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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20 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}...
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1answer
29 views

Geodesics using Euler-Lagrange

thanks for taking a look at my question. This is a homework problem from a section covering Euler-Lagrange equations. I'm asked to consider the arc length formula: $S = \int\limits_{{t_1}}^{{t_2}} {...
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1answer
49 views

Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of ...
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1answer
27 views

Is it possible to use the convolution theorem on a finite interval integral ? (Laplace)

Say I have the following equation : $$\int_{0}^{1}\cos(t-\tau)x(\tau) d\tau = t\cos(t)$$ if we replace 1 in the integral for t it is easily solvable using the convolution of Laplace and the answer ...
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24 views

If $g(x) = \int_{x}^{\infty}y\arccos\left(\frac{x}{y}\right)f(y)dy$ then $f(x) = \frac{2}{\pi}\int_{0}^{\infty}g''\left(x\cosh(t)\right)dt$

I can prove that the solution is correct, but it involves using the solution written in a different form and then showing that it is indeed the solution. What I'm after is an argument that leads to ...
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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)...
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21 views

Integral equation with elliptic integral kernel

Let $$K[k]=\int_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}\theta}{\sqrt{1-k^2 \sin^2\theta}}$$ be the complete elliptic integral of the first kind. Is there a simple solution of the integral equation: $$...
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1answer
41 views

Solution of an integral equation

Consider a simple Wiener-Hopf integral equation of the first kind with unknown function $\phi(x)$ for $x\geq 0$: $$f(x)=\int_0^\infty \phi(y)\min\{x,y\}\,\mathrm{d}y$$ where $f(x)=x-a$ and $a \geq 0$...
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1answer
26 views

Fredholm equation with symmetric kernel

I have the following equation : $$ \phi(x) = \frac{x}{2} + \frac{\pi^2}{4}\int_{0}^{1}K(x,t)\phi(t)dt $$ where $$ K(x,t)= \left\{ \begin{array}{ll} \frac{x(2-t)}{2} & \mbox{if } 0 \leq x \...
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1answer
48 views

Solution to Integral Equation and Conditions for Existence

Show that under certain circumstances, the integral equation $$\int_0^t (t^2-u^2)^{\alpha-1} \chi (u) du = \phi (t)$$ Possesses the solution: $$\chi (t) = \frac{2}{\pi} \sin (\alpha \pi) \Big[ \phi ...
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26 views

Combining two integral equation. Hankel transform

I have this two integral equations which are an Hankel transform pair. I gotta combine them to find $A(k)$ $$ R^{n+{1}/{4}} \, \Sigma(R,t=0) \, = \, \int_0^\infty [A(k)\, k^{-1}] \, J_l(ky) \, k \, ...
3
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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{\...
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1answer
81 views

How to find the function such that $\int_0^1f(x)\ \mathrm dx=e^{-4n^{2}{\pi}}$ [closed]

Find $f(x)$ where: $$ \int_{0}^{1}f(x,n)\ \mathrm dx=e^{-4n^{2}{\pi}} $$ Is it possible that question contains infinitely many answers? How to solve this ? Please provide me a hint.
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2answers
52 views

Integral equation of the form $\int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4}$

How to solve an integral equation of the following form \begin{align} \int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4} \end{align} where $a$ and $b$ are some positive constants. I am not ...
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1answer
45 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
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36 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 $f(x)$ ...
0
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0answers
58 views

Is it possible to find $g(\kappa)$ in this equation

I have ran into the following integral equation as part of my research. For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$. I have the following equality $$\int\limits_0^\infty g(\...
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36 views

Is it possible to solve these simultaneous equations with integrals?

I have the following two equations: \begin{align} 2-3\int_{-\infty}^{y_0} f(x_0,y)\,\mathrm{d}y +\varepsilon x_0=0\\ 2-3\int_{-\infty}^{x_0} f(x,y_0)\,\mathrm{d} x+\varepsilon y_0=0 \end{align} where ...
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0answers
94 views

Determining whether the extremal problem has a weak minimum or strong minimum or both

The extremal of the functional $\int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ ...
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1answer
43 views

Multiple choice differential equation question.

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuously differentiable function satisfying $$y(t)=y(0)+\int_{0}^{t}y(s)ds,t\geq 0$$ Then $1.y^{2}(t)=y^{2}(0)+\int_{0}^{t}y^{2}(s)ds$ $2.y^{2}(t)=y^...
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1answer
36 views

How to solve Fredholm Integral Equation of the Second Kind in $C[0,1]$

I need to solve, in $C[0,1]$, the equation $\displaystyle x(t) - \lambda \int_{0}^{\pi}(\sin t \cos s)x(s) ds = \sin t$. Adding the integral part to both sides, I obtain $x(t) = \sin t + \lambda \...
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2answers
48 views

Solving Volterra integral equation

I would like to solve $4u(t)+\int_0^t\sin(t-s)u(s)ds=5t, \ t\geqslant 0$. Any ideas on how to approach this equation?
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22 views

Solution of the following Fredholm integral of the second kind

$H(s,x)=\int_0^{\infty } \frac{e^{(-s-1) (u+x)} \left(2 e^{(s+2) u+s x}+s\right) }{2 s}H(s,u) \, du+2 e^{-(1+s) x}$ Is there any chance to obtain the solution ($H(s,x)$) of this equation? I managed ...
3
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2answers
31 views

How to solve this integral equation with the Dirichlet kernel?

It is $$ S (t) = 1 - i g \int_0^t d \tau \left( \sum_{n=-M}^M e^{-i n (t- \tau )} \right) S(\tau) . $$ The kernel is the Dirichlet kernel. Numerical result is shown in the figure. The $M\...
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1answer
23 views

Show that there is a unique function on the interval that solves:

$$f(x) = e^{-2x} + \int_0^\infty e^{-2x-2y} \sin(x-y)f(y) \, dy$$ I can't get a good bound on $e^{-2x-2y} \sin(x-y)$ so that I can apply Banach.
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1answer
35 views

Solving a separable integral equation: $y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$

Solving integral equation. My answer is wrong. Where do I make a mistake? $$y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt $$ $$ y'(x) = \frac{d}{dx} \int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$$ $$ \frac{...
2
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0answers
27 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 ...
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22 views

Banach Theorem on Metric Space for Integral Equations

My instructor said that Banach doesn't apply in this case: f(x) = sin(x) + $\int_0^x$$f^2$(z)dz f(0) = 0; f'(0) = 1 > 0 f'(x) = cos(x) +$f(x)^2$, which is positive on (0,$\pi$) so f is positive ...
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1answer
37 views

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

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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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 ...
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3answers
708 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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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: ...
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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( \...
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1answer
59 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
24 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} f_n(...
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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 ...
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1answer
30 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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0answers
45 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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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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30 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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25 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) \,...
1
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23 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 ...
1
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0answers
23 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 ...
0
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0answers
64 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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2answers
51 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 f(s)ds=\frac{...
0
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0answers
22 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 ...
0
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0answers
51 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 [0,1]...
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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: $y\...
1
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1answer
51 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 ...
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 ...