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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10
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848 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 ...
8
votes
0answers
216 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
125 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) ...
4
votes
0answers
146 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 ...
4
votes
0answers
142 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) = ...
3
votes
0answers
53 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 ...
3
votes
0answers
82 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
3
votes
0answers
29 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 $g(x)$ ...
3
votes
0answers
66 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
195 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: ...
3
votes
0answers
80 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 ...
3
votes
0answers
83 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
117 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
105 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
36 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
34 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 ...
2
votes
0answers
46 views

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
47 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 ...
2
votes
0answers
62 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
28 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 ...
2
votes
0answers
46 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
89 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
87 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
83 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
49 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
124 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.
2
votes
0answers
339 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
0answers
37 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...
1
vote
0answers
137 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
1
vote
0answers
35 views

Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
1
vote
0answers
47 views

Integral equation involving magnitude/modulus squared

I wish to solve the following integral equation that has popped up in my studies of focused light. If you notice, it looks almost like a homogenous Fredholm integral equation of the first kind. ...
1
vote
0answers
40 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
1
vote
0answers
77 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
1
vote
0answers
83 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
1
vote
0answers
42 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
1
vote
0answers
44 views

A solid rocket model: a differential equations set with ending time unknown

I am modelling a rocket model. Consider a solid rocket motor, (let us for sake of simplicity assume that the propellant is distributed in the case with a cylindrical shape: see shape in fig.1 of the ...
1
vote
0answers
25 views

asymptotics of the solution of an integral equation

Suppose we are given the integral equation $$ u(x;a) =v(x)+\int_0^a K(x,y)\,u(y;a)\,dy, $$ where $K(x,y)$ and $v(x)$ are known functions, and $a>0$ is a constant. What I am interested in is the ...
1
vote
0answers
33 views

Does this integral equation have a name, and is there a reference that deals with it?

The equation is $$S(z,t)=\int_a^b \int_c^d f(z,u) g(t,s) S(u,s) \, du \, ds$$ where $S$ is the unknown function and $f,\ g$ are fixed from the outset. I can approximate solutions in some special ...
1
vote
0answers
43 views

Solution for a Fredholm integral equation

I have an integral equation: $\int_0^T( t^\alpha + s^\alpha -|t-s|^\alpha) \phi(s)ds=\lambda\phi(t)$ for $\alpha\in(0,2)$. I think this is a Fredholm equation but I am not sure how to solve it. ...
1
vote
0answers
36 views

eigenvalues of homogeneous integral equation of second kind, with singular kernel

There is a homogeneous integral equation of second kind with a singular kernel(non-symmetric). The equation has the form: $\int_{a}^{b} k(x,t)Γ(t)dt =λΓ(x).$ It's 2-norm is infinity, $||k(x,t)||_2 ...
1
vote
0answers
31 views

Smoothness of solutions to Fredholm integral equation

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel ...
1
vote
0answers
94 views

Multidimensional (Fredholm) integral equation (of first kind)

Suppose, \begin{align*} g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s)h(t_1) + K_2(t_1,t_2,s)\right]ds %\\ %g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s) + K_2(t_1,s)h(t_2)\right]ds \end{align*} The problem is ...
1
vote
0answers
37 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
1
vote
0answers
172 views

solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
1
vote
0answers
80 views

Solution of Volterra convolution-type integral equation

The convolution-type Volterra integral equation of the first kind $f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$ can be solved (at least formally) by applying the Laplace ...
1
vote
0answers
51 views

Are there methods to solve coupled integral and integro-differential equations?

I have one fredholm integral equation $$ y(x)=f(x)+\int_0^1 K_1(x,g(x),t)y(x(t))dt$$ and an integro-differential equation $$ \frac{dg(x)}{dx}=h(x)+\int_0^1 K_2(x,y(x),t)g(x(t))dt$$. Are there any ...
1
vote
0answers
36 views

Are there methods to solve a system of coupled integral equations?

I was wondering if there were methods to solve a system of coupled integral equations. The example case I am thinking about is $$f(x)=g(x)+\int_a^xf(x^\prime)h(x^\prime) dx^\prime$$ ...
1
vote
0answers
62 views

Definite Integral of Periodic Function Multiplied by another Function

For one part of a problem I am working on, I am trying to show that $y'(t) \geq 1$ for all $t \geq 0$ when $y'= 1- \int^t_0 g(s)y(s) ds$ When $g(t)$ is periodic, $g(t) <0$ for all $t$, and ...
1
vote
0answers
37 views

Can we get a general solution to the following system of equations?

A system of equations $$\frac{df}{dt}(t)=-txg(t)-x, \frac{dg}{dt}(t)=(1-t)xf(t)+x$$ which was constructured from the integral equation $$ F_t (t,x)= x \int_0^{1-t} (1-s)F(s,x) ds + (1-t)x, $$ This ...
1
vote
0answers
27 views

An integro differential equation involving $f$,$f_h$ and second derivative of $f$.

Let $f_h$ be the Hilbert transform of the real function $f$. I need some help solving this integro differential equation : $$\alpha f_h(x) + \beta f''(x) = f(x)$$ A simple sinusoid doesn't seem to ...