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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3
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1answer
199 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 ...
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2answers
284 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?
6
votes
2answers
2k 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 ...
7
votes
1answer
200 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 ...
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votes
2answers
265 views

Integral eigenvectors and eigenvalues

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 $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
12
votes
1answer
354 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 ...
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4answers
308 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 ...
3
votes
2answers
681 views

Volterra integral equation of secong 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
2answers
445 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
154 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 ...
3
votes
1answer
500 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 ...
1
vote
1answer
1k 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 ...
6
votes
2answers
98 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
451 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: ...
3
votes
2answers
357 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
293 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 ...
3
votes
2answers
200 views

Can we bound from above sub-solutions of Volterra integral equations?

Let us review Gronwall's lemma. If $v \in C^0([\tau, T])$ is such that $$v(t) \le c + \int_{\tau}^t u(s)v(s)\, ds,\qquad t \in [\tau, T]$$ where $c$ is a real constant and $u \in C^0([\tau, T])$ is ...
1
vote
1answer
64 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 ...
0
votes
2answers
335 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
147 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 ...