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

1
vote
3answers
77 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
12 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
48 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
17 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
43 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
votes
1answer
27 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
36 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 ...
0
votes
2answers
49 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
363 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
50 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}\ ...
1
vote
1answer
542 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 ...
0
votes
1answer
21 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} ...
0
votes
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 ...
0
votes
0answers
15 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
31 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
69 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
0answers
15 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
56 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
164 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
40 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
22 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). ...
0
votes
0answers
14 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: ...
1
vote
1answer
59 views

How to find the characteristic number of a given integral equation?

How to find the characteristic number of the following integral equation? $$y(x)= \lambda \int_{0}^{1} (3x-2)ty(t)dt$$
1
vote
0answers
26 views

Derivation of an integral equation

I have the following system $$\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=f, \text{ for } x \in (0,1)$$ with boundary conditions $u_x(0)=0$ and $u(1)=0$. For $a(x)>0$, and $b(x)=\frac{1}{a(x)}$, I ...
1
vote
0answers
17 views

Integral equation: existence

Let $H$ and $h$ be smooth functions of one and two variables respectively. Consider equation $$ H(x) = \int_\Bbb Rh(x,y)f(y)\mathrm dy \qquad \forall x\in \Bbb R. $$ When does it have a solution, ...
0
votes
2answers
19 views

Find the Laplace transform for this function

Find the Laplace transform for this function $$f(x)=(1+2ax)x^{-\frac{1}{2}}e^{ax}$$ Please, help me see my answer below Thank you for your participation
0
votes
1answer
25 views

Solve this integral equation using Laplace transform

Solve this integral equation using Laplace transform $$f(x)=x^2 + \int_{0}^{x}f^{\prime}(x-t) e^{-at} dt ,f(0)=0 $$ Please Help see mu answer below Thank you for your participation
0
votes
1answer
18 views

Solve this integral equation using Fourier transform

Solve this integral equation using Fourier transform $$\int_{-\infty}^{\infty} \frac{f(t)}{(x-t^2)+a^2} dt= \frac{\sqrt{2} \pi}{x^2 + b^2}$$ for $b> a > 0 $ Please Help see my answer ...
2
votes
0answers
60 views

Integral equation: averaging

Let $X, Y, Z$ be Borel spaces, for simplicity we can assume that they are $\Bbb R$. Consider an equation $$ \int_{X\times Y}f(x,y,z) \kappa(x,\mathrm dy)\mu(\mathrm dx) = \int_{X\times Y}f(x,y,z) ...
0
votes
1answer
39 views

Solve the integral equation with symmetric kernel

I have the following integral equation with symmetric kernel $$g(x)=\cos \pi x +\lambda \int_{0}^{1} k(x,t)g(t)\,dt $$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= \begin{cases} (x+1)t, ...
0
votes
0answers
32 views

Solving a Fredholm equation using Neumann series technique

It is a very simple kind of Fredholm equation: $$f(x)=x+\int_0^1(1+xt)f(t)\,dt.$$ I solved it and I know that the answer is $f(x)=-2$. But how can i solve this equation by the Neumann series ...
3
votes
2answers
926 views

What is a hypersingular integral kernel?

While reading literature about boundary element and finite element methods I have repeatedly seen that some integral kernels are singular and others are hypersingular. Could you explain what is the ...
0
votes
0answers
16 views

A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$ 0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt $$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
0
votes
1answer
22 views

Optimal Control of Integral Equations of the first kind

I have managed to find a couple of papers which deal with the optimal control of systems governed by integral equations of the second kind (e.g. here: ...
0
votes
0answers
20 views

Solving a weighed integral equation

I am struggling in solving a specific type of equation which simply pops out when we're dealing with weighed functions. I think the general context is worth mentioning. Let's say we have a discrete ...
2
votes
0answers
48 views

Optimization Problem Involving an Integral Equation

I have an integral equation for a probability current $j(t)$ given by: $$ \large\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} ...
1
vote
0answers
45 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...
1
vote
4answers
106 views

What is the function that satisfies $\int_0^x f(t) dt=constant$ [closed]

$$\int_0^x f(t) dt=constant$$ What is the function that satisfies this condition ? Thank you!
0
votes
0answers
45 views

The resolvent kernel

Good morning. How can we prove that $$ \Gamma (s,t, \Lambda ) - \Gamma (s,t, M ) = \Lambda - M \times \int { \Gamma(s,t, \Lambda ) \times \Gamma (s,t,M) \, dx }$$ I tried to use the formula of ...
0
votes
0answers
21 views

Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
2
votes
1answer
551 views

What is the difference between resolvent kernel and iterative kernel of an integral equation?

As we have different methods to find resolvent kernel, which is more suitable among all those methods? And what is the difference between resolvent and iterative kernels?
4
votes
2answers
2k views

Volterra integral equation of second type solve using resolvent kernel

Solve the integral equation $$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I ...
4
votes
2answers
59 views

Check whether a functional has an extremal or NOT

Find the extremal of the functional $$J(y)=\int_a^b F(x,y,y')\,dx$$where , $F(x,y,y')=y'+y$ , for admissible functions $y$. From Euler-Lagrange equation , $\displaystyle ...
0
votes
0answers
57 views

An MCQ for the solution of a non homogegeous Volterra's equation $y(x) = \int _ {0}^{x}(x - s)y(s)ds = \frac{x^3}{6}$

Let $y : [0, \infty) \rightarrow R$ be a twice continuously differentiable and satisfy $$y(x) = \int _ {0}^{x}(x - s)y(s)ds = \frac{x^3}{6}.$$ Then $y(x) = \frac{1}{6}\int_{0}^{x}s^3 sin(x - ...
2
votes
2answers
85 views

derivative over nested integrals

I have the following problem. First an example for two-variable functions Let $A=A(t,\tau)$ and $B=B(t,\tau)$ If I want to compute the following $\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t ...
1
vote
0answers
30 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume ...
1
vote
1answer
51 views

Which method solves this integral equation? $\int_{-1}^{1}w(x)\,e^{tx}\,dx=6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$

Today I encountered this integral equation wrt. $w(x)$: $$\int_{-1}^1 w(x)\ e^{t x}dx = 6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$$ I never solved such equations, and when I tried to ...
2
votes
3answers
52 views

Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
0
votes
1answer
25 views

Checking whether the function $u(x)=e^x$ solves the integral equation $u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$

The function being $u(x)=e^x$ and the integral equation is $$u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$$ I can't do the integration and I'm confused about how to deal with the parameter $\lambda$ ...