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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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 ...
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1answer
27 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
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30 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 $h$ ...
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25 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
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2answers
45 views

Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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0answers
41 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
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2answers
81 views

Integral equation/ODE

I have to find all the functions $f(x)$ such that $$f(x)=xe^{(1-x^{2})/2}-xe^{-x^{2}/2}\int_{1}^{x}t^{-2}e^{t^{2}/2}dt$$ which satisfies $$f(x)=1-x\int_{1}^{x}f(t)dt$$ I tried to equal both, but when ...
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18 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 $g(x)$ ...
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0answers
13 views

Solving a nonlinear volterra integral equation with two integrals each with a non separable kernel

I am trying to solve the nonlinear volterra integral equation ...
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4answers
51 views

Solve $y = 2 + \int^x_2 [t - ty(t) \,\, dt]$

While working on some differential equation problems, I got one of the following problems: $$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$ I have no idea what an integral equation is however, the hint ...
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3answers
33 views

Solving integral equation using Laplace transform

$x(t)+\int_0^t(t-\tau)x(\tau)=t^2$ Is $x(\tau)$ the equivalent of $d\tau$? How do I solve this particular equation?
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0answers
18 views

Writing $y'' + P(x)y' + Q(x)y = g(x) $ as a Fredholm Integral Equation

Is it possible to convert a general linear second order boundary value ode $$y'' + P(x)y' + Q(x)y = g(x), y(a) = y_a, y(b) = y_b$$ to a Fredholm integral equation, explicitly determining the Kernel ...
3
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0answers
59 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 ...
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0answers
19 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 ...
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0answers
21 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 ...
2
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2answers
66 views

solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$ |f(t,x)|\leq \phi(t)|x|. $$ Suppose ...
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1answer
72 views

How can i solve the integral equation

How can i solve the integral equation $$z(t) =\int_0^t z(q)(1-(t-q))\,dq+C?$$ Solving for the function $z$. I have access to Mathematica.
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0answers
38 views

Integral equation with convolution

I need to solve the following integral equation for $\phi(t)$: $$ \ln \phi(t) - c_2\int\limits_{-\infty}^{\infty} k(t-\tau) \, g(\tau,\phi(\tau)) \, d\tau = c_1 $$ On the web I found a solution for ...
2
votes
1answer
42 views

Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
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0answers
33 views

Integral Equation of convolution type

given is the following integral equation: All variables and functions are given, except for n(x). I need to find n(x). Does anybody have an idea how to approach this problem? Many thanks in ...
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4answers
54 views

A little doubt with integral equation

I have the next equation: $$\int_{0}^{t}h(\tau)e^{-(t-\tau)}\mathrm{d}\tau=10e^{-t}\cos(4t) \tag{1}$$ Derivating both sides, I get: $$h(t)e^{-(t-t)}=h(t)=10[(-1)e^{-t}\cos(4t)+e^{-t}(-4)\sin(4t)] ...
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0answers
32 views

Integral equation solution in power series

Given the integral equation $$ g(x)= \int_{-\infty}^{\infty}K(x-y)f(y) \, dy$$ for a known function $ g(x) $ and kernel $ K(x)$. Of course I know this is a Wieener-Hopf integral equation but I ...
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0answers
27 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 ...
3
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1answer
90 views

Showing that a sequence of Picard iterates converges

I have a sequence of functions: $$y_{n}(x) = 1 + \int \limits_0^x 1 + t^2 + y_{n-1}^2(t)\,\mathrm dt$$ With $y_0 = 1$. I'm trying to show that this converges in a box $-1 \le x \le 1$ and $-10 \le ...
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1answer
118 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
0
votes
0answers
13 views

inequality for compact operator

Let $K(x)$, $x\ge0$ be a nonnegative-valued continuous function with support $(0,\infty)$ and such that $\int_0^\infty K(x)\,dx=1$. Let $\mathcal{K}$ be an integral operator given by $$ ...
2
votes
2answers
59 views

Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
2
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1answer
54 views

An integral equation $x(t)=a-\left(1-x(0)\right)e^{-\int_{t_1=0}^tx(t_1)dt_1}$

Consider ${\rm x}\left(0\right)$ is a fix positive real number, and we have following equation: $$ {\rm x}\left(t\right) =a - \left[1 - {\rm x}\left(0\right)\right]\exp\left(-\int_{0}^{t}{\rm ...
6
votes
1answer
67 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
1
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0answers
29 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. ...
2
votes
1answer
21 views

Homogeneous Integral Equations

In Arfken (3rd ed) ex. 16.5.1 he derives the integral equation for a one dimensional linear oscillator that includes the Green function (eq. 16.148). This equation is a homogeneous integral equation. ...
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23 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
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3answers
132 views

Solving a non-linear integro-differential equation

I am trying to solve the following equation $$ f^2(x) - g^2(x) = \alpha\int_0^x f(u) (x-u)du $$ For $\alpha=0$ we get $f=g$. I would like to see how the solution moves away from $g$ when I increase ...
2
votes
1answer
37 views

Solving $g(x)=\int_{3}^{x} g(t) dt$

The question is what set of continuous functions solves the problem $g(x)=\int_{3}^{x} g(t) dt$. My answer so far: g(3)=0, g'(x)=g(x)-g(3) therefore g(x)=g'(x)=$ce^x$. Obviously $ce^x=ce^x-ce^3$ ...
0
votes
2answers
41 views

Integral equation involving Binomial distribution

I am trying to find the form of a function $u^{(n)}(p)$ which satisfies $\forall k \in [0,n] \int_0^1 dp\, u^{(n)}(p) \binom{n}{k} p^k(1-p)^{n-k} = \frac{1}{n+1}$. This is a private case of a more ...
1
vote
1answer
36 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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1answer
28 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
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votes
1answer
19 views

Simple integral equation

There is a simple integral equation: $$\int_0^\infty p(s) ds = 1- \frac{1}{\lambda}p(0).$$ Do you know how to solve it for $p(0)$ given we know $p(s)$ for $s>0$? Or maybe some references to methods ...
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0answers
27 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 ...
4
votes
0answers
79 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 ...
0
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1answer
28 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
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1answer
27 views

Is it true that $\displaystyle (Tf)(x)=g(x)+\int_{a}^b k(x, y)f(y)\ dy$ is a contraction in $(C^0([a, b]), d_\infty)$?

In $C^0([a, b])=\{f:[a, b]\longrightarrow \mathbb R: f\ \textrm{continuous}\}$ consider the metric $$d_\infty(f, g)=\sup_{x\in [a, b]}|f(x)-g(x)|.$$Let $T:C^0([a, b])\longrightarrow C^0([a, b])$ given ...
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0answers
72 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 ...
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0answers
35 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 $z=0$. ...
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0answers
95 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: ...
0
votes
0answers
43 views

How to solve an integral equation?

Consider the following integral equation: $$\log f(x)+\frac{\displaystyle 1}{\displaystyle 2\,\pi}\displaystyle\int_{\mathbb{R}^2} f(y)\log|x-y| dy+f(x)=0\tag{*}$$ How to prove the existence of ...
1
vote
0answers
29 views

Existence and uniqueness of an integral equation

Does this equation $$f(u)=1+\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{1}{(u-v)^2+1}f(v)dv$$ has a bounded continuous solution? Is this solution unique? Here $f$ is defined over $\mathbb{R}$ and ...
0
votes
0answers
31 views

Regulairty of eigenfunctions of singular integral equations

Can you provide a proof or a reference, to study from, for the following problem: Let $\Gamma$ be a real analytic rectifiable closed curve in the plane, $ds$ is the arc-length , and kernel $K(z,w)$ ...
0
votes
1answer
86 views

Computation Method to solving Homogeneous Fredholm Integral Equation of Second Kind with Symmetric Kernel

I am attempting to write a program that will be able to numerically solve a homogeneous Fredholm Integral Equation of Second Kind, with a Symmetric Kernel. I have been looking through textbooks and ...
0
votes
0answers
32 views

Resolution of numerical solution of first-kind Volterra integral equation

This is a Volterra integral equation of the first-kind with a convolution-type kernel $k(t-t')$: $f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$ It is well known and expected that when ...