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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12
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1k 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 ...
9
votes
0answers
231 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 ...
7
votes
0answers
159 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) \...
6
votes
0answers
379 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
5
votes
0answers
62 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 ...
5
votes
0answers
119 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $...
4
votes
0answers
50 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)\; dy=\frac{1}{...
4
votes
0answers
84 views

Integral equation $f(x) - \int_0^x f(t)dt = 0$

I'd like to know the solution (if solvable) to the following integral equation $$f(x) - \int_0^x f(t)dt = 0$$ Also I'd like to know what is the required mathematical background to be able to find a ...
4
votes
0answers
65 views

Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
4
votes
0answers
144 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) = \frac{e^{-\sqrt{|x|}}\sin(\...
3
votes
0answers
22 views

What is the value of C in this integral equation?

Consider the following integral equation: $$\int_0^1 С(yW(x))W^3(x)\,dx=F(y),$$ $$ W(x)=0.5( \cosh(kx)-\cos(kx)-A( \sinh(kx)-\sin(kx))) $$ $F$ is known. $A= \frac{ \cosh(k)+\cos(k)}{ \sinh(k)+\sin(k)...
3
votes
0answers
43 views

How to solve a Volterra integral equation of the second kind

I have the following equation \begin{equation} F(\theta) + (c)^{\frac{1}{c}-1}\sqrt{\frac{c}{2\pi}}\int\limits_0^\theta \frac{\theta^{\frac{1}{c}} - \tau^{\frac{1}{c}}}{(\theta - \tau)^{3/2}}\exp{\...
3
votes
0answers
58 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}\ t^{x-...
3
votes
0answers
45 views

Solvability of an integral equation

Is the following integral equation solvable ? $$ F(x)-\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\frac{\sin \gamma(x-y)}{\pi(x-y)}$$ and $$f(x)=e^{i\gamma x}$$ and $\gamma$ is a parameter.
3
votes
0answers
76 views

Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is $$ψ(x)=\...
3
votes
0answers
91 views

Contraction mapping principle application

I'm to prove that the following equation has a unique solution: $$f(x) = \int_0^1 e^{-sx} \cos(\alpha f(s)) ds.$$ (Here, $\alpha \in (0,1)$.) The form of the exercise screams to apply the ...
3
votes
0answers
141 views

Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
3
votes
0answers
58 views

Uniqueness of solution to integral equation for operator valued functions

Let $X$ be a Banach space. Suppose I have a 2 parameter family of bounded operators on $X$: $V(t,s)$, $0\leq s\leq t \leq T$, such that $V(t,s)x=U(t,s)x+\int_s^t V(t,r)H(r)U(r,s)x\,dr$ and $(t,s)\...
3
votes
0answers
214 views

Solving an integral equation.

Let $K(x,y)=(x+y)e^{-y^2/2-xy}$. I need a constructive way (not simply verifying it is a solution) to show that $f(x)=e^{-x^2/2}$ is the solution to the integral equation: \begin{equation}f(y)=\int_0^\...
3
votes
0answers
96 views

does f(x) have unique fixed point?

Let $g$ be a probability density function. We can assume about $g$, whatever we like (Only important thing, we know about random variable Y,which has $g$ as p.d.f is $P(Y<0)>0$.) Next, let $V(x)$...
3
votes
0answers
103 views

Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$ \int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t) $$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
3
votes
0answers
121 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
3
votes
0answers
108 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 ...
2
votes
0answers
28 views

Strong Solutions to Nonlinear ODE by Contraction Mapping

Consider the $1$-d ODE $$-u_{xx}+u-\epsilon u^{2}=f, \tag{1}$$ where $f$ is a nice RHS, say $f\in\mathcal{S}(\mathbb{R})$, and $\epsilon>0$. By using the Bessel potential, one looks for solutions ...
2
votes
0answers
52 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 ...
2
votes
0answers
63 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) \...
2
votes
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}(...
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 ...
2
votes
0answers
28 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.
2
votes
0answers
46 views

Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
2
votes
0answers
67 views

Solution of Abel type integral equation

I would like to know when (for what functions $f$) and how I can find integrable solution of equation \begin{align} f(x)=\int_x^{\infty}\frac{u(y)}{\sqrt{y-x}} \ dy, \end{align} where $u$ is unknown ...
2
votes
0answers
131 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
2
votes
0answers
44 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
2
votes
0answers
243 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 \end{align*}...
2
votes
0answers
56 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t \...
2
votes
0answers
79 views

FM signals and non-trivial solutions to a homogeneous Fredholm integral equation of the first kind

I am looking for any non-trivial solution to the following integral equation. That is, find any function $q(\theta)\neq 0$ which satisfies the following equation: $$\int_0^a \sin(x\cos\theta)q(\theta)...
2
votes
0answers
73 views

Reciprocal Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...
2
votes
0answers
88 views

reference request: a graduate level textbook on viscosity solutions of IPDEs

I am particularly interested in IPDE which arises from optimal stopping/control problems so I would like some references on integral-partial differential equations, which are related this area. I have ...
2
votes
0answers
65 views

Analytic continuation of an integral equation

Consider the (possibly singular or hypersingular) integral equation $$Z\int_{-\infty}^\infty B(x,x^\prime) \ A(x^\prime) dx^\prime = A(x)$$ where $Z$ indicates the finite part of the integral should ...
2
votes
0answers
123 views

Unicity of solution to an integral equation

Suppose we have the following integral equation $$f(x)=\int_0^\infty f(y)(x+y)e^{-x^2/2-xy}\text{d}y$$ where $f: [0,\infty)\rightarrow[0,\infty)$. The function $f(x)=Ce^{-x^2/2}$ solves the equation. ...
2
votes
0answers
96 views

applications integrals

I have the following problem to solve: $\int_{0}^{1}K(x,y)\phi(y)dy$ where: $K(x,y)=x(1-y), 0\leq x\leq y\leq 1$ and $K(x,y)=y(1-x), 0\leq y\leq x\leq 1$ already tried using the methods suggested ...
2
votes
0answers
92 views

What is the meaning of the definition below? Taken from a 1909 book on Integral Equations.

Definition: We say that the discontinuities of a function of $(x, y)$ are regularly distributed in $S$ or in $T$ if they all lie on a finite number of curves with continuously turning tangents, no one ...
2
votes
0answers
61 views

What kind of numerical methods are best applicable to this?

I'm wondering: what would be the best numerical method for solving a nonlinear integral equation of the form $$f(x) = a(x) + \int_{-A}^{A} K(x, t, f(t)) dt$$ where $f$ is the unknown function, a ...
2
votes
0answers
180 views

Fredholm integral equation of the first kind

Can we solve the following specific integral equation: $$ \int_0^1v^n(1-v)^{x-1}K(v)dv=f(x) ,x\in[0,1) $$ If it is solvable, I wonder whether its solution can be represented in a closed form.
1
vote
0answers
33 views

Is there a general method to go about deriving a definite integral for a given result?

I was reading a blog post earlier about the Sophomore's Dream and a question came to mind: Say we wanted to find a definite integral that gives the following result $$\sum_{n=1}^\infty \left(\frac{a}...
1
vote
0answers
37 views

Existence of solution of equation involving normal distribution

I've tried to show that the following equation has a solution: \begin{equation*} g(x)=\left[1-\left(2\int _{\mu}^{x}f(y)dy\right)^2\right]-8xf(x)\int _{\mu}^{x}f(y)dy=0, \end{equation*} where $f(x)$ ...
1
vote
0answers
14 views

For which $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $

For which values of $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $. This is a generalization of Solve ...
1
vote
0answers
23 views

integral equation and laplace transform

Solve the following integral equation $ u(x)= \cos x - \int_{0}^{x} (x-y)cos(x-y)u(y) dy $ I applied Laplace transforms to the above integral equation and so the initial equation is written as: ...
1
vote
0answers
10 views

How to determine if a homogeneous integral equation has non-trivial solutions?

The equation is $f(p) = \int_{0}^{\Lambda} dk \; \mathcal{M}(k,p;E) \, f(k)$, where the kernel $\mathcal{M}$ is $\mathcal{M}(k,p;E) = \frac{2/ m\pi}{1 - \sqrt{E + 3p^2/4}} \frac{k}{p} \log\left( \...
1
vote
0answers
37 views

Applied Mathematics Book on Integro-Differential Equations

I'm interested in teaching a course on integro-differential equations and their applications. I was wondering if anyone could suggest a decent book on the subject. I'm currently looking at "Nonlocal ...