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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29 views

Show, that $T\colon C([a,b])\to C([a,b])$

I have a question concerning an integral equation that is written as an fixed point equation, namely $$ u(x)+\int_a^x F(x,y,u(y))\, dy=f(x,u(x)),~~x\in [a,b] $$ with $$ ...
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1answer
42 views

Using Banachs fixpoint-theorem to show the uniqueness of a solution of a non-linear integral equation

Show, that under the conditions (1) $-\infty<a<b<\infty, f\in C([a,b]\times\mathbb{R}), F\in C([a,b]\times [a,b]\times\mathbb{R})$ (2) $\exists~1>c\geq 0~\forall x\in [a,b]~\forall ...
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1answer
72 views

Solving an Integral Equation

Here is the Question; Solve the integral equation, $$\int_0^tY(u)Y(t-u)du = \frac12 (\sin t-t\cos t)$$ Really not sure how to go about this, took the Laplace transform of the right side getting, ...
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1answer
33 views

Finding a y(x) that satisfies $ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $

I'm having problem with finding a y(x) that satisfies $$ y(x) = \int_0^x \! \left(\frac{t} {y(t)+1}\right)^2 \, \mathrm{d}t $$ Here is what I tried to do. $$ y(x) = \int_ \! \left(\frac{x} ...
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0answers
29 views

How to Solve This Special Case of Multidimensional Integral Operator?

I'm dealing with an integral equation of the following form: $1 = f(x)\int dy f(y)B(x,y)$ where $B(x,y)$ is a known function, and I want to solve for $f(x)$. If I treat $f(x)f(y)$ as one big unknown ...
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1answer
54 views

Integral equations that can be solved elementary

Solve the following integral equations: $$ \int_0^xu(y)\, dy=\frac{1}{3}xu(x) \tag 1 \label 1 $$ and $$ \int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1. \tag 2 \label 2 $$ Concerning $\eqref 1$, I read ...
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3answers
120 views

How to solve $y'+6y(t)+9\int_0^t y(\tau)d\tau=1$, $\,y(0)=0$

I want to solve this equation. It reminds me something about Laplace transform. I am sure that I must use it order to solve this: $$y'+6y(t)+9\int_0^t y(\tau)d\tau=1,\qquad y(0)=0$$
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0answers
47 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
2
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1answer
134 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
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2answers
85 views

Differential Equation with Integral

Determine the unique solution of: $$y'+4y+5\int_0^x y\,dx = e^{-x},$$ given that $y(0)=0$. [Hint: Take the derivative of both side of the given equation before you start solving.] Please I need ...
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0answers
56 views

Integral Equations $\phi(x), K(x,y)$

Suppose that $K(x,y)=g(x)h(y)$ and that $\int_a^b g(x)h(x)dx=0$. Let $\phi_0(x)=f(x))$. Show that all iterates equal the first iterate and find a simple formula for the solution. Basically, I ...
0
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0answers
35 views

Finding a nonzero continuous function that satisfies this integral equation, but not unique?

If $h(x)$ is such that it satisfies $$ h(x) = \lambda \int_a^b K(x,y) h(y)\, dy $$ Then for $\phi(x)$ a solution of $$ \phi(x) = f(x) + \lambda \int_a^b K(x,y)\phi(y)\, dy $$ it is true that $\phi + ...
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0answers
18 views

How to make sense of this condition on $K$

When considering the integral equation $$ \phi(x) = f(x) + \lambda \int_a^b K(x,y) \phi(y) \, dy $$ suppose I am given the condition that $K(x,y) = g(x)h(y)$ and that $\int_a^b g(x)h(x) \,d x = 0$. ...
1
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1answer
95 views

Integral Equation-Volterra 2nd kind

Given $$ f(x) = \sqrt{x} + \lambda\int_0^x\sqrt{xy}f(y)dy. $$ I found the derivative of $f$ to be $$ f'(x) = \frac{1}{2\sqrt{x}} + \lambda\left(xf(x) + ...
1
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0answers
31 views

Solve an integral equation

How can I solve the following problem for $s(x,T)$, $s(\cdot)$ is continuous and strictly increasing in $x$. $\int_0^T\frac{-x^3+x^2-T(1-T)x}{(s(x,T)-x)^2}dx=0$ s.t. $s(T,T)=T$
2
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0answers
54 views

reference request: a graduate level textbook on viscosity solutions of IPDEs

I am particularly interested in IPDE which arises from optimal stopping/control problems so I would like some references on integral-partial differential equations, which are related this area. I have ...
2
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0answers
25 views

Uniqueness of solution to integral equation for operator valued functions

Let $X$ be a Banach space. Suppose I have a 2 parameter family of bounded operators on $X$: $V(t,s)$, $0\leq s\leq t \leq T$, such that $V(t,s)x=U(t,s)x+\int_s^t V(t,r)H(r)U(r,s)x\,dr$ and ...
2
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1answer
122 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...
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0answers
31 views

numerical implementation of the resolvent kernel of an integral equation

I started exploring implentation of Volterra equations only recently. The iterative kernel for my problem looks like this: $$L_i(x,y) = \int\limits_x^y L_1(y,t)L_{i-1}(t, x)dt. $$ I have been trying ...
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0answers
38 views

The integral equation $\int_{-\infty}^{\infty}f(x) + af(g(x)) dx = b$

How to solve for $f(z)$ in the equation $\int_{-\infty}^{\infty}f(x) + af(g(x)) dx = b$ where 1) $f(x),g(x)$ are holomorphic near the real line. 2) $x$ is considered real here. 3) $a$ is a given ...
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0answers
37 views

Solution of an integral equation

Consider the integral equation: $$\int_{0}^{\infty}\left(1-\left(\frac{u}{2\pi x} \right )^{1/4}-\exp{\left(\frac{u}{2\pi x} \right )^{1/4}} \right )\eta (x,y)dx=u\delta(u-y)$$ $\delta(\cdot)$ is the ...
2
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1answer
40 views

Understanding solution to first-order differential equation written in integral form, $y(x)=\int_0^xy(t)dt + x + 1$

I have a differential equation and I'm trying to understand the solution printed in the back of the book. I will specify what part I'm struggling to understand: Problem statement: Solve the following ...
3
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0answers
194 views

Solving an integral equation.

Let $K(x,y)=(x+y)e^{-y^2/2-xy}$. I need a constructive way (not simply verifying it is a solution) to show that $f(x)=e^{-x^2/2}$ is the solution to the integral equation: ...
1
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0answers
41 views

Analytic continuation of an integral equation

Consider the (possibly singular or hypersingular) integral equation $$Z\int_{-\infty}^\infty B(x,x^\prime) \ A(x^\prime) dx^\prime = A(x)$$ where $Z$ indicates the finite part of the integral should ...
1
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1answer
46 views

The trace of an integral equation?

I am reading a paper about spectroanalysis and encountered the following integral equation: $$\int_{-1}^{1}\frac{\sin A(x-x')}{\pi(x-x')}\psi(x')dx'=\lambda\psi(x)$$ Then the paper gives without proof ...
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0answers
94 views

What does this notation regarding integral equation kernels and norms mean?

I am attempting to understand what types of kernels the standard theory of Fredholm Type-2 integral equations applies to, but I've never taken a course in analysis. Basically, given a kernel, ...
2
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1answer
92 views

Solving integral equation

Solve the following integral equation: $${u(x)}={x}+{e^{x}}+\int_{0}^{1}\left({5}{x}^{2}{t}^{2} -{3}{t}^{2}+{t}\right) {u(t)}dt. $$
2
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1answer
162 views

Question about integral equations

Consider the equation $$g(t) = \int_a^b K(t,s)f(s) ds $$ where $g$ and the kernel $K$ are known and $f$ is to be determined. Suppose that the equation has a solution. Under what conditions on the ...
4
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1answer
125 views

Integral equation $u(t)=f(t)+a\int_0^t u(s)ds\quad t\geq 0$

Let $a\in\mathbb R$ and $f\colon [0,1]\to\mathbb R$ a continuous function. Solve the integral equation $$u(t)=f(t)+a\int_0^t u(s)\mathrm ds,\quad t\geq 0$$ and find an explicit formula for the ...
2
votes
0answers
83 views

Unicity of solution to an integral equation

Suppose we have the following integral equation $$f(x)=\int_0^\infty f(y)(x+y)e^{-x^2/2-xy}\text{d}y$$ where $f: [0,\infty)\rightarrow[0,\infty)$. The function $f(x)=Ce^{-x^2/2}$ solves the equation. ...
3
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0answers
71 views

does f(x) have unique fixed point?

Let $g$ be a probability density function. We can assume about $g$, whatever we like (Only important thing, we know about random variable Y,which has $g$ as p.d.f is $P(Y<0)>0$.) Next, let ...
2
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1answer
241 views

The integral equation $y(x)=x-\int_1^x xy(t)dt$ [closed]

The integral equation $$ y(x)=x-\int_1^x xy(t)dt \tag{$y\in C^1[1,\infty)$}$$ has the solution $y=x(1-\log x)$ $y=xe^{\left(x-\frac{1}{2}\right)}(x-1)+x$ $y=xe^{1-x^2}+x$ $y=x-x \cdot e^{x^2}+ex$ ...
1
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0answers
70 views

A question on convergence of solution of an integral equation.

In Pipkin's "A Course on Integral Equations", on page 24 problem 2, he asks us to find out whether or not iteration will converge uniformly for an integral equation of the second kind, i.e $u=f+Ku$ on ...
3
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2answers
70 views

A function/distribution which satisfies an integral equation. (sounds bizzare)

I think I need a function, $f(x)$ such that $\large{\int_{x_1}^{x_2}f(x)\,d{x} = \frac{1}{(x_2-x_1)}}$ $\forall x_2>x_1>0$. Wonder such a function been used or studied by someone, or is it just ...
0
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1answer
42 views

Unicity (or not) of the solution of an integral equation

Given the integral equation: $$\int_0^a f(x)\left[ \frac{d^2}{dx^2}f(x) \right]dx=a$$ with the condition: $$\lim_{x\to\infty}f(x)=0$$ how can I find its solution? Is the solution (if any) the only one ...
12
votes
4answers
354 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 - ...
0
votes
4answers
89 views

Find $f(x)$ such that $2 \int_0^x f(t) \,\mathrm dt = x(f(x)+2000)$

Let $f: \Bbb R \to \Bbb R$ be such that $$2 \int_0^x f(t) \,\mathrm dt = x(f(x)+2000)$$ for every $x$. Find $f(x)$.
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2answers
53 views

Different formulations for multiple integral equation

On several papers, I found the following model for a multiple integral equation: $$g(s)=\int\limits_{\Omega} h(s,t)f(t)\,\mathrm{d}t$$ where $s,t \in \mathbb{R}^3$, and $\Omega \subseteq ...
5
votes
3answers
413 views

Prove there is a unique continuous function satisfying this integral equation

This is a question from an old real analysis qual: Prove that there is a unique continuous function $f:[0,1] \to \mathbb{R}$ such that $$f(x) = \cos x + \int_0^x f(y)e^{-y}dy$$ for $x \in [0,1]$ I ...
3
votes
1answer
152 views

If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$

If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$ solution:- $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$ $\int^x_0 f (t) dt =x+ \int^0_x t f (t) dt$ + ...
3
votes
1answer
245 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 ...
4
votes
3answers
101 views

Find $f(x)=?$ functional equation

I would appreciate if somebody could help me with the following problem: Q: Find $f(x)$ ($f'(x)$: conti-function , $x \in\mathbb{R}$) $$f(x)=\sin ^2x+\int_{0}^{x}tf(t)dt$$
2
votes
0answers
84 views

applications integrals

I have the following problem to solve: $\int_{0}^{1}K(x,y)\phi(y)dy$ where: $K(x,y)=x(1-y), 0\leq x\leq y\leq 1$ and $K(x,y)=y(1-x), 0\leq y\leq x\leq 1$ already tried using the methods suggested ...
0
votes
1answer
76 views

Reducing an integral equation to a differential one

In my course about differential equations I have the following problem: Find all the functions $f:\mathbb{R} \longrightarrow \mathbb{R}^+$ such that the area below the graphic of the function in an ...
-2
votes
2answers
122 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
7
votes
1answer
212 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 ...
-4
votes
2answers
295 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 $$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
1
vote
2answers
443 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 ...
3
votes
2answers
169 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 ...
2
votes
0answers
82 views

What is the meaning of the definition below? Taken from a 1909 book on Integral Equations.

Definition: We say that the discontinuities of a function of $(x, y)$ are regularly distributed in $S$ or in $T$ if they all lie on a finite number of curves with continuously turning tangents, no one ...