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

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} (\...
4
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2answers
79 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
34 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
20 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): $$...
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1answer
71 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 y(s)ds\right)^2-...
1
vote
1answer
42 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
26 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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0answers
19 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: $$p(\tau_b,\...
1
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0answers
30 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
20 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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votes
1answer
28 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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votes
2answers
20 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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1answer
23 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
62 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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1answer
51 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
37 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
19 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 ...
0
votes
1answer
26 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: https://hal.inria.fr/inria-00473952/file/RR-7257....
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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 ...
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0answers
52 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'}} e^{-\frac{\left[\bar{x}(t)-\bar{x}(...
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0answers
64 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
114 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
47 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 ...
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votes
0answers
23 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
143 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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3answers
92 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
86 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 \mathrm{...
1
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0answers
31 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 $f(x,y)=g(x/y)/...
4
votes
2answers
70 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 \frac{d}{dx}\left(\frac{\...
1
vote
1answer
58 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 ...
0
votes
0answers
43 views

Second kind Volterra integral equation with weakly singular.

Consider the second kind Volterra integral equation. $$ f(t)=g(t)+\int_0^t f(t) K(t,s)ds $$ where $g(t)$ is continuous and $K(t,s)$ is a weakly singular kernel, i.e. $$ K(t,s)\leq \frac{C}{(t-s)^\...
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
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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 $\lambda$ here....
2
votes
1answer
43 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
13 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
38 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
56 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
52 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 ...
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vote
0answers
28 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
76 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
1
vote
2answers
94 views

Solve integral equation by converting to differential equation

One has the following integral equation: $$y(x)=\frac{1}{1+x^2}+\int_{0}^{x}\sin(x-t)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation?
2
votes
2answers
69 views

prove $ \int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$

prove that for every $ p,q \gt 0$ $$ \int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$$ I tried to start from one side and change variables to get something similiar to the ...
2
votes
0answers
27 views

A Gronwall type inequality for Fredholm operators

I am interested in finding out the result of Gronwall type inequality for Fredholm operators. What will be the form? How one can show such inequality for fredholm operators.
5
votes
0answers
61 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
votes
3answers
65 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} - \...
0
votes
0answers
44 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
0
votes
0answers
37 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), y(0,r)=1-\rho(r)...
0
votes
1answer
41 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
294 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 I(t,\omega)}{\...
4
votes
1answer
78 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? ...