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

3
votes
1answer
406 views

Volterra integral equation with variable boundaries

$$\phi (x)=x+\lambda \int_{a}^{x}(x-y)\phi (y)dy$$ I'm also Trying to solve this integral equation like she does Solving an integral equation with a separable kernel. and I also have some doubts ...
10
votes
2answers
4k views

Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not kow how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow ...
1
vote
2answers
423 views

Solution of an integral equation $\phi(x)+\int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $

Solve the following integral equation: $\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $ I need to solve the integral equation above. Can anyone help me please?
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 ...
7
votes
1answer
256 views

Fredholm Equations

I have the following problem to solve $$\phi (x)=\lambda \int_{o}^{\pi }\cos(2x+y)\phi (y) dy+ \sin x$$ following the instructions from the following link early to conclude that: $$\phi (x)=\lambda ...
-3
votes
2answers
487 views

Integral eigenvectors and eigenvalues [closed]

I need to find the eigenvalues e eigenvectors of this integral. a) $$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$ b)- Solved thanks $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
14
votes
2answers
445 views

if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant, show that $f(x)$ is constant; or ...
1
vote
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 ...
1
vote
4answers
332 views

Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$

I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$. This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
13
votes
4answers
551 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
3
votes
1answer
741 views

Eigenvalues of an operator

I think this question isn't that hard, but I am a bit confused: Define $$(Af)(x):=\int_{0}^{1}\cos(2\pi(x-y))f(y)dy.$$ Then $A$ is an operator on functions. Find the eigenvalues and the ...
7
votes
2answers
566 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: ...
3
votes
2answers
314 views

Solve integral equation of second kind using Fredholm method

I need to solve this integral equation $$\phi (x)=(x^2-x^4)+ \lambda \int_{-1}^{1}(x^4+5x^3y)\phi (y)dy$$ Using the Fredholm theory of the intergalactic equations of second kind. I really don't ...
1
vote
3answers
105 views

Solve $y'-\int_0^xy(t)dt=2$

I have not idea how to approach this differential equation. $$y'-\int_0^xy(t)dt=2$$. Basically, I did, $$F''(t)-F(x)+F(0)=2 \;\;\;\;\;\;\; F'=y$$ I am stuck. Thank You.
1
vote
1answer
2k views

How to find eigenfunctions of a linear operator (follow-up question)

This is a follow up to this question. My original question was answered by @oeanamen but for convenience here I state the follow up question completely. I am interested in calculating characteristic ...
39
votes
2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
11
votes
2answers
640 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 ...
0
votes
1answer
41 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
6
votes
2answers
115 views

Does this condition on the sum of a function and its integral imply that the function goes to 0?

Consider a bounded real-valued function $S:\mathbf{R}\to\mathbf{R}$ so that $$\lim_{x\to\infty} \left( S(x) + \int_1^x \frac{S(t)}{t}dt\right)$$ exists and is finite. Can one say that ...
3
votes
1answer
679 views

How to find eigenfunctions of a linear operator

I wonder if there is a general way of finding characteristic values and eigenfunctions of a given linear operator described by an integral. As a special case suppose I am interested in this function: ...
6
votes
2answers
262 views

Can we bound from above sub-solutions of Volterra integral equations? (Nonlinear Gronwall's Lemma)

Gronwall's lemma says the following. Assume that $v\in C^0([t_0, T])$ is a nonnegative function. If $u \in C^0([t_0, T])$ satisfies the integral inequality $$u(t) \le c + \int_{t_0}^t u(s)v(s)\, ...
4
votes
1answer
457 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
3
votes
4answers
323 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
3
votes
2answers
477 views

Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

Integral equation $$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has: a unique solution for $\lambda \neq \frac{4}{\pi +2}$; a unique solution for $\lambda \neq ...
3
votes
1answer
360 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
2
votes
0answers
238 views

Solve integral(convolution) equation

I have been trying to find a solution to the following convolution equation: \begin{align*} e^{-ax^2/2}*\ln \left(f(x)*e^{-ax^2/2}\right)+e^{-bx^2/2}*\ln \left(f(x)*e^{-bx^2/2}\right)=cx^2 ...
2
votes
1answer
582 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?
1
vote
2answers
74 views

How can I solve this integral equation by converting it to a differential equation

Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation
1
vote
1answer
247 views

Homogeneous Fredholm Integral Equation

I'm having problem obtaining the solution of the homogeneous Fredholm Integral Equation of the 2nd kind, with separable kernel. I always get a zero if I use the normal method i was taught for the ...
1
vote
1answer
73 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 $y(t)$
1
vote
1answer
150 views

Differential equations, integral equations

Is there an analytical way of proving that if $\phi$ is a solution to \begin{equation} y(t)=e^{it}+a\int_{t}^{\infty}\sin (t-s)y(s)s^{-2}ds, \end{equation} then $\phi$ would be a solution to the ...
1
vote
1answer
2k views

Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
0
votes
1answer
176 views

Solving integral equation $a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$

I've got this nasty-looking integral equation involving taking two minimums: $$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$ where $\delta(\cdot)$ is the ...