t)dt$ be solved via noise terms phenomenon? I made $u_0(x)=-1+x+\frac{x^2}2+2e^x $ $u_{k+1}(x)=-\int_0^xu_k(t)dt$, $k\geq0$ $u_1(x)=-\int_0^...
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1answer
78 views

Stability, critical points and similar properties of solutions of nonlinear Volterra integral equations

I've posted also on MathOverflow, but repost here once I think it's an important question and would like to get more attention. Thank you! I have a system of nonlinear Volterra integral equations of ...
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1answer
107 views

Is there a way to exactly solve this integral? [closed]

Is there a way to determine $\theta_c$ in the following integral equation: $$\int_{-\theta_c}^{\theta_c} d\theta \, \exp (a \cos\theta)=1,$$ where $a$ belongs to positive reals?
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0answers
62 views

Solving a Fredholm equation

I'm trying to solve this form of Fredholm equation: $$ g(v)=f(v)+\int\limits_{0}^{a} g(v_s)K(v,v_s)\mathrm{d} v_s, $$ where, $f, K$ is a given function $K(v,v_s)=K_1(v-bv_s)+K_2(v+bv_s)$, where $b$ ...
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0answers
34 views

Integral functional equation involving product

This question arises in an application in Statistics when studying prediction aggregation. Suppose one has a collection of $J$ probability density functions defined on some space $\mathcal{Y}$, $\...
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2answers
33 views

Solving for $g(y)$ in the following integral equation

Given the following integral equation where $f, h$ are known: $$ f(t) = \int_{t-1}^t{g(y)\cdot h(y)\ \textrm{d}y}$$ Is it possible to solve this for $g(t)$, i.e. get an equation of the form $g(t) = \...
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0answers
42 views

Kernel of Linear Functional/ Solution to Integral Equation

I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int_0^1 \int_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in ...
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0answers
61 views

solve for x(t) under integral

the following integral equation must be solved for x(t). $$x(t) + \mu(t) \frac{1}{T} \int_{T}x(\tau)h(t-\tau)d\tau = y(t)$$ where $y(t)$, $\mu(t)$, and $h(t)$ are all known functions. All functions ...
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0answers
39 views

Reference request to understand the “Linear integral Equation” from Rainer Kress's book

I am reading "Linear integral Equation" by Rainer Kress. Mean while he is using many terms from differential geometry and several variable calculus for example volume elements in multi ...

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