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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35 views

Are there methods to solve a system of coupled integral equations?

I was wondering if there were methods to solve a system of coupled integral equations. The example case I am thinking about is $$f(x)=g(x)+\int_a^xf(x^\prime)h(x^\prime) dx^\prime$$ ...
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1answer
198 views

Find all continuous functions such that $f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$

From Spivak Find all continuous functions such that $$f(x)^2 = \int_0^x f(\xi)\frac{\xi}{1+\xi^2}d\xi.$$ $f(x)^2$ is clearly differentiable. I'd like to write $\dfrac{d}{dx}f(x)^2=2f(x)f'(x).$ ...
3
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2answers
67 views

Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$

From Spivak Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$ My approach: differentiate both sides to get $f''(t) = f'(t)$, giving $f'(t) = Ce^t$, implying $f(t) = Ce^t + ...
2
votes
3answers
93 views

Uniform convergence of matrix integral sequence

I was given recursively defined: $$ M_k(t)=I+\int_{t_0}^tA(s)M_{k-1}(s)~ds $$ and $M_0=I$ and that $A(t)$ is a matrix with entries that are continuous functions on $t_0\leq t\leq t_1$. By induction ...
3
votes
4answers
210 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 ...
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2answers
36 views

Integral equation and constant rules

I have an integral equation of the form: $$f(x)=3+4\int_a^bf(t)~dt$$ How can I put the constants inside the integral to get something where I can apply the fundamental theorem of calculus?
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0answers
68 views

Series solution for coupled PDEs

So I have this system for $f(z, v, t)$ and $\Psi(z, t)$, $$ \frac{\partial f}{\partial t} + v \frac{\partial f}{\partial z} - g(v) \frac{\partial \Psi}{\partial z} = 0 \tag{1} $$ $$ \frac{\partial^2 ...
0
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1answer
79 views

How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
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0answers
61 views

Definite Integral of Periodic Function Multiplied by another Function

For one part of a problem I am working on, I am trying to show that $y'(t) \geq 1$ for all $t \geq 0$ when $y'= 1- \int^t_0 g(s)y(s) ds$ When $g(t)$ is periodic, $g(t) <0$ for all $t$, and ...
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2answers
80 views

Question about integral is equal to zero

Suppose we have the following equation $$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$ where the function $g(x)$ does not depend on $r$, while $f(x,r)$ is function of ...
1
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0answers
37 views

Can we get a general solution to the following system of equations?

A system of equations $$\frac{df}{dt}(t)=-txg(t)-x, \frac{dg}{dt}(t)=(1-t)xf(t)+x$$ which was constructured from the integral equation $$ F_t (t,x)= x \int_0^{1-t} (1-s)F(s,x) ds + (1-t)x, $$ This ...
3
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1answer
54 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
2
votes
1answer
251 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) + ...
2
votes
1answer
180 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
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1answer
57 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
108 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
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1answer
281 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. ...
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0answers
54 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
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3answers
207 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 ...
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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 ...
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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 ...
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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
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1answer
122 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
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120 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
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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
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1answer
48 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
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1answer
76 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 ...
1
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1answer
59 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
133 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
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0answers
48 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
173 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
47 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 + ...
1
vote
1answer
103 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
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. $s(T,T)=T$
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0answers
61 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
28 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
137 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
33 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
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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
48 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: ...
2
votes
0answers
46 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 ...
0
votes
0answers
100 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
96 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
167 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
128 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 ...