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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2
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1answer
161 views

How to differentiate this equation involving an integral expression?

I want to differentiate the Volterra integral equation $\phi(t) + \int_0^t (t - \xi) \, \phi(\xi) \, \mathrm{d}{\xi} = \sin{2t}$. Am I right in thinking that the integral can just be removed like so? ...
0
votes
1answer
56 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
1
vote
1answer
103 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
1answer
239 views

Convert an integral equation in an initial value problem of an ODE of degree 2

The following exercise is a part of a bigger exercise. Therefore I first give you the setting of the whole exercise and then (in the grey box below) the part of the exercise which I mean here. ...
0
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0answers
52 views

System of integral equations for a unimodal symmetric probability distribution

Let $f(x)$ be a symmetric unimodal probability distribution on $\mathbb R$, with mean $\mu=0$. By unimodal, I mean that $f(x)$ is strictly increasing for $x<\mu$ and strictly decreasing for ...
4
votes
3answers
201 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
0
votes
1answer
27 views

Clarification of intergal equations.

I feel like asking a question that show how long I have to go. Clarification. In differential equations, I start with a rate of change and find the indefinite integral to find the function. In ...
1
vote
0answers
27 views

An integro differential equation involving $f$,$f_h$ and second derivative of $f$.

Let $f_h$ be the Hilbert transform of the real function $f$. I need some help solving this integro differential equation : $$\alpha f_h(x) + \beta f''(x) = f(x)$$ A simple sinusoid doesn't seem to ...
0
votes
0answers
65 views

Numerically solving delay differential equations

I understand that the Dickman Function can be solved numerically by converting it to a delay differential equation. The Mathworld link above describes how this may be achieved for $$\rm ...
2
votes
1answer
107 views

Integral equation corresponding to initial value problem

Question is to : Form an integral equation corresponding to the initial value problem $$\frac{d^2y}{dx^2}+y=0 ~; ~ x>0$$ with initial conditions $y(0)=1$ and $y'(0)=0$ What i have tried so far is ...
7
votes
0answers
118 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) ...
0
votes
0answers
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 $$ ...
1
vote
1answer
46 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 ...
1
vote
1answer
73 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, ...
1
vote
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} ...
0
votes
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 ...
1
vote
1answer
55 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 ...
0
votes
3answers
125 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: $$y'+6y(t)+9\int_0^t y(\tau)d\tau=1,\qquad y(0)=0$$
0
votes
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
votes
1answer
156 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 ...
0
votes
2answers
87 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 ...
0
votes
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
votes
0answers
44 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 + ...
0
votes
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
vote
1answer
100 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
vote
0answers
32 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
votes
0answers
58 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
votes
0answers
27 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
votes
1answer
132 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 ...
1
vote
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 ...
1
vote
0answers
39 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 ...
2
votes
1answer
43 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
votes
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
vote
0answers
44 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
vote
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 ...
0
votes
0answers
99 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
votes
1answer
93 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
votes
1answer
163 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
votes
1answer
127 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
84 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
votes
0answers
74 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
votes
1answer
246 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
vote
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
votes
2answers
72 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
votes
1answer
44 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 ...
13
votes
4answers
370 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 $f(x)$.
0
votes
2answers
55 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 ...
6
votes
3answers
467 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
159 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:- $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$ $\int^x_0 f (t) dt =x+ \int^0_x t f (t) dt$ + ...