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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1answer
53 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 ...
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1answer
27 views

What is the difference between single and double layer potential

I want to know the difference between single-layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
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1answer
20 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
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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 ...
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1answer
19 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 ...
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0answers
18 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 ...
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0answers
24 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 ...
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0answers
22 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
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2answers
31 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
75 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 ...
-1
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0answers
39 views

Solution Method for Volterra equation

I need to get this equation solved. \begin{align} \phi(u)=& \frac{2c}{\left(\sigma\varphi\right)^2u}\phi(0)\\ &+\frac{2}{(\sigma\varphi)^2} \int_{0}^{u} \left[ \frac{ ...
3
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0answers
38 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
22 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
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0answers
22 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
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2answers
20 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
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0answers
41 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
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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
28 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
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0answers
33 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
53 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
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1answer
39 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
26 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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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
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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
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1answer
61 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
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2answers
42 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
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1answer
27 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
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0answers
33 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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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
42 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
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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 ...
0
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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 ...
0
votes
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 ...
1
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1answer
56 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
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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
132 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
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0answers
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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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
49 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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0answers
63 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
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1answer
81 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
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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 ...
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0answers
74 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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0answers
34 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 ...
1
vote
1answer
45 views

How can I solve the following exercise

How can I solve the following exercise $$φ_1(x)=e^x-\int_{0}^{x}φ_1(t)dt+4\int_{0}^{x}e^{x-t}φ_2(t)dt$$ $$φ_2(x)=1-\int_{0}^{x}e^{-x+t}φ_1(t)dt+\int_{0}^{x}φ_2(t)dt$$
0
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0answers
23 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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2answers
49 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
votes
0answers
65 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 ...
1
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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 = ...
0
votes
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 ...