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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Help solving this integral equation

I've got the following relation: For any $t_m \in (0, t_f)$, $$I(k | t_0,t_f,x_0) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(k_m | t_0, t_m, x_0) I(k-k_m|t_m,t_f,x_m) dk_m dx_m $$ I want ...
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58 views

Solving x from$\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$?

I found something strange when I try to solve this equatiin of $x$: $\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$, where $t$ and $c_0$ are constants. $f(\tau)$ is a known polynomial ...
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1answer
243 views

Difficult integral equation with function

Suppose $f(x)\in C^1([0,1])$ and $f(0)=0$. Let $$\phi(x)= \begin{cases} \int_0^x\frac{f(t)}{\sqrt{x-t}}dt &\quad\text{if}\quad x\in(0,1]\\ 0&\quad\text{if}\quad x=0 \end{cases} $$ (a) Prove ...
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1answer
151 views

Recurrence relation for a function with an integral of the function?

Pardon my lack of tex skills, but what is the recommended procedure in the following scenario: $$g(f) = 1+\int_0^{1-f} g\left(\dfrac{f}{1-x}\right)\,dx$$ I am not sure how to proceed in such a ...
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1answer
742 views

How to solve Volterra's integral equation of second kind with numerical solution

The problem occurs to me when I tried to solve \begin{align}E(x)=1+2(1-x)^2\int_{x}^{1}(1-t)E\left(\frac{x}{t}\right)dt\end{align} with $E(1)=1$ and $\lim_{x\to 0^+}E(x) \to +\infty$. I'd like to ...
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1answer
164 views

Numerical solution of fractional integro-diffrential equ. using collocation method?

problem comes from "Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan" ...
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1answer
134 views

General question about solving equations involving a definite integral

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, ...
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3answers
306 views

Continuous solutions of $f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy$ [duplicate]

Possible Duplicate: Eigenvalues of an operator Find all the functions $f \in C([0,\frac{\pi}{2}])$ which are solutions of $$ f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad ...
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2answers
294 views

Solving for unknown functions

I am not a mathematician, so excuse if my question is silly or badly stated. I have the following problem. I have 2 conditions on two unknown continuously differentiable functions: ...
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3answers
208 views

Integral equation with a constraint

I am stuck on the following problem: given the following Volterra integral inhomogeneous equation: $$\phi(x)=\exp(-x)+\lambda\int_0^x\frac{1}{x^2+t^2}\phi(t)dt$$ is it possible to solve it given the ...
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1answer
159 views

Asking solutions for the integral equations

This is from Berkeley Problems in Mathematics, Spring 86. It asks for $\lambda\in \mathbb{R}$, find all solutions of the following two equations: $$\phi(x)=e^{x}+\lambda\int^{x}_{0}e^{x-y}\phi(y)dy; ...
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2answers
102 views

Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$

For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both ...
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1answer
42 views

Need help with a differential equation -like problem.

$\forall y \in \mathbb{R}, \int_{-\infty}^{\infty} f(x)f(x-y)dx=f(y)$ I also know that $\int_{-\infty}^\infty f(x) dx$ converges and that $f$ is symmetric about the origin. What does $f$ look like? ...
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1answer
91 views

Matrix inversion of an analytical function

Following problem: I have a function $f(x_1,x_2)$ and Im looking for the inverse $finv(x_1,x_2)$ of the function which is defined through: $\int f(x_1,y)\cdot finv(y,x_2) d y =\delta(x_1,x_2) $ ...
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101 views

integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related

given two functios $ f(x) $ and $ g(x) $ related by $$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$ what ...
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2answers
60 views

Integral Hammerstein-like equation solution

In signal processing theory, I found this integral equation which I suppose to be of Hammerstein type: $$u(t)-\int_0^1\frac{\cos(\omega t+\phi)}{u(\phi)}d\phi=0$$ I didn't find anything in literature ...
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1answer
81 views

Laplace transform of convolution with modified limits

I have an expression such as $\int_0^{x+l}y(z)g(x-z) dz$ and I want to evaluate its Laplace transform w.r.t $x$ in terms of the Laplace transform of $y(x)$. I know that I can substitute $t=x+l$, and ...
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1answer
160 views

Laplace transform having this unusual property in convolution?

Here is the problem Solve $y'(t) = 1 - \int_{0}^{t} y(t - v)e^{-2v}dv$ The solution sets $\mathcal{L}(y) = Y(s)$ and does the following Notice that in step 1, they have $$Y(s)\dfrac{1}{s+2}$$ ...
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2answers
117 views

Solutions of an integral equation

Given the integral equation: $$\sqrt{f(x)}\int_{0}^{x}f(\tau)d\tau=g(x)$$ with g(x) known function, in what cases and how is it possible to solve it? Thanks
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463 views

Eigenvalues and eigenfunction of integral operator

Suppose we are given an integral operator $ g(x)=f(x)+ \lambda \int_{0}^{\infty}K(x,t)f(t)dt $ with the kernel $ K(x,t)=K(t,x)$. According Hilbert-Schmidt theory then, the function can be obtained ...
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2k views

Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow ...
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1answer
141 views

Integral equation $\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}}f(R \cos(x)) d x = 1$

Can we prove that there does not exist a function $f$, which satisfies this equation for all $R>0$: $$\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}} f(R \cos(x))\, dx= 1.$$
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2answers
686 views

Volterra integral equation of secong type solve using resolvent kernel

Solve the integral equation $$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I ...
2
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1answer
226 views

Volterra integral equation of second type

Solve the Volterra integral equation of second kind $$ y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds $$ I know two methods for such integral equations: Picard's method The mthod of ...
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1answer
261 views

Partial integro-differential equation

I don't know if there is a method to solve this following integro - differential equation: $$\partial_t{u(x,t)}\int_{0}^{x}u(\zeta,t)d\zeta=\partial_{xx}u(x,t)$$ Can someone give me some hint? ...
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1answer
61 views

can we have $u=0$ from the integral value 0?

If u is a bivariate function and we have $\int_\theta^{\theta+1}{\int_\theta^y{u(x,y)(y-x)^{n-2}}dx}dy=0$ for all $\theta\in\mathbb R$, here $n>2$ is a constant, can we infer that $u=0$ a.e. on the ...
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1answer
295 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
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76 views

Integral equation solution hint.

I am looking for the family of distributions that satisfy the following condition: $$\int_{-1}^{+\infty}f(x)x d x=0$$ and with this other conditions on $f(x)$: $$f(x)\ge 0 \text{ in }(-1,+\infty]$$ ...
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1answer
1k views

How to find eigenfunctions of a linear operator (follow-up question)

This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ...
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1answer
454 views

How to find eigenfunctions of a linear operator

I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral. As a special case suppose I am interested in this function: ...
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141 views

Finding a series of functions orthogonal to all $x^{2n}$ but one

We need to find a series of functions $f_m$ verifying the property $$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$ We already have found $$f_0(x) = ...
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2answers
72 views

homogeneous linear differential equation question

I was wondering if there is an analytical solution to the following homogeneous linear differential equation $$\dfrac {dM} {dt}=\dfrac {M} {\alpha \left( t\right) }e^{\beta\left( t\right) t}$$ which ...
5
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1answer
110 views

Anomalous integral equation

I'm trying to solve the following equation: $$\int_0^{f(x)}f(t)dt=g(x)$$ Differentiating under integral I obtain: $$f[f(x)]\frac{d}{dx}f(x)=\frac{d}{dx}g(x)$$ I know the function $g(x)$. Is there a ...
1
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1answer
54 views

Hammerstein stochastic integral equation

I'm in trouble with the following integral equation: $$\phi(t)=\rho\int_0^1 t^2 s \phi(s)^2 ds+\nu(t)$$ where $\nu(t)$ is a white gaussian noise with variance $\sigma$ and mean value $\mu$. Is it ...
5
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1answer
142 views

Integral equation and existence: $g(x)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$

I'd like to know how one would go about showing that the following function, $f$, that is almost everywhere positive exists: $$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$ ...
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1answer
150 views

Solving integral equation $a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$

I've got this nasty-looking integral equation involving taking two minimums: $$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$ where $\delta(\cdot)$ is the ...
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1answer
180 views

How does one solve this integral equation $1+ax=\int_{-\infty}^xf(x-t)dt$

I've run into having to solve this equation for $f(x)$: $$1+ax=\int_{-\infty}^xf(x-t)dt$$ Unfortunately, I am not familiar with solving integral equations. Can anyone help? Is is even soluble? ...
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47 views

How to show existence of two functions satisfying certain conditions? [duplicate]

Possible Duplicate: Finding two functions (density) $g,f$ satisfying some conditions I've asked this board before if they knew of a clever way to construction two functions $f$ and $g$ ...
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1answer
123 views

Comparison between solutions of ODE

Could anyone help on the following problem? Let R(t) be the solution to the integral equation: $R(t)=1+\int_{0}^{t}\frac{1}{R(s)}ds$, namely $R(t)=\sqrt{2t+1}$. Assume that X is continuous and ...
9
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696 views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
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2answers
509 views

solution of Fredholm integral equation of the first kind with symmetric rational kernel

How can be solved this Fredholm first kind integral equation: $$f(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{g(y)}{x+y}dy$$
2
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2answers
205 views

Evaluate the definite integral: $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$

We were recently asked to evaluate this - $y(x) = \int_{0}^{\pi} \sin(x+y(x)) dx$ I think we can start by breaking up the integral as $y(x) = \int_{0}^{\pi} \sin(x)\cos(y(x)) dx + \int_{0}^{\pi} ...
4
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1answer
281 views

How can I solve this integral equation in terms of Hermite polynomials?

It must be proven that the solution of the integral equation $$f(x)=\int_{-\infty}^{+\infty} e^{-(x-t)^2} g(t)dt$$ is $$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$ ...
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3answers
386 views

How can I solve this integral equation using characteristic values and eigenfunctions?

$$ f(x)= \int_0^1 e^{|x-t|} f(t) \, dt+x-1 $$ I can't solve it, because I can't find the boundary conditions?
4
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1answer
464 views

How to solve $t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$

I want to solve this equation. It reminds me something about Laplace transform. I am sure that I must use it order to solve it. $$t-2f(t) = \int_0^t(e^\tau- e^{-\tau})f(t-\tau)d\tau$$ How to do it? ...
7
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2answers
452 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: ...
8
votes
0answers
206 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
2
votes
0answers
283 views

Homogeneous Fredholm Equation of Second Kind

I'm trying to show that the eigenvalues of the following integral equation \begin{align*} \lambda \phi(t) = \int_{-T/2}^{T/2} dx \phi(x)e^{-\Gamma|t-x|} \end{align*} are given by \begin{align*} ...
1
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4answers
309 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
3
votes
1answer
105 views

Find $f(x)$ such that $f(x) = 1 + \frac1{x} \int_1^x f(t) \mathrm{d}t$. What am I doing wrong?

I need to find a continuous function defined for real and positive $x$ such that $f(x) = 1 + \frac1{x} \int_1^x f(t)\ \mathrm{d}t$. What I did is the following: $$\begin{align*}f(x) &= 1 + ...