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 - ...

learn more… | top users | synonyms

0
votes
0answers
43 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 ${\alpha}$ ...
1
vote
1answer
28 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 $1.y^{2}(t)=y^{2}(0)+\int_{0}^{t}y^{2}(s)ds$ ...
-1
votes
0answers
34 views

Need help with this equation [closed]

I have been trying to evaluate $(1.102 ^ {12} – 1)/2$. 
0
votes
1answer
35 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 ...
2
votes
2answers
45 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?
0
votes
0answers
21 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
votes
2answers
27 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 ...
0
votes
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.
3
votes
1answer
31 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? $$y(x) = 1+\int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt $$ $$ y'(x) = \frac{d}{dx} \int_{1}^{x} \frac{y(t)dt}{t(t+1)}dt$$ $$ ...
2
votes
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 ...
0
votes
0answers
19 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 ...
0
votes
1answer
36 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.
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 ...
13
votes
3answers
691 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 ...
1
vote
0answers
18 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( ...
2
votes
1answer
58 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) ...
0
votes
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} ...
1
vote
0answers
30 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 ...
1
vote
1answer
27 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), ...
0
votes
0answers
43 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. ...
1
vote
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
1
vote
0answers
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 ...
0
votes
0answers
22 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
vote
0answers
21 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
vote
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 ...
0
votes
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 ...
1
vote
2answers
47 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 ...
0
votes
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
votes
0answers
47 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 ...
1
vote
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: ...
1
vote
1answer
46 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
vote
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 ...
0
votes
1answer
20 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
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)\; ...
0
votes
1answer
52 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 ...
2
votes
1answer
25 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) \, ...
0
votes
1answer
73 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
votes
0answers
44 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 ...
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 ...
0
votes
3answers
129 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)} ...
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
votes
0answers
23 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
votes
1answer
28 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 ...
0
votes
1answer
32 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} ...
4
votes
2answers
77 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 ...
0
votes
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 - ...
0
votes
0answers
20 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
64 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
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 ...