Questions tagged [integral-equations]
This tag is about questions regarding the integral equations. 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.
944
questions
0
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29
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Fredholm integral equation, exercise 12 Functional analysis Kreyszig
I'm trying to do a exercise of Kreyszig book of functional analysis but I'm stuck, I'm trying to solve the integral equation
\begin{equation}
x(s)-\mu \int_{0}^{2\pi}sin(s)cos(t)x(t)dt =\hat{y}(s)
\...
- 93
0
votes
0
answers
31
views
Seeking Detailed Explanation for Transforming an Integral Equation Using Euler's Formula and Error Function
I am working on understanding the transformation of a specific integral equation into a simpler form using Euler's Formula and the Error Function. The original equation is:
$$
u(x, t) = u_0\left\{1 - ...
- 1
1
vote
3
answers
137
views
Solve $f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau$
The question:
$$\begin{equation*} f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau \end{equation*}.$$
I believe we need to take the Laplace transform of all terms. I am getting stuck with this part:
...
- 21
1
vote
2
answers
49
views
Dimension of null space of an operator $T$
Let, $K(x,y)$ be a kernal in $[0,1]\times [0,1]$ defined as $K(x,y)=\sin(2\pi x)\sin(2\pi y)$. Consider the integral operator
$$T(u)(x)=\int_0^1 u(y)K(x,y)\,dy$$
where, $u\in C[0,1]$. Which of the ...
- 12.9k
1
vote
0
answers
45
views
Is there a general procedure to solve numerical integral equations with non-elementary integrals?
I'm feeling a little lost in trying to solve equations of the form:
$$f(x)+\int_a^b\phi(x,t)\mathrm{d}t=0$$
Where the integral in the LHS is non-elementary, and the variable $x$ is the unknown.
If the ...
0
votes
0
answers
123
views
Fredholm equation of the second kind with a quotient kernel
I'm trying to find a solution to a Fredholm equation of the second kind of the form
$$f\left(x\right)=g\left(x\right)+\lambda\intop_{a}^{b}\mathcal{K}\left(\frac{t}{x}\right)f\left(t\right)\mathrm{d}t....
- 1
0
votes
0
answers
19
views
Solving for generalized eigenfunction of weakly divergent integral operator
I'm interested in solutions to the generalised integral eigenfunction equation
$$
f(z^2,d-1)=2 \int_{z}^\infty \frac{f(r^2,d)}{\sqrt{1-\frac{z^2}{r^2}}}dr = \int_{z^2}^\infty \frac{f(y,d)}{\sqrt{y-z^2}...
1
vote
0
answers
59
views
Is it possible to write any arbitrary partial differential equation as an integral equation?
Note that I am not a mathematician; I am simply deducing using the very fallible means of deduction via intuition, which by no means is rigorous.
My question concerns the possibility for any PDE to be ...
- 123
3
votes
1
answer
104
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Fredholm Integral Equation of the Second Kind in $L_2[0,1]$
Given space $L_2[0,1]$ and the equation
$$\displaystyle x(t) + \lambda \int_{0}^{1}(\frac{1}{2} - |t-s|)x(s) ds = \cos(\pi*t)$$
And I want to find a solution to the equation for all values $\lambda \...
- 107
2
votes
0
answers
142
views
Eigenfunctions of the integral kernel 1/(x^2 + x'^2)
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
- 21
2
votes
0
answers
65
views
Cauchy integral equation with derivative
Does anybody know the solution of this singular Cauchy-like integral equation:
$$
y(x) = \int_{-\infty}^{\infty} \frac{y'(x')}{x-x'}dx'\\
y(0) = 1, \lim_{ \lvert x \rvert \to \infty } y(x)= 0
$$
The ...
1
vote
0
answers
40
views
How to calculate the kernel of an integral given the original function and its product [closed]
I am trying to solve for the kernel of the following integral.
$\int_{-\infty}^{\infty}K(x,t)f(t)dt = g(x)$
I know g(x) and I know f(x) but I am unsure of how I may solve for the kernal. I am trying ...
0
votes
0
answers
72
views
Solve the integral equation for $f$. [duplicate]
Find all functions $f$ that satisfy:
$$\int f(x)dx \cdot \int (1/f(x))dx = -1$$ So far, I have tried a handful of methods. I have substituted a variable $u$ for $f(x)$, I have substituted an ...
0
votes
0
answers
48
views
Existence of solution to "weird" integral equation
in my current work I come across an integral of the form
\begin{align}
x(a) = \int_{\Omega} f(a, u) x(u) du
\end{align}
where $\Omega$ is $\Omega \subseteq \mathbb{R}$, e.g. $\Omega = (0,1)$. The ...
- 36
1
vote
1
answer
79
views
About a counterexample for an integral-functional equation in number theory.
I was reading
http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf
How did the counterexample for the equation on page 8 look like ??
Specificly :
(quote)
“Tur´an’s lecture (probably a quite ...
- 15.3k
1
vote
1
answer
61
views
Asymptotic solution of an integral equation
Consider an integral equation of the form
$$\sigma_B(\Lambda)+\int_{-B}^{B} K\left(\Lambda-\Lambda^{\prime}\right) \sigma_B\left(\Lambda^{\prime}\right) d \Lambda^{\prime}=f(\Lambda)$$
where the ...
- 685
0
votes
2
answers
90
views
Fredholm integral equation of the second kind with constant kernel
I'm trying to read Kress' Linear integral equations, and I'm stuck at the first example. There must be something obvious I'm missing, and to that end, should I read something before this text?
$f(x)=\...
0
votes
1
answer
112
views
Solution of a simple integral equation [closed]
I have the following integral equation:
$$
f(x)=\exp{\left(-\int_{-x}^{\infty}f(y)\,dy\right)}\,\,,
$$
where a condition on $f(x)$ holds:
$$
f(x=0)=\frac{1}{2}\,.
$$
I know that the solution is:
$$
f(...
4
votes
1
answer
78
views
An analytical solution of the integral equation $ \int_0^\rho \left( \frac{s}{\rho} \right)^3 f(s) \, \mathrm{d}s +\int_\rho^1 f(s)\,\mathrm{d}s=1$
While elaborating on the solution for the Green's function of a mechanics problem involving disks moving on an interface, I came across the following integral equation for the unknown function $f(s)$:
...
- 395
0
votes
1
answer
38
views
Does this integral equation have a non-zero solution?
Let $C[0,\pi]$ denote the vector space of all continuous functions on the closed interval $[0,\pi]$. Does there exist a non-zero function $x(t)\in C[0,\pi]$ such that the equation $$\int_0^\pi (\sin(s)...
- 915
4
votes
0
answers
83
views
Derivative of singular integrand
I am trying to differentiate this integral with respect to $x$:
$$T(x,t) = {1\over\sqrt\pi} \int_0^t {g(s) \over \sqrt{t-s} } e^{-{x^2\over 4(t-s)}} ds$$
According to this paper the derivative with ...
- 141
2
votes
1
answer
81
views
Solution Integro differential equation
I try to solve the two integro differential equations
$f(at)=\frac{df(t)}{dt}$ and $f(at)=\int_{0}^{t}f(\tau)d\tau$ $a\gt0$.
Do you have an idea to suggest to me.
Thank you very much for your kind ...
- 159
2
votes
1
answer
68
views
Prove existence and uniqueness $f\in C([0,1])$ such that $f(t)+\frac{t}{2}\cos(f(t))=\int_0^1f(ts)s^2ds$
Prove there is a unique $f\in C([0,1])$ such that
$$
f(t)+\frac{t}{2}\cos(f(t))=\int_0^1f(ts)s^2ds.
$$
Any idea how to approach the problem ?
I considered to use banach-theorem(fixed-point) but I am ...
- 2,292
7
votes
1
answer
463
views
Solving $\frac{\partial}{\partial t} f =hf+ h \int \mathrm {d} i\, h f$
I'm looking for the solution of partial differential equation
$$\frac{\partial}{\partial t} f(i,t) =\left(a f(i,t) + b\int_0^\infty \mathrm {d} i\, h(i) f(i,t)\right)h(i)$$
Where
$$f(i, 0)=1, \int_0^\...
- 4,873
0
votes
0
answers
19
views
How to determine two functions from an equation relating their Fourier series coefficients
I'm trying to solve a boundary value problem (Laplace's equation within a rectangle), with a peculiar combination of boundary conditions: one side of the rectangle has a homogeneous Dirichlet boundary ...
- 121
0
votes
0
answers
24
views
Solving for a distribution function in Python
I am trying to solve the following equation for $f(E)$
$g(E)=\int_{E}^{E+a}f(E')dE'$
where $a$ is a constant, and $g(E)$ is known numerically. I used Leibniz's integral rule to solve for:
$\frac{dg(E)}...
3
votes
0
answers
74
views
Filling functions on hypercubes given the function's integrals over each coordinate
$\newcommand\dif{\mathop{}\!\mathrm{d}}$
Suppose $f:[0,1]^2\rightarrow\mathbb{R}_{>0}$ and that:
$$\int_0^1{f(x,y)\dif x}=a(y),\hspace{1cm}\int_0^1{f(x,y)\dif y}=b(x),$$
for some known functions $a,...
- 715
1
vote
0
answers
56
views
Looking for an integral operator with eigenvalues $n+\frac{1}{n^2}$
We have the following integral operator
$$
Ku(t)=\int_0^1 G(t,s)\ u(s)\ ds,\quad u\in L^2[0,1],
$$
where
$$G(t,s)=
\begin{cases}
s(1-t)& 0\leq s\leq t\leq 1\\
t(1-s)& 0\leq t\leq s\leq 1
\...
- 7,757
0
votes
1
answer
63
views
Find closed formula for integral equation with two variables
I have found a recursive solution to a problem which I am seeking a closed formula. The problem is equivalent to obtaining a function of two variables $P(a,b)$ with the following conditions:
$$0 \le a ...
- 623
0
votes
1
answer
43
views
Solution to homogeneous Fredholm equation of the second kind with asymmetric kernel
I am trying to solve the homogeneous Fredholm equation of the second kind:
$w(r) = \frac{c}{2}\frac{1}{\frac{\gamma + a}{v \sigma b} + 1 - \frac{\gamma}{f\sigma} r} \int_{-1}^1 d r' w(r'),$ $\quad\...
- 1
0
votes
0
answers
32
views
Solving System of Integral Equations
I was able to reduce an optimization problem to a system of 8 integral equations.
But I am not sure whether it is possible to solve them:
The equations are of the form (no Einstein summation not ...
- 711
0
votes
0
answers
55
views
Integro-functional equation
I'd like to solve (preferably analytically) the following integro-functional equation or at least to reduce it to some kind of a differential equation if possible:
$\varepsilon \left( {\omega ,q} \...
- 61
0
votes
0
answers
28
views
Solving acausal power-law convolution on a finite interval
I'm trying to solve the following integral equation for x(t):
$$
\int_0^1 x(\tau) \ G(\tau - t) \ d\tau = y(t)
$$
with acausal $G$ defined for the entire real line:
$$
G(t) = |t|^{-a}
$$
Some $y$ of ...
- 1
0
votes
1
answer
59
views
Does the Riemann-Lebesgue Lemma Apply to $L^1$ or $L^2$ Space?
In the literature on inverse problems, the Riemann-Lebesgue lemma is often used to demonstrate the ill-posedness of integral equations with square-integrable kernels. For example, in Groetsch (1984),
...
- 155
0
votes
0
answers
20
views
Numerical method to solve integral equation with a twist
I'm trying to find some Nash equilibrium in games with continuous action space.
It all comes down to finding some probability distribution $p$ in $[0, 1]$ such that for every $0< x < 1$, either $...
- 271
0
votes
2
answers
78
views
Minimizing an integral subject to terminal equality constraints
How can I determine the minimum value that the following integral can take
$$ J (y) = \int_0^1 \left( x^4 \left(y''\right) + 4 x^2 \left(y'\right)^2 \right) {\rm d} x $$
knowing that $y$ is not ...
4
votes
0
answers
78
views
Application of Kannan Fixed Point theorem to the integral equations
Let $(X,d)$ be a complete metric space. The map $T:X\longrightarrow X$ is called a Kannan type contraction if
$$d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], ∀x, y ∈ X,\alpha\in[0,\frac{1}{2}).$$
Kannan's ...
- 1,250
0
votes
0
answers
87
views
Find eigen functions corresponding to the same eigen value $1$ for the given integral equation
Find the eigen values and eigen functions of the integral equation :
$$\phi(x)-\lambda \int_{-1}^1\cos[\pi(x-t)]\phi(t)\,dt=f(x)$$
Consider the homogeneous integral equation
\begin{align*}
\phi(x)&...
- 12.9k
4
votes
2
answers
252
views
Prove the equation $\int_0^4 f(x(x-3)^2) \,dx=2\int_1^3 f(x(x-3)^2) \,dx$
Let $f:\mathbb{R}→\mathbb{R}$ be a continuous function. Prove the equation $$\int_0^4 f(x(x-3)^2) \,dx=2\int_1^3 f(x(x-3)^2) \,dx.$$
I have tried substituting $$x(x-3)^2 = u.$$
But after that I couldn'...
- 181
4
votes
2
answers
129
views
question about a trick that is used in Landau Mechanics
In Landau's Mechanics, there is a special trick for determining the potential energy from the period of oscillation.
Landau calculates the integral $\int_0^\alpha \frac{T(E) dE}{\sqrt{\alpha - E}}$ ...
3
votes
4
answers
168
views
Solving a non-homogeneous Volterra integral equation of the second kind
Q. If $y(x)=1+\displaystyle\int_0^x e^{-(x+t)}y(t)\,dt,$ then $y(1)$ equals:
(a) $0$, (b) $1$, (c) $2$, (d) $3$.
I tried the successive approximations method (starting with $y_0=0$), conversion to a ...
- 425
2
votes
1
answer
94
views
How can we solve this integral equation?
The following equation seems extremely simple:
$$
g(b)=\int_1^{\frac{1}{b}} \frac{X f(X)}{\sqrt{1-b X}} \, dX
$$
But how to solve it, that is, restore the function $f(X)$ from the known function $...
- 133
4
votes
1
answer
107
views
How to solve Volterra Integral Equation
I need help solving an Integral Equation that my professor showed without teaching to us. I've tried emulating the textbooks example but I still can't crack it.
$$ f(t) = \cos\, t + 4 \, e^{-2t} - \...
- 41
1
vote
1
answer
165
views
Fredholm Alternative Definition
I'm reading Introductory functional analysis by Kreyszig and it discuss Fredholm alternative in the following way:
8.7-1 Definition (Fredholm alternative). A bounded linear operator $A:X \rightarrow X$...
- 29
1
vote
0
answers
77
views
Proof of Campbell's formula for compound Poisson process
I am having difficulty proving the final bit of Campbell's formula, that is, given $C_t = H_1 + \cdots + H_{N_t}$ a compound Poisson process with iid jumps $H_k ~ \mu$ and an independent Poisson ...
- 5,233
3
votes
1
answer
97
views
How to solve a Fredholm equation with known $\lambda$?
I have the Fredholm equation,
$$\phi(x)=\sin x+\lambda\int_0^\pi\cos(x/2-3y)\phi(y)dy$$
and would like to solve it.
First, I found using the precondition for contraction of the Fredholm operator: $$|\...
- 2,574
0
votes
1
answer
221
views
Converting an Voltera equation to an IVP
I have the following Voltera equation
$$ u(x) = e^x + \int_{0}^{x}u(t)dt. $$
To convert it on an ODE, I differentiate and find
$u'(x) = e^x + u(x)$ where $u(0) = 1$ but I see the solution is
$$ u'(x) ...
- 236
2
votes
0
answers
42
views
Proof of the Peano existence theorem using a sequence of piecewise linear functions
If $f(y, t)$ is continuous in a rectangle $D = \{0 \le t - t_0 \le a$, $|y - y_0| \le b \}$ then the initial value problem $y' = f(y, t)$ with $y(t_0) = y_0$ has at least a solution in $[t_0, t_0 + \...
- 56
2
votes
1
answer
90
views
Inverting integral relation [duplicate]
Let's consider the following equation,
$$g(y) = \int_0^1 \frac{f(x)}{1-xy} \, dx$$
Let's assume that $g(y)$ is known. Is there a general way to determine $f(x)$? In other words, is there a way to ...
- 31
1
vote
0
answers
27
views
Fredholm integral equation --- almost of the first kind
I have an equation on $f(x)$ which has the form
$$\int_a^b K(x, y)f(y)dy = g(x) + f'(x=b)\times h(x).$$
The value of $f'(x=b)$ (first derivative of $f$, evaluated at $b$) is finite and given by a ...
- 111