The integral-equations tag has no wiki summary.
2
votes
1answer
108 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, ...
5
votes
3answers
195 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
$$
f(x) = \lambda \int_0^{\pi/2} \cos({x-y)}f(y)\,dy, \qquad ...
9
votes
2answers
162 views
Solving for unknown functions
I am not a mathematician, so excuse if my question is silly or badly stated. I have the following problem. I have 2 conditions on two unknown continuously differentiable functions:
...
5
votes
3answers
168 views
Integral equation with a constraint
I am stuck on the following problem:
given the following Volterra integral inhomogeneous equation:
$$\phi(x)=\exp(-x)+\lambda\int_0^x\frac{1}{x^2+t^2}\phi(t)dt$$ is it possible to solve it given the ...
1
vote
1answer
124 views
Asking solutions for the integral equations
This is from Berkeley Problems in Mathematics, Spring 86. It asks for $\lambda\in \mathbb{R}$, find all solutions of the following two equations:
$$\phi(x)=e^{x}+\lambda\int^{x}_{0}e^{x-y}\phi(y)dy; ...
0
votes
0answers
199 views
Numerical solution of an integro-differential equation with convolution
I have an integro-differential equation that I need to solve numerically. The equation is of the form:
$$\frac{dX}{dt} = cX - X\left(b + q\frac{dX}{dt}\right),$$
where $q\frac{dX}{dt}$ denotes the ...
2
votes
2answers
93 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 ...
0
votes
1answer
38 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? ...
2
votes
1answer
69 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) $
...
3
votes
0answers
72 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 ...
0
votes
2answers
54 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 ...
1
vote
1answer
68 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 ...
1
vote
1answer
137 views
Analytic integration formula?
Has it been proven that there is no formula for analytic integration? Is this still open, could there be a formula?
2
votes
1answer
110 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 $$Y(s)\dfrac{1}{s+2}$$
...
3
votes
2answers
102 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
1
vote
0answers
261 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 ...
4
votes
2answers
892 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 ...
2
votes
1answer
129 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 $R>0$:
$$\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}} f(R \cos(x))\, dx= 1.$$
0
votes
0answers
57 views
Solving intergration equation
I am trying to determine $\gamma$ such that $$\int_0^T\exp\left(\frac{3t(T-t/2)\gamma\sigma}{T^2}+\alpha t+\frac{\sigma^2}{2}\left[t-\frac{3t^2}{T^3}\left(T-\frac{t}{2}\right)^2\right]\right)\;dt=K,$$ ...
1
vote
1answer
289 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
votes
1answer
153 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 ...
1
vote
1answer
164 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?
...
0
votes
1answer
57 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 ...
3
votes
1answer
257 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:
$$\int_0^1 ...
1
vote
0answers
74 views
Integral equation solution hint.
I am looking for the family of distributions that satisfy the following condition:
$$\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]$$
...
1
vote
1answer
622 views
How to find eigenfunctions of a linear operator (follow-up question)
This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely.
I am interested in calculating characteristic ...
3
votes
1answer
298 views
How to find eigenfunctions of a linear operator
I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral.
As a special case suppose I am interested in this function:
...
0
votes
0answers
59 views
Finding a series of functions orthogonal to all $x^{2n}$ but one
We need to find a series of functions $f_m$ verifying the property
$$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$
We already have found $$f_0(x) = ...
1
vote
2answers
66 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
votes
1answer
93 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
vote
1answer
50 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
votes
1answer
135 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:
$$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$ ...
0
votes
1answer
99 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 ...
1
vote
1answer
151 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?
...
1
vote
0answers
45 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 $g$ ...
1
vote
1answer
84 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
votes
0answers
459 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 ...
1
vote
2answers
370 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
votes
2answers
181 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} ...
1
vote
1answer
108 views
Proving $\int_0^\infty \frac{w}{1+w^2} \sin wx dw=\frac{\pi}{2}e^{-x}$ with the Laplace transformation (and/or Fourier transformation)
Can anyone help me prove this with the help of the Laplace transformation?
$$\int_0^\infty \frac{w}{1+w^2} \sin wx dw=\frac{\pi}{2}e^{-x}$$
where $x>0$
EDIT: So I was wondering if you could ...
4
votes
1answer
212 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
$$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$
...
0
votes
3answers
299 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
votes
1answer
233 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
votes
2answers
321 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: ...
7
votes
0answers
184 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
191 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
vote
4answers
271 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
98 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 + ...
2
votes
1answer
492 views
How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)
I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\ $ in $\ L_2[0,1]\ $ space.
And I want to understand how to solve it, not just obtain an answer.
1
vote
0answers
239 views
What is the difference between resolvent kernel and iterative kernel of an integral equation?
As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
