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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25 views

Is differentiating both sides a valid way to solve an integral equation?

I'm trying to solve this integral equation for $P(k)$: $\frac{(\int_0^\infty k^{N+1}P(k)dk)^2}{\int_0^\infty k^{N-1}P(k)dk\int_0^\infty k^{N+3}P(k)dk}=1$ Not being familiar with integral equations, I'...
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47 views

Numerically solving the integral equation $b(t) = c \int_0^\infty K( t, u, b(t), b(t+u)) du$

I am reading an article where the author numerically solves the following integral equation for $b(t)$ (sec 7, p. 25): $$ b(t) = c \int_0^\infty K( t, u, b(t), b(t+u)) du, \qquad t \in [0, \infty), \...
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10 views

How to extend interval when function is increasing at one point

I am reading a research paper and got stuck on one point. Please refer above image.7th equation is given as $$\nu^{2}\frac{d^{2}v}{dx^{2}}=F(x)v(x)$$ I need to prove a claim that $v(x)<0$ and $v'(...
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36 views

Solving coupled integral equations

I would like to solve coupled integral equations of following form: $$ \begin{cases} f(0,n) = 1 + \displaystyle\int\limits_{0}^{\infty} K(n,p)f(1,p)dp \\ f(1,n) = g(n) + \displaystyle\int\limits_{0}^{...
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1answer
27 views

Changing a double integral into a single integral - Volterra-type integral equations

I have a question regarding a calculation that i stumbled upon when proving that a Cauchy problem can be converted in a Volterra-type integral equation. Specifically, this equality: \begin{equation*} \...
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1answer
90 views

Solve the integral equation $\int_0^{-x}f(x')dx'= f(x) + x$. [closed]

Like it says, I'm playing around with even and odd functions and require a function such that $$\int_0^{-x}f(x')dx'= f(x) + x\,.$$ I can't think how to go about it, any help appreciated.
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29 views

Existence of the integral equation solution

In the following equation: $$\int_{-\infty}^\infty g(x) \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)} dx = \frac{1}{\mu^2 - 1} , $$ The function $g(x)$ is unknown and doesn't depend on $\mu$....
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13 views

Is it possible to find a solution to this equation, reducing it to the Fredholm equation?

To continue the questions: https://math.stackexchange.com/questions/3743374/approximation-of-the-convolution-operator?noredirect=1#comment7697800_3743374 https://math.stackexchange.com/questions/...
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37 views

Integral equation $\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$

In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ ...
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14 views

Integral equation problem

Consider the integral equation $\phi(x)-\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t=f(x) .$ Then there exists a continuous function $f:[-1,1] \rightarrow(0, \infty)$ for which solution exists there ...
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39 views

Solution to the following Fredholm integral equation?

I have the following integral equation $$1 = \int_{-L}^{L} K(s-t)f(t)\mathrm{d}t,$$ where we are solving for $F = \int_{-L}^{L}f(s)\mathrm{d}s$. That is, $f(t)$ is an unknown function that we do not ...
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1answer
55 views

Solutions of Differential Equations and Integral Equations

Are integral equations and differential equations allways equivalent, i.e. the set of solutions is equal? The following are my thoughts about it. Are they correct? Let's define the function $x:\Bbb ...
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1answer
155 views

Turning a definite integral equation into a differential one.

Consider the following equation where $p(u)$ is a probability distribution and where $g$ is the unknown : \begin{equation} g = \int_{-\infty}^\infty p(u)f(u,g) \mathrm{d} u \end{equation} Using the ...
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43 views

Solve for $z$, $ z = \int\frac{p(x)}{x-z^*}\mathrm{d}x $

Let $p(x)$ be a probability distribution. Without further information is it possible to solve the following equation for the unknown $z\in \mathbb{C}$ ? $$ z = \int\frac{p(x)}{x-z^*}\mathrm{d}x $$ ...
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17 views

Solve integral equation with Laplace transforms (proving no solution exists)

This questions pertains to a lingering open problem in my research that I have had issues with solving. Let $Y_i\sim\operatorname{Gamma}(\alpha_i,\beta_i=\alpha_i/\kappa_i)$ for $i=1,2$ be ...
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14 views

condense integral expression or find its upper bound

Suppose that $f_{kj}:\mathbb{R}\mapsto\mathbb{R}$ is a sequence of delay kernels, $k,j=1,\ldots,n$, $a_{kj}\in\mathbb{R}$, and $y_{k}(t)$, $t>0$, a solution (unknown) to a delay differential ...
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22 views

Solving a second order Fredholm's Integral equation

I am trying to solve the following equation, $f(x)-\mu\int_{0}^1\cos(2\pi(x-y))f(y)dy=g(x)$ but I am having some trouble. I know that if we look at this as an operator we get $(I-T)(f)=g(x)$ with $T(f)...
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21 views

Solution to a differential equation with final value

I've seen the following differential equation with a final value in a book: $\dot{x}=J(y) \, e^{-r(t-t_f)}$, $x(t_f)=P$ where $t_f$ is a fixed final time. The solution given is: $x(t)=P-\int_t^{t_f} ...
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10 views

Fitting and Solving Integral Equations

Say I have a collection of data, $d_i$, that come from pixels $i$ arranged in a single row on a detector. Each pixel has a different field of view, which I'll say goes from $a_i$ to $b_i$ (angle from ...
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2answers
29 views

Solution of the following integral Equation $\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$

Consider that the following equation is solvable then analyze with respect to $\lambda$ $$\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$$ Can someone tell me how can I solve it ?
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46 views

Solution of the following integral Equation $\varphi(x) - \dfrac {\pi^2} {4}\int\limits_0^1 K(x,t)\varphi(t) \: dt=\dfrac{x}{2}$

$$\varphi(x) - \dfrac {\pi^2} {4}\int\limits_0^1 K(x,t)\varphi(t) \: dt=\dfrac{x}{2}$$ $$K(x,t) = \begin{array}{c} \dfrac{x(2-t)} {2} \quad 0\leq x <t\\ % just use \\ \dfrac{t(2-x)} {2} \quad t\...
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1answer
56 views

Solution to the integral equation with constant parameter

I wanted to solve this integral equation, I'll be grateful for explanation with step by step solution. $y(x)=1+α\int_{0}^xdp sin(x-p)y(p)$
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1answer
23 views

Can a vector of random variables be separated into dependent and independent variation?

Is it possible to uniquely decompose a vector $\underset{d_x \times 1}{x}$ of $d_x$ random variables into dependent and independent sources of variation? Suppose we know the distribution $P_x$ of a ...
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23 views

Integral equation with varying interval

I'm trying to find a function $S$ that would yield a known function $f$ when integrated over a interval on varying size. More specifically I'm varying $x$ sinusoidally around $x_0$ so $x = x_0 + \...
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10 views

Solution of Hilbert type Fredholm integral equation

Is there a formula similar to Hilbert inverse relation that would solve following integral equation in general form for $\phi$ as an integral? $$f(x) = \int_1^\infty \frac{\phi(y)dy}{x-y}$$ Provided ...
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18 views

General approaches/methods solving integral equations (Fredholm, Wiener-Hopf, etc.)

I have recently been exploring the Karhunen–Loève theorem and implied eigenfunctions as a method to construct low-rank approximations to Gaussian process covariance matrices. In general, discovering ...
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24 views

How can I determine functions satisfying the following conditions?

I am trying to understand mathematically what happens with the following system: A uniform, inextensible string of length $2s$ hanging at its ends from two poles of equal height $h.$ We know that the ...
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51 views

Resolvent Kernel of Fredholm integral equation $R(x,t;\lambda)$ is bijective in $\lambda$

We have the integral Equation: $$y(x)=f(x)+λ\int_{a}^{b} K(x,t)y(t) dt ,x\in [a,b],K\in L^{2}([a,b]\times[a,b]),f\in L^{2}([a,b])$$ Prove that $R(x,t;\lambda)$ is a one-to-one function in $\lambda$. ...
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1answer
25 views

solving integral equation, differentiation method

I have trouble with solving Volterra integral equation by using differentiation method $\varphi(x)=x-\int_0^x e^{x-t}\varphi(t)dt.$ picture of task so, I guess I need to find derivative of $\varphi(...
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13 views

Eigenvector of a “potential” operator

Let $H=L^2(\Omega)$ with $\Omega \subset \mathbb R^n$ bounded. Define $K:H\to H$ as $$Kf(\vec y):=\int_{\Omega} \frac{f(\vec x)}{<\vec x,\vec y>^\alpha}d\vec x\qquad 0<\alpha<0.5$$ Then $K$...
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1answer
23 views

How to solve the following integral equation [closed]

Is there any method to solve the following integral equation, either analytically or numerically: $$A(t) cos(\omega t) + \int_0^t \omega A(\tau) sin(\omega \tau) d\tau = f(t)$$ Where: $$A(t): ...
3
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2answers
140 views

Find the value of $f\left(\frac{1}{4}\right)$.

We have a continuous function $f$ on $[0,1]$ such that $$\int_0^{1} f(x) dx=\frac{1}{3}+ \int_0^{1} f^2(x^2) dx$$ Then find $f(1/4)$. I tried to find such function. Suppose that $f(x)=a+bx$, now we ...
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19 views

Solving linear integral equations using Schmidt's Method

Solve exactly or approximately using Schmidt's method: \begin{align*} u(x)+\int_{0}^{1} \frac{1}{1+xt}u(t)dt=1 \end{align*} I have done problems using Schmidt's method, but here I am stuck on how to ...
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1answer
28 views

Exercise on eigenvalues of an Hilbert-Schimdt operator

I have the following exercise: "Find the values $\lambda\in\mathbb{C}\setminus\{0\}$ s.t. the integral equation $$ \lambda\phi(x)-\int_0^1 e^{x+t}\phi(t)dt=f(x) $$ admits a unique solution $\phi_f$ ...
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241 views

How do I solve this integral equation?

Case 1: We will look at an easier problem first. Let $|\alpha|, |\beta| \leq \alpha_c, \alpha_c \leq \pi$. I want to solve for $\rho(\beta)$ in the following equation, where $P$ denotes the principal ...
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1answer
63 views

Solve the following integro-differential equation by Laplace transform

guys, I can solve this by using the convolution theorem however when it comes to Laplace I'm stuck somehow. Can someone help me with this, please? $$ \frac{dy}{dt}+2\int_{0}^{t}y(\tau)cosh(t-\tau))d\...
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1answer
116 views

How to prove that $u(r)=k \frac{1}{r}$ is the only solution for the integral equation $\int_{V'}\rho'\ u(r)\ dV' = constant$?

Consider a hollow spherical charge with density $\rho'$ continuously varying only with respect to distance from the center $O$. $V'=$ yellow volume $k \in \mathbb {R}$ $\forall$ point $P$ inside ...
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19 views

application of infinite Hilbert transform to an integral equation

Using infinite Hilbert transform pair, find the solution to the integral equation $$\frac{1}{1+x^2}=\int_{-\infty}^\infty\frac{Y(t)}{x-t}dt$$ Modifying the RHS a bit we have $$\frac{1}{\pi(1+x^2)}=\...
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0answers
21 views

Integral operator $A$ such that $A^2=I$

Let $A$ be the linear integral operator, $$ (Af)(x)=\int_{-\infty}^{\infty}K(x,y)f(y)dy, $$ such that $A^2=I$ where $I$ is the identity operator. My calculations: if $Af=g$ then $Ag=f$; $$ (Af)(x)=\...
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0answers
32 views

Injectivity for an integral operator

Consider the operator $$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$ with $k\in L^2((0,1)\times(0,1)).$ I want to know under what assumption the kernel is reduced to zero. i....
2
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2answers
236 views

Exponentiating a differential operator - “Pseudo-Schrodinger” equation

I am struggling to follow a calculation presented in the paper Statistical Mechanics of one-dimensional Ginzburg-Landau fields. An analogous calculation is presented inthe thesis A Study of the ...
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1answer
19 views

Boundedness of solution of second kind Volterra integral equation (Proof verification )

Consider the following second kind Volterra integral equation $$x(t)=\int_0^tk(s,t)x(s)ds+f(t),$$ where $f$ is continuous on $[0,T]$ and $k$ is continuous kernel for all $0\leq s\leq 0 \leq t \leq T$. ...
2
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0answers
37 views

Integral equation related to the Beta distribution

Suppose $z$ is drawn from a continuous distribution on $[0,1]$ with c.d.f. $F(z)$. Suppose that given $z$, $x$ is drawn from a Beta distribution with $\alpha=z\nu$ and $\beta=(1-z)\nu$, where $\nu>...
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0answers
22 views

Eigenvalues of second kind Fredholm integral operator

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
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1answer
32 views

Solving a Volterra Integral Equation with a Bessel Function

So, I am doing a project on Bessel Functions and one of the questions is: Solve the following Volterra Integral Equation of the First Kind. $$ \int_{0}^{x} J_{0}(x-t)y(t) dt = sin(x) $$ where, $J_{...
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0answers
22 views

Nonlinear Volterra integral equations, when kernel is constant

I am interested in the solution uniqueness, existence and positivity of such equation: $$ u(x)=u_0+k_2g(x)+k_1\int_0^xf(u(s))ds, $$ where $k_1,k_2\in\mathbb{R}$ and $g(x)$ is not known Hölder ...
1
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1answer
28 views

How to find Resolvent Kernel?

Find the resolvent kernel associated with the kernel $K(x,t) = |x-t|$ in the interval $(0,1)$ ? I tried solving it but I am stuck at the part where we have to split the integral.
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1answer
16 views

Proving Boundness of Two Linear Operators

I have that $K:C[0,1] \rightarrow C[0,1] $ and $K_N:C[0,1] \rightarrow C[0,1]$ where: $$K \phi (x) = \int_0^1 k(x,t) \phi (t) dt $$ $$K_N \phi (x) = \int_0^1 k_N(x,t) \phi (t) dt $$ Where $k(x,t):= ...
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0answers
15 views

Approximating the Solution of a Second Kind Integral Equation

I am defining the approximation $k_N$ to $k$ by the following construction: Let $h:=1/N $ and, $n=$ {$0, \frac{1}{2}, 1 , \dots , N$} let $t_n := nh = \frac{n}{N}$ And for $0 \leq x \leq 1$ and $n=1,...
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2answers
101 views

When is it the case that $\sum_{k=0}^nf(n,k) = \int_{\mathbf{R}}f(n,x) \, \mathrm{d}x$?

Some context: over the past two weeks, I have been solving an integral. I completed it last night. However, I just realized that I have essentially shown $$\sum_{k=0}^nf(n,k) = \int_{\mathbf{R}}f(n,x) ...

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