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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11 views

Unique solution of an integral equation in $L^1[0,1]$

Let $h\in L^1[0,1]$. Prove that there is a unique solution (almost everywhere) of the following integral equation: $$f(x)=h(x)+\frac{1}{2}\int_0^x\log(1+f(y)^2)dy$$ The idea is to use the fixed-point ...
4
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0answers
45 views

Looking for a function that satisfies some kind of mean value property

Given $a<b\in (0,1)$ and $\delta<1/2$, I need to find an integrable function $\gamma :(a-\delta,b+\delta)\to [0,1]$ such that $$\frac{1}{2\delta}\int_{x-\delta}^{x+\delta}\gamma(y)\; ...
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1answer
41 views

Solving integral equation with Fourier transform?

I'm trying to solve the following integral equation using Fourier transforms: $$u(t)+ \int_{-\infty}^{t} e^{\tau-t} u(\tau)\,d\tau=e^{-2|\tau|}$$ I tried to transform both sides of the equation using ...
2
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1answer
17 views

Express the solution of the integral equation in the resolvent form

Express the solution of the integral equation $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \cos(x+t)f(t) \, dt$$ in the resolvent form $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \Gamma(x,t;\lambda)\phi(t) \, ...
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1answer
20 views

Show that the nonlinear integral equation has a unique solution.

Show that the nonlinear integral equation $$f(x) = \int_0^1 e^{-sx}\cos{(\alpha f(s))}ds,$$ $0\leq x\leq 1$, $0<\alpha<1$, has a unique solution. I originally thought was some form of Fredholm ...
2
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0answers
36 views

Separable Kernel in Volterra integral equation

I can't get my head around why the kernel in the Volterra integral equation can't be separable. $$u(x) = f(x) + \int_a^x K(x,s)u(s)ds, x \in [a,b]$$ A separable kernel $K(x,s)$ is the one that can be ...
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vote
1answer
33 views

Given integral equation, find $y(1)$

Let $y(t)$ be a continuous function on $[0,\infty)$ whose Laplace transforms exists. If $y(t)$ satisfies $$\int\limits_0^t(1-\cos(t-\tau))y(\tau)d\tau=t^4\to(1)$$ then $y(1)=$ I was able to find ...
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2answers
48 views

Resolvent Kernel of Volterra Integral Equation

The resolvent kernal $R(x,t,\lambda)$ for the Volterra integral equation $$\phi(x)=x+\lambda\int\limits_a^x\phi(s)ds$$ is $\begin{array}1 1. e^{\lambda(x+t)} && 2. e^{\lambda(x-t)} ...
3
votes
0answers
50 views

Same values for Gamma Function

I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function. Using the integral representation, namely $$\Gamma[x] = \int_0^{+\infty}\ ...
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0answers
15 views

General solution of Fredholm integral equation of second kind with these conditions

Show that if $x_0(s)$ satisfies the equation $$x(s) = y(s) + \lambda \int_0^s K(s,t)x(t) dt \tag{$*$}$$ and the kernel $K$ has characteristic value $\lambda$ of rank $p$,that is, corresponding to ...
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votes
1answer
24 views

Ill-posed integral equation problem using Fourier Transfom

By using the Fourier Transform, show that the following equation $$\int_{-\infty}^{+\infty} K(x-y) g(y) dy = f(x), \qquad -\infty < x < \infty$$ is ill-posed. For overcoming ill-posedness of ...
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votes
1answer
21 views

Eigenfunctions for the symmetric kernel of an integral equation

The solution of the symmetric integral equation below: $$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{$*$}$$ with separable kernels method is $$g(s) = f(s) + \lambda \int_{-1}^{1} ...
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votes
2answers
68 views

Solve the integral equation $f(x) = x + \lambda \int_0^1 f(z)\,dz$

Find a closed-form solution for $f(x)$ in the following equation $$ f(x) = x + \lambda \int_0^1 f(z)\,dz $$ where $\lambda$ is a constant I tried integrating both sides from $0$ to $1$ but wasn't ...
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1answer
31 views

solve integral equation using adomian decomposition.

i am trying to solve a few integral equation problems prior to the exams. This particular one, however, doesn't to converge. or am I going about it the wrong way? The equation: $u(x) = 1 - x^2 - ...
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0answers
15 views

How to classify this integro-differential equation?

I have a system of three coupled integral equations for three unknowns $j(t), \bar{x}(t)$ and $\lambda(t)$ to be solved between $t=0$ and $t=T$ (b.c. are $\bar{x}(0)=x_0$ and $\bar{x}'(T)=0$): (1): ...
2
votes
1answer
56 views

Squaring a integral equation

If $y:[0,\infty)\to[0,\infty)$ is a continuously differentiable function satisfying $$y(t)=y_0-\int\limits_0^t y(s)ds$$ for $t\ge0$, then $y^2(t)=y_0^2+\left(\int\limits_0^t ...
1
vote
1answer
40 views

convert differential equation to Integral equation

$$ y''(x) + y(x) = x$$ with b.v conditions $$ y(0) = 1, y'(1) = 0 $$ Integrating $$ y'(x) - y'(0) + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$ $ let y'(0) = c_1 $ $$ y'(x) - c_1 + \int \limits _0 ^x ...
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1answer
22 views

Some good texts for integral equations

I am really interested in the theory of integral equations and I am just starting out on the reading. However, I am wondering what could be the best texts to look for (and from where, if possible). ...
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13 views

numerical solution of integral equation with unknown bound

I am reading a paper on High Harmonics Generation (HHG) and a Lewenstein model The paper is here. I would like to reproduce some results but I am stuck at the following problem. I have: ...
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0answers
26 views

Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
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0answers
17 views

Integral equation: existence

Let $H$ and $h$ be smooth functions of one and two variables respectively. Consider equation $$ H(x) = \int_\Bbb Rh(x,y)f(y)\mathrm dy \qquad \forall x\in \Bbb R. $$ When does it have a solution, ...
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1answer
23 views

Solve this integral equation using Laplace transform

Solve this integral equation using Laplace transform $$f(x)=x^2 + \int_{0}^{x}f^{\prime}(x-t) e^{-at} dt ,f(0)=0 $$ Please Help see mu answer below Thank you for your participation
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2answers
19 views

Find the Laplace transform for this function

Find the Laplace transform for this function $$f(x)=(1+2ax)x^{-\frac{1}{2}}e^{ax}$$ Please, help me see my answer below Thank you for your participation
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votes
1answer
18 views

Solve this integral equation using Fourier transform

Solve this integral equation using Fourier transform $$\int_{-\infty}^{\infty} \frac{f(t)}{(x-t^2)+a^2} dt= \frac{\sqrt{2} \pi}{x^2 + b^2}$$ for $b> a > 0 $ Please Help see my answer ...
2
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0answers
60 views

Integral equation: averaging

Let $X, Y, Z$ be Borel spaces, for simplicity we can assume that they are $\Bbb R$. Consider an equation $$ \int_{X\times Y}f(x,y,z) \kappa(x,\mathrm dy)\mu(\mathrm dx) = \int_{X\times Y}f(x,y,z) ...
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votes
1answer
38 views

Solve the integral equation with symmetric kernel

I have the following integral equation with symmetric kernel $$g(x)=\cos \pi x +\lambda \int_{0}^{1} k(x,t)g(t)\,dt $$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= \begin{cases} (x+1)t, ...
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0answers
32 views

Solving a Fredholm equation using Neumann series technique

It is a very simple kind of Fredholm equation: $$f(x)=x+\int_0^1(1+xt)f(t)\,dt.$$ I solved it and I know that the answer is $f(x)=-2$. But how can i solve this equation by the Neumann series ...
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0answers
16 views

A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$ 0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt $$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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1answer
22 views

Optimal Control of Integral Equations of the first kind

I have managed to find a couple of papers which deal with the optimal control of systems governed by integral equations of the second kind (e.g. here: ...
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0answers
20 views

Solving a weighed integral equation

I am struggling in solving a specific type of equation which simply pops out when we're dealing with weighed functions. I think the general context is worth mentioning. Let's say we have a discrete ...
2
votes
0answers
48 views

Optimization Problem Involving an Integral Equation

I have an integral equation for a probability current $j(t)$ given by: $$ \large\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} ...
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0answers
44 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...
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vote
4answers
106 views

What is the function that satisfies $\int_0^x f(t) dt=constant$ [closed]

$$\int_0^x f(t) dt=constant$$ What is the function that satisfies this condition ? Thank you!
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0answers
45 views

The resolvent kernel

Good morning. How can we prove that $$ \Gamma (s,t, \Lambda ) - \Gamma (s,t, M ) = \Lambda - M \times \int { \Gamma(s,t, \Lambda ) \times \Gamma (s,t,M) \, dx }$$ I tried to use the formula of ...
0
votes
0answers
21 views

Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
1
vote
1answer
58 views

How to find the characteristic number of a given integral equation?

How to find the characteristic number of the following integral equation? $$y(x)= \lambda \int_{0}^{1} (3x-2)ty(t)dt$$
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vote
3answers
76 views

Solution of the integral equation $y(x)+\int_{0}^{x}(x-s)y(s)ds=x^3/6$

So the question is to solve the integral equation to find out $y(x)$. $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\dfrac{x^3}{6}$$ So I find out the Resolvent kernel for $(x-s)$ and got $\sin{(x-s)}$, so I ...
2
votes
2answers
85 views

derivative over nested integrals

I have the following problem. First an example for two-variable functions Let $A=A(t,\tau)$ and $B=B(t,\tau)$ If I want to compute the following $\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t ...
1
vote
0answers
30 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume ...
4
votes
2answers
59 views

Check whether a functional has an extremal or NOT

Find the extremal of the functional $$J(y)=\int_a^b F(x,y,y')\,dx$$where , $F(x,y,y')=y'+y$ , for admissible functions $y$. From Euler-Lagrange equation , $\displaystyle ...
1
vote
1answer
50 views

Which method solves this integral equation? $\int_{-1}^{1}w(x)\,e^{tx}\,dx=6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$

Today I encountered this integral equation wrt. $w(x)$: $$\int_{-1}^1 w(x)\ e^{t x}dx = 6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$$ I never solved such equations, and when I tried to ...
2
votes
3answers
52 views

Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
0
votes
1answer
25 views

Checking whether the function $u(x)=e^x$ solves the integral equation $u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$

The function being $u(x)=e^x$ and the integral equation is $$u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$$ I can't do the integration and I'm confused about how to deal with the parameter $\lambda$ ...
2
votes
1answer
28 views

Solving a Volterra Integral Equation of the 2nd Kind

Can anyone help me in finding a closed-form solution to the integral equation $$x\left(t\right)=1-\lambda \int _{0}^{t}e^{-\alpha \left(t-\tau \right)} \cos ^{2} \left(k\tau \right) x\left(\tau ...
0
votes
0answers
11 views

when the solution of a volterra integral equation is a probability distribution?

I have a Volterra integral equation looking like this $$F(t) = \int_0^t \lambda(s)ds - \int_0^t{F(s)\lambda(t-s)ds}$$ is there any condition I can ask on $\lambda$ for the solution $F$ to be upper ...
1
vote
0answers
36 views

Love's equation $f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$

Let us consider Love's equation: $$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$ Is $f(x)$ a two times differentiable function?
2
votes
0answers
54 views

Cosine Fourier series solution of semi-major axis nonlinear integral equaton

Consider an integral equation $$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$ I am required to solve for $z(t)$. I approached this problem by considering the integral on right hand ...
0
votes
1answer
49 views

How to find exact solution of this volterra equation?

I was working on numerical solution of this equation (by block pulse). $$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ...
1
vote
0answers
22 views

How to solve this special case of Fredholm Integral Equation of the First Kind

General form of 'Fredholm Integral Equation of the First Kind' $f(x) = \int_a^b{K(x,t)\phi(t)} dt$ Where $\phi(t)$ is the unkown My special case is $1 = \int_a^b{k(t)\phi(t)} dt$ A trivial ...
1
vote
2answers
66 views

How can I solve this integral equation by converting it to a differential equation

Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation