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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31 views

Integral equation: $x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t$ [on hold]

Would you please find the function $f$ such that $$x f(x) = \int _0 ^x \int _0 ^t f(u) \ \Bbb d u \ \Bbb d t \quad ?$$ Thank you.
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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 ...
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2answers
45 views

On the solution of Volterra integral equation

I got stuck with some strange point, solving Volterra integral equation: $$ \int_0^t (t-s)f(s) ds =\sqrt{t}. $$ The solution can be obtained by ssuccessive differnetiation $$ \int_0^t ...
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0answers
10 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 ...
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630 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
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14 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: ...
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0answers
6 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( ...
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2answers
578 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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1answer
53 views

How to Numerically Solve an integral equation.

First I really doon't have any background with integral equations! That said, I would like to solve the following: $$\int_a^b \frac{K(t)}{t-x} \phi(t) dt=f(x) , a<x<b$$ where$$\int_a^b \phi(t) ...
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1answer
22 views

Find $f$ explicitly when $f_n$ is defined recursively and $\lim_{n\to\infty} f_n = f.$

Given that $f_1(x) = 0$ and \begin{equation} f_{n+1}(x) = e^{-2x} + \int_0^x e^{-2t}f_n(t) \; dt, \; \text{ where }n = 1,2,\dots \end{equation} identify $f(x)$ explicitly where $\lim_{n\to\infty} ...
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0answers
25 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 ...
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1answer
19 views

Use contraction mapping theorem to show that the integral equation has a unique continuous solution on $t \in [0,3]$

I have to use the contraction mapping theorem to prove that the integral equation with continuous functions $K(t,s)$ and $f(t)$, $\begin{equation*} x(t) = \lambda \int_{0}^{3} K(t,s) x(s)ds + f(t), ...
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40 views

Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$ x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds, $$ with $K$ given. ...
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0answers
17 views

Solve y′−∫x0y(t)dt=2 [duplicate]

How i solve this? I have not idea how to approach this differential equation. y′−∫x0y(t)dt=2 I get the probably familiar DE: y′′−y=0. please someone can solve this exercise
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3answers
59 views

Different formulations for multiple integral equation

On several papers, I found the following model for a multiple integral equation: $$g(s)=\int\limits_{\Omega} h(s,t)f(t)\,\mathrm{d}t$$ where $s,t \in \mathbb{R}^3$, and $\Omega \subseteq ...
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0answers
29 views

How do I solve this integral equation $x(t)-\mu \int_a^bcx(\xi)d\xi=v(t)$?

Also, how the corresponding Neumann series to this equation help obtain a convergence condition for a general integral equation like $$x(t)-\mu\int_a^bk(t,\xi) x(\xi) d\xi=v(t).$$ I think the second ...
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0answers
20 views

Collocation method for integral equation with monotone increasing kernel

Is it possible to approximate the solution ($f(x)$) of this type of integral equation if the kernel ($k(x,t)$) is a strictly monotone increasing function? $f(x-T)= g(x)+\int_0^{\infty } k(x,t) f(t) ...
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0answers
17 views

System of linear Volterra integral equation

Consider the Volterra integral equation $$ f(t) = g(t) +\int_0^t K(t,s) f(s) ds $$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
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0answers
21 views

Boundedness of the solution of the integral equation associated to the heat kernel

(Cross-posting http://mathoverflow.net/questions/232720/boundedness-of-the-solution-of-the-integral-equation-associated-to-the-heat-kern ) Let $\Omega$ be a bounded open set of $\mathbb R^n$ ($n\geq ...
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0answers
62 views

Integral Equation from zero to infinity

Is there anyone could help to solve the following problem: Suppose $\,h\left(x\right)\,$ is a known function and $\,y\left(x\right)\,$ is unknown, you may assume these two are nice functions. I am ...
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0answers
22 views

$n-$dimension integral

Is there anyone could help me solve the following equation: $$\int_0^t \int_0^{t-v_1} \cdots \int_0^{t-\sum_{i=1}^{k-1} v_i} \prod_{j=1}^k f(\sum_{i=1}^{j} v_i) dm(v_k) \cdots dm(v_2)dm(v_1)$$ You ...
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0answers
63 views

integral equation into differential equation

I have the equation $$ E = \alpha \iint_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because $dS$ is an area, ...
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0answers
41 views

Uniqueness of solution, Volterra integral equation of first kind

I am looking for a reference/proof under which conditions the following Volterra integral equation of the first kind has a unique solution: $$H(x)=\int_0^xK(x,y)g(y)dy$$ with $x\in[0,1]$, $y\in ...
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0answers
43 views

Solution of a partial integro-differential equation system

I'm trying to solve the following system: $\frac{d P_0(t,x,y)}{\partial t}+\frac{\partial P_0(t,x,y)}{\partial x}+\frac{\partial P_0(t,x,y)}{\partial y}=-\left(Q_1(t)+Q_2(t)\right) P_0(t,x,y)+\mu _1 ...
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1answer
33 views

How does the following differential equation imply the integral equation?

The original differential equation: $\begin{cases} y''+\left(1+t^2 \right )y=0, &t>0 \\ y\left ( 0 \right )= 1, y'\left ( 0 \right )=0 \end{cases}$ The corresponding integral equation: ...
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1answer
42 views

Direct numerical solutions for first kind Volterra integral equations

For clearly deliver my purpose, I rewrite this question. Consider first kind Volterra integral equations $$ \int_0^t k(t,s)f(s)ds=g(t) \quad 0\leq t\leq T $$ where $k(t,s)$ is continuous but not ...
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0answers
38 views

Second kind Volterra integral equation with weakly singular.

Consider the second kind Volterra integral equation. $$ f(t)=g(t)+\int_0^t f(t) K(t,s)ds $$ where $g(t)$ is continuous and $K(t,s)$ is a weakly singular kernel, i.e. $$ K(t,s)\leq ...
1
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1answer
26 views

Show a bound on the error

Suppose $g \in C[a,b]$, and $M(b-a) < 1$, and let $y$, $y_N$ $\in C[a,b]$ be the unique solutions of the equations, $y = g + Ky$ and $y_N = g + K_N y_N$. $y_N$ has been previously defined as an ...
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3answers
82 views

Solution of the integral equation $y(x)+\int_{0}^{x}(x-s)y(s)ds=x^3/6$

So the question is to solve the integral equation to find out $y(x)$. $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\dfrac{x^3}{6}$$ So I find out the Resolvent kernel for $(x-s)$ and got $\sin{(x-s)}$, so I ...
0
votes
1answer
18 views

Unique solution of an integral equation in $L^1[0,1]$

Let $h\in L^1[0,1]$. Prove that there is a unique solution (almost everywhere) of the following integral equation: $$f(x)=h(x)+\frac{1}{2}\int_0^x\log(1+f(y)^2)dy$$ The idea is to use the fixed-point ...
4
votes
0answers
49 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)\; ...
2
votes
1answer
23 views

Express the solution of the integral equation in the resolvent form

Express the solution of the integral equation $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \cos(x+t)f(t) \, dt$$ in the resolvent form $$f(x) = \phi(x)+\lambda\int_0^{2\pi} \Gamma(x,t;\lambda)\phi(t) \, ...
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1answer
47 views

Solving integral equation with Fourier transform?

I'm trying to solve the following integral equation using Fourier transforms: $$u(t)+ \int_{-\infty}^{t} e^{\tau-t} u(\tau)\,d\tau=e^{-2|\tau|}$$ I tried to transform both sides of the equation using ...
0
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1answer
57 views

Show that the nonlinear integral equation has a unique solution.

Show that the nonlinear integral equation $$f(x) = \int_0^1 e^{-sx}\cos{(\alpha f(s))}ds,$$ $0\leq x\leq 1$, $0<\alpha<1$, has a unique solution. I originally thought was some form of Fredholm ...
2
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0answers
43 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 ...
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2answers
80 views

Resolvent Kernel of Volterra Integral Equation

The resolvent kernal $R(x,t,\lambda)$ for the Volterra integral equation $$\phi(x)=x+\lambda\int\limits_a^x\phi(s)ds$$ is $\begin{array}1 1. e^{\lambda(x+t)} && 2. e^{\lambda(x-t)} ...
1
vote
1answer
33 views

Given integral equation, find $y(1)$

Let $y(t)$ be a continuous function on $[0,\infty)$ whose Laplace transforms exists. If $y(t)$ satisfies $$\int\limits_0^t(1-\cos(t-\tau))y(\tau)d\tau=t^4\to(1)$$ then $y(1)=$ I was able to find ...
3
votes
1answer
380 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 ...
3
votes
0answers
56 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}\ ...
0
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1answer
29 views

Eigenfunctions for the symmetric kernel of an integral equation

The solution of the symmetric integral equation below: $$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{$*$}$$ with separable kernels method is $$g(s) = f(s) + \lambda \int_{-1}^{1} ...
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1answer
24 views

Ill-posed integral equation problem using Fourier Transfom

By using the Fourier Transform, show that the following equation $$\int_{-\infty}^{+\infty} K(x-y) g(y) dy = f(x), \qquad -\infty < x < \infty$$ is ill-posed. For overcoming ill-posedness of ...
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0answers
21 views

General solution of Fredholm integral equation of second kind with these conditions

Show that if $x_0(s)$ satisfies the equation $$x(s) = y(s) + \lambda \int_0^s K(s,t)x(t) dt \tag{$*$}$$ and the kernel $K$ has characteristic value $\lambda$ of rank $p$,that is, corresponding to ...
0
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1answer
34 views

solve integral equation using adomian decomposition.

i am trying to solve a few integral equation problems prior to the exams. This particular one, however, doesn't to converge. or am I going about it the wrong way? The equation: $u(x) = 1 - x^2 - ...
4
votes
2answers
76 views

Solve the integral equation $f(x) = x + \lambda \int_0^1 f(z)\,dz$

Find a closed-form solution for $f(x)$ in the following equation $$ f(x) = x + \lambda \int_0^1 f(z)\,dz $$ where $\lambda$ is a constant I tried integrating both sides from $0$ to $1$ but wasn't ...
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votes
0answers
19 views

How to classify this integro-differential equation?

I have a system of three coupled integral equations for three unknowns $j(t), \bar{x}(t)$ and $\lambda(t)$ to be solved between $t=0$ and $t=T$ (b.c. are $\bar{x}(0)=x_0$ and $\bar{x}'(T)=0$): (1): ...
2
votes
1answer
60 views

Squaring a integral equation

If $y:[0,\infty)\to[0,\infty)$ is a continuously differentiable function satisfying $$y(t)=y_0-\int\limits_0^t y(s)ds$$ for $t\ge0$, then $y^2(t)=y_0^2+\left(\int\limits_0^t ...
1
vote
1answer
228 views

Convert IVP to an equivalent Volterra integral equation

Convert the following initial value problem to an equivalent Volterra integral equation: $ \begin{cases} u'' -u' \sin x + \Bbb e ^x u= x \\ u(0)=1\\ u'(0)=-1\\ \end{cases} $ I ...
1
vote
1answer
42 views

convert differential equation to Integral equation

$$ y''(x) + y(x) = x$$ with b.v conditions $$ y(0) = 1, y'(1) = 0 $$ Integrating $$ y'(x) - y'(0) + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$ $ let y'(0) = c_1 $ $$ y'(x) - c_1 + \int \limits _0 ^x ...
0
votes
1answer
25 views

Some good texts for integral equations

I am really interested in the theory of integral equations and I am just starting out on the reading. However, I am wondering what could be the best texts to look for (and from where, if possible). ...
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0answers
16 views

numerical solution of integral equation with unknown bound

I am reading a paper on High Harmonics Generation (HHG) and a Lewenstein model The paper is here. I would like to reproduce some results but I am stuck at the following problem. I have: ...