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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34 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
4
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3answers
54 views

Fourier-like integral equation

(This question inspired by question A specific 1st order PDE which looks almost like a linear PDE.) Solve integral equation $$ g(x)=\int\limits_{-\infty}^\infty \rho(\omega)\left[e^{i\omega x} - ...
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0answers
16 views

Numerical Solutions of Fredholm Integral Equations of the First Kind

Can anyone recommend me some papers about numerical solutions of Fredholm integral equations of the first kind? Thanks in advance
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0answers
25 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
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1answer
21 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
0
votes
1answer
57 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
4
votes
1answer
65 views

Explicit solution for equation

The claim is that this equation has an explicit solution. $$\frac{\partial}{\partial t}c(x,t)=\frac{a}{\pi}\int_{\mathbb{R}}\frac{c(y,t)-c(x,t)}{(y-x)^2}dy.$$ What can one do to find this solution? ...
2
votes
0answers
19 views

Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
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0answers
42 views

Angular Integration

How does polar integrating works? It don't seems to be common because with my google research I couldn't find anything. Maybe there's another word for it? It is about solving the black body ...
3
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1answer
60 views

transform integral to differential equations

I found a similar system of integral equations in a paper. It says that it can be solved by differentiating and then using standard techniques. My question is, how can I differentiate such a system in ...
0
votes
1answer
21 views

Singular Integral Equation

I need to find an approximate solution $u(z)$ of the following equation: $\int_{-H}^0 q(s,z)\,u(s)\,ds = -2\,\rm{i}\,\xi_0(z)$ where $q(s,z) = ...
0
votes
1answer
24 views

Solving for a function inside an integral

Is there a way to solve for $f(x)$ when $$ g(x)=\int_0^x dx' W(x,x') f(x') $$ If it weren't for the x-dependence in $W(x,x')$, I could write for example, $$ f(x)=\frac{1}{W(x)}\frac{\partial ...
1
vote
1answer
23 views

Integral Equation Unknown Limits

What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution? The simplest example is the ...
1
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0answers
21 views

solve the integral equation using Resolvent kernel (plz help)

show that solution of the integral equation $y(x) + \int_0^x (x-s)y(s)ds={x^3/6}$ is $y(x) = \int_0^x s. sin(x-s)ds$ I'm only a beginner on this topic...Using Resolvent kernel I've got ...
1
vote
0answers
26 views

Numerically finding eigenvalues of a Volterra operator of first kind

I'm looking for a solution to the following problem - $\int_{-\infty}^{\infty} K(x-y) f(y) = \lambda f(x)$ Consider $K(x-y) = \left\{ \begin{array}{lr} e^{-(x-y)} & : x > y \\ 0 & : x ...
0
votes
0answers
24 views

which kernel has finite rank?

in integral equations $$x(s)=y(s)+\lambda\int_a^b k(s,t)x(t) dt\\k \in L^2[a,b]$$ which one of listed kernels has finite rank ? how to show (or proof)? ...
4
votes
2answers
33 views

Need solution to Volerra integro-diff equation

I need to solve a system of Volterra integro-diff equation of form $$ y(t) = x(t) - \int_{0}^{t} k(t-\tau) y'(\tau) \;\mathrm{d}\tau $$ where kernel is of form $$ k(t-\tau) = P(t)Q(\tau) $$ Is it ...
1
vote
1answer
78 views

Convert IVP to an equivalent Volterra integral equation

Convert the following initial value problem to an equivalent Volterra integral equation: $ \begin{cases} u'' -u' \sin x + \Bbb e ^x u= x \\ u(0)=1\\ u'(0)=-1\\ \end{cases} $ I ...
3
votes
0answers
39 views

Solvability of an integral equation

Is the following integral equation solvable ? $$ F(x)-\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\frac{\sin \gamma(x-y)}{\pi(x-y)}$$ and $$f(x)=e^{i\gamma x}$$ and $\gamma$ is a parameter.
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0answers
24 views

Solve integral equation that involves hyperbolic cosine

I'm trying to find a weight function w(x) that makes this integral 0 $\int_0^1 w(x) \cosh((\alpha_n+i\omega_n)x) \cosh((\alpha_n-i\omega_n)x)=0$, where where $\omega_n=(2n+1)\frac{\pi}{2}$ and ...
1
vote
0answers
23 views

Integro-differential eigenvalue problem

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
1
vote
2answers
21 views

Transforming the integral equation $u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b$ into its equivalent differential equation

Let $u \in C^2[0, 1]$ satisfy for some $ \lambda \neq 0$ and $a \neq 0,$ $$u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b.$$ Then show that u also satisfies $\frac{d^2u}{dx^2} + \lambda u ...
0
votes
0answers
50 views

Integral equation with exponential

I would like to solve the following integral equation for $u(t)$, where $\theta, \gamma, \lambda, \kappa$ and $\sigma$ are parameters, but I haven't managed to obtain a solution so far. Any hints on ...
0
votes
1answer
34 views

solve integral equation using the theory of compact operator

Find solutions of $$u(x)-\lambda\int^{2\pi}_0\sum_{j=1}^n\frac{1}{j}cos(jy)cos(jx)u(y)dy=sin^2x$$ for all values of $\lambda$. Find the resolvent kernel for this equation. (Find the least squares ...
0
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0answers
33 views

An MCQ for the solution of a non homogegeous Volterra's equation $y(x) = \int _ {0}^{x}(x - s)y(s)ds = \frac{x^3}{6}$

Let $y : [0, \infty) \rightarrow R$ be a twice continuously differentiable and satisfy $$y(x) = \int _ {0}^{x}(x - s)y(s)ds = \frac{x^3}{6}.$$ Then $y(x) = \frac{1}{6}\int_{0}^{x}s^3 sin(x - ...
0
votes
0answers
34 views

Galerkin method for the following integral equation

I have the following integral equation that I want to approximately solve for $u$ $$ u(x)=G(x_0,x)-\int\limits_{\partial D} \left\{ \frac{\partial G(y,x)}{\partial n(y)} +ik\beta(y) G (y,x) ...
2
votes
1answer
55 views

Extremizing the boundary value problem $I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$

Extremizing the boundary value problem $$I[y]=\int_0^1y'^2(x)\,dx+y^2(0)-2y^2(1)$$ My Thought: First, we use Euler-Lagrange equation and solving we get , $y(x)=C_1x+C_2$. Then we put it in ...
2
votes
1answer
40 views

The kernel $k(x,y)=\frac{y}{y^2+x^2}$ is a solution of which equation?

The kernel $$k(x,y)=\frac{y}{y^2+x^2}$$is a solution of (A) Heat equation (B) Wave equation (C) Laplace equation (D) Lagrange equation Which are correct ? I tried through ...
1
vote
1answer
28 views

How can I solve the following exercise

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
1
vote
0answers
23 views

Is there a solution to this integral equation?

Consider the following equation $$ H(y) = \int_{0}^{\infty} G\left(\frac{y-\phi_2(v)}{\phi_1(v)} \right) \exp(-v) ~\mathrm{d}v $$ where $\phi_i$ are well-behaved differentiable functions on the ...
1
vote
0answers
45 views

Integral Identity

A question from a multivariable calculus exam: I have tried lots of methods like integrating the RHS by parts. Any help would be appreciated. Find $w(y)$ such that the identity $$ ...
1
vote
1answer
65 views

Extremizing the following boundary value problem

Consider the functional $$J(y)=y^2(1)+\int_0^1y'^2(x)\,dx$$ with $y(0)=1$ , where $y\in C^2[0,1]$. If $y$ extremizes $J$ then find the value of $y(x)$. I tried through Bolza problem. Firstly ...
0
votes
2answers
45 views

Solve the following Fredholm Integral Equation

Solve the Integral Equation :$$y(x)=\frac{6}{5}(1-4x)+\lambda\int_0^1(x\ln t-t\ln x)y(t)\,dt$$ Let , $$y(x)=\frac{6}{5}(1-4x)+\lambda xC_1-\lambda\ln x C_2$$where, $$C_1=\int_0^1\ln ...
0
votes
1answer
32 views

For what value(/s) of $\lambda$ , solution of the following Integral Equation does not exist?

For what value(/s) of $\lambda$ , solution of the following Integral Equation does not exist ?$$y(x)=1+\lambda\int_0^1(1-3xt)y(t)\,dt$$ Let , $$y(x)=1+\lambda C_1-3\lambda xC_2$$where , ...
0
votes
0answers
35 views

how to deal with this integral equation

I posted this question before in the Physics stack exchange, but it was recommended to post it better here. While reading a paper I saw the following integral equation. $\frac{1}{g} = ...
0
votes
0answers
18 views

Determine whether the function $y(t) = T \{x(t)\} = \frac{1}{T}\int_{t-\frac{T}{2}}^{t+\frac{T}{2}}x(\tau)d\tau $ is Casual,Linear,Time-invariant

I am trying to determine whether the following function is Casual Linear Time Invariant $$y(t) = T \{x(t)\} = \frac{1}{T}\int_{t-\frac{T}{2}}^{t+\frac{T}{2}}x(\tau)d\tau $$ I know that Casual ...
1
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0answers
45 views

Find all solutions to integral equation

Let $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and $F:\mathbb{R}\rightarrow \mathbb{R}$ be given functions such that $\int_\mathbb{R} F(x) dx = 0$. Find all $h:\mathbb{R^2}\rightarrow \mathbb{R}$ ...
1
vote
1answer
46 views

Solutions of $u(x)=\int_{\mathbb R^n} |x-y|^p u(y)^{-q} dy$ are bounded away from zero

In one of research papers I am interested in, see this link or this if you cannot access, there is a lemma, Lemma 5.1, saying that if $n\geq 1$, $p,q>0$ and $u$ a non-negative Lebesgue measureable ...
0
votes
1answer
20 views

How can one solve this equation in $Z^2$?!

Ho can one solve the egality $2x+3y=xy$ ? I have to find a value of $x$ in fonction of $y$ so ? I have to add somthing and substrate it I added -2xy then $2x(1+y)-3y(1+x)=0$ Here im suck Can ...
0
votes
1answer
17 views

Evaluation of non-solvable/solvable x(?)

Here goes the question Assuming "x" is a real number,such that the equation is given $(x+\frac {1} {x})^2=3$,to evaluate $x^3+\frac {1} {x^3}$ And here goes my working,taking 2+ hours,only to ...
1
vote
1answer
63 views

Uniqueness of homogeneous Fredholm equation of the first kind

Suppose $K(x,t)$ is known and $$ \int f(x)K(x,t)dx=0 $$ Are there some known sufficient and \ or necessary conditions on $K(x,t)$ such that the only solution is $f(x)=0$ a.s.? ($f$ can be in a space ...
0
votes
0answers
12 views

How to solve for the prior probability distribution in this integral equation?

I've obtained the Bayesian posterior probability for a problem and found it to be equal to $$ z = \frac{\int_{0}^{1} p^{h'} p^{h} (1 - p)^t\Pr(p)\,dp}{\int_{0}^{1}\hspace{1.35em}p^h (1 - p)^t ...
4
votes
2answers
134 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
2
votes
0answers
45 views

Solution of Abel type integral equation

I would like to know when (for what functions $f$) and how I can find integrable solution of equation \begin{align} f(x)=\int_x^{\infty}\frac{u(y)}{\sqrt{y-x}} \ dy, \end{align} where $u$ is unknown ...
1
vote
0answers
15 views

How to numerically solve an integral equation with a Cauchy principle kernel?

Consider such a Fredholm equation of $f(x)$: $$ f(x) = g(x) + \lim_{\epsilon \rightarrow 0^+ }\int_{-\infty}^{+\infty} \frac{d y V(x-y)}{a^2+ i \epsilon - y^2} f(y) .$$ Here $V(y)$ is a nice ...
2
votes
2answers
50 views

Find continuous $f$ with period $1$ such that $f(x) =\int_0^1 f(x-t)f(t) dt$

The problem is to find all $f : \mathbb{R} \to \mathbb{C}$ that is continuous and has a period of $1$ (not necessarily smallest period) such that the following equality holds: $$f(x) = \int_0^1 ...
1
vote
0answers
71 views

numerical analysis of partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
1
vote
1answer
93 views

Homogeneous Fredholm Integral Equation

I'm having problem obtaining the solution of the homogeneous Fredholm Integral Equation of the 2nd kind, with separable kernel. I always get a zero if I use the normal method i was taught for the ...
0
votes
1answer
54 views

Fredholm integral equations

I'm having problem obtaining the solution of the homogeneous Fredholm integral equation of the 2nd kind with a separable kernel. I always get a zero if I use the normal method I was taught for the non ...
1
vote
0answers
80 views

Association of PDE's with Integral Equations?

We know the following associations : Volterra Integral Equations $\leftrightarrow$ Initial Value Problems Fredholm Integral Equations $\leftrightarrow$ Boundary Value Problems My questions are : ...