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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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 ...
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2answers
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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 ...
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34 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 ...
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1answer
31 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 ...
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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 - ...
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30 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) ...
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1answer
31 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 ...
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24 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 ...
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1answer
24 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 : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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21 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 ...
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41 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 $$ ...
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1answer
46 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 ...
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2answers
38 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 ...
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1answer
20 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 , ...
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32 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} = ...
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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 ...
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40 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 $h:\mathbb{R^2}\rightarrow \mathbb{R}$ ...
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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 ...
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1answer
19 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 ...
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1answer
16 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 ...
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1answer
37 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 ...
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11 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 ...
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2answers
126 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 ...
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44 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 ...
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14 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 ...
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2answers
48 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 ...
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48 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 ...
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1answer
58 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 ...
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1answer
47 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 ...
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0answers
62 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 : ...
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31 views

Solution of a partial integro-differential equation

I want to find a solution for the following equation $$ \partial_y u(x,y) + \partial_x u(x,y) + \int_{0}^{x} r(x-x') u(x',y) dx' $$ $$ u(x,0)=0 \quad u(0,y)=\delta(y) $$ in $(x,y) \in ...
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1answer
45 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
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22 views

How did Poisson discover his integral formula?

I am quite curious about the history behind it. His derivation should be different from those on today's textbooks.
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47 views

Numerical solution of the Volterra equation with an exponential factor

Given : $$u(x)=x+2 \int_0^x e^{x-t}u(t)dt$$ Solve the Volterra Equation numerically using Trapezoidal Rule in $(0,5)$ choosing $n=8$ and compare with the exact values. The Exact Solution I ...
2
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0answers
52 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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0answers
41 views

Can my wrong derivation of the Gamma function be fixed?

I found the following simple but wrong derivation of the Gamma function: We start from the definition of the exponential function $$ e^x = \sum_{k=0}^{\infty}\frac{x^k}{k!} \\ \Rightarrow 1 = ...
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1answer
55 views

Show that $\int_0^1 \phi^2(x)dx$ does not exist where $\phi(x)=x^{x-1}$

iI am currently studying integral equations from the book "Integral Equations" by Harry Hochstadt. In its second exercise (page $42$) it is asked to (Q.No $2$) show that $\displaystyle \int_0^1 ...
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50 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
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1answer
40 views

Solving the following Volterra Integral Equation?

How do I solve the following Volterra, non-homogeneous, $1st$ kind Integral Equation : $$ \dfrac{x^2}{2}=\int_0^x (1-x^2+t^2)u(t) dt$$ I know I cannot apply Laplace Transform because the kernel is ...
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17 views

Why is this restriction necessary?

I have to prove that if $\omega$, $\phi$ are 2 solutions to the equation $$\alpha(t)=\int_{0}^{t}f[s,\alpha(s)]ds$$ That then for $t\geq 0$ $$\phi(t) - \omega(t) = ...
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0answers
42 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
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1answer
81 views

Check whether the extremal has weak minima or weak maxima

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its weak maximum on all its extremals weak minimum on all its extremals weak maximum on some, but not on all of its extremals weak ...
3
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1answer
101 views

Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
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0answers
41 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
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1answer
47 views

Solving an equation by Laplace transform

Consider the following equation: $$ y^{\prime\prime}(x) +x = \int _0 ^x (x-u)y(u)du \qquad y(0)=0 \quad y^{\prime}(0)=1$$ I solved it by Laplace transform and got $-\sinh x$ as a solution. It is ...
2
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1answer
83 views

Fredholm integral equation [closed]

How can I solve the following fredholm integral equation $$ψ(x)=x+λ\int_{0}^{2π}|x-t|ψ(t)dt$$ The kernel contains absolute value
2
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0answers
207 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 ...
3
votes
0answers
71 views

Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is ...
3
votes
1answer
51 views

Integral $\int_\tau^\infty e^{\frac{-g_m}{\bar\gamma_m}}\frac{dg_m}{1+Pg_m}$ [closed]

$$I=\int_\tau^\infty e^{\frac{-g_m}{\bar\gamma_m}}\frac{dg_m}{1+Pg_m}$$ As you know exponential integral define in [0 inf], but I want to calculate it in [thu inf]. I'm really appreciating everyone ...
0
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1answer
27 views

Solution to Fredholm equation of the second type with symmetric Gaussian kernal

Is something known about the solution to Fredholm equations of the 2nd type of the following form: $\displaystyle f(x) = g(x) + \int_{-k}^k f(y) h(x-y) dy$ where $f: [-k, k] \to \mathbb{R}$, $g(x)$ ...