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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33 views

Is it true that $\displaystyle (Tf)(x)=g(x)+\int_{a}^b k(x, y)f(y)\ dy$ is a contraction in $(C^0([a, b]), d_\infty)$?

In $C^0([a, b])=\{f:[a, b]\longrightarrow \mathbb R: f\ \textrm{continuous}\}$ consider the metric $$d_\infty(f, g)=\sup_{x\in [a, b]}|f(x)-g(x)|.$$Let $T:C^0([a, b])\longrightarrow C^0([a, b])$ given ...
1
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0answers
119 views

Multidimensional (Fredholm) integral equation (of first kind)

Suppose, \begin{align*} g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s)h(t_1) + K_2(t_1,t_2,s)\right]ds %\\ %g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s) + K_2(t_1,s)h(t_2)\right]ds \end{align*} The problem is ...
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0answers
40 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
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0answers
217 views

solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
2
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1answer
66 views

Existence and uniqueness of an integral equation

Does this equation $$f(u)=1+\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{1}{(u-v)^2+1}f(v)dv$$ has a bounded continuous solution? Is this solution unique? Here $f$ is defined over $\mathbb{R}$ and ...
0
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1answer
263 views

Computation Method to solving Homogeneous Fredholm Integral Equation of Second Kind with Symmetric Kernel

I am attempting to write a program that will be able to numerically solve a homogeneous Fredholm Integral Equation of Second Kind, with a Symmetric Kernel. I have been looking through textbooks and ...
2
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2answers
96 views

To solve non-linear Integro-differential equation

I am just begin to study integral equations, in which i come with following problem regarding second kind Volterra non-linear integro-differential equation, $$u'(x)=-1+\int_{0}^{x}u^{2}(t)dt$$ with ...
14
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2answers
437 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 ...
32
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2answers
2k 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?
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0answers
113 views

Solution of Volterra convolution-type integral equation

The convolution-type Volterra integral equation of the first kind $f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$ can be solved (at least formally) by applying the Laplace ...
1
vote
2answers
38 views

Reading Speed for Constant Time to Finish

You open a very long new book on your e-reader and read a few pages. It helpfully informs you that based on your reading speed you have 16 hours of reading left until you are done. You read the rest ...
2
votes
1answer
94 views

Solve: $\int_{0}^{2\pi}g \psi e^{i n \theta}\,\text{d}\theta = n/(n-i\alpha) \int_{0}^{2\pi}\psi e^{i n \theta}\,\text{d}\theta$

For $\alpha>0$, I want to find a $g(\alpha, \theta)$ such that $$ \int_{0}^{2\pi}g(\alpha, \theta)\psi(\theta)e^{i n \theta}\,\text{d}\theta = \frac{n}{n-i\alpha} \int_{0}^{2\pi}\psi(\theta)e^{i n ...
0
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1answer
117 views

Prove existence and uniqueness of differential/integral equation

This is a homework question, so I'm essentially asking for hints and not answers due to academic honesty concerns. The course is in real analysis (baby rudin) and it is essentially chapter 9 which ...
1
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0answers
66 views

Are there methods to solve coupled integral and integro-differential equations?

I have one fredholm integral equation $$ y(x)=f(x)+\int_0^1 K_1(x,g(x),t)y(x(t))dt$$ and an integro-differential equation $$ \frac{dg(x)}{dx}=h(x)+\int_0^1 K_2(x,y(x),t)g(x(t))dt$$. Are there any ...
1
vote
1answer
60 views

Integral equation question

If f(x) and f(t) both have the same domain and range, is there a general way to find $\int_{0}^{x^2} f(t) dt = f(x)$ given t? The actual problem tells that t = 9 and f(x) = $5 e \exp{x cos ...
6
votes
2answers
107 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 ...
1
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0answers
46 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 $$f(x)=g(x)+\int_a^xf(x^\prime)h(x^\prime) dx^\prime$$ ...
5
votes
1answer
240 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
votes
2answers
76 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
125 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
254 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 ...
0
votes
2answers
39 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?
0
votes
1answer
95 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 ...
1
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0answers
75 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 ...
0
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2answers
87 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
39 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
votes
1answer
56 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 ...
3
votes
1answer
310 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
296 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
69 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
123 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
522 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. ...
4
votes
3answers
224 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
30 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
31 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 ...
2
votes
1answer
196 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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0answers
136 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
31 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
54 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
90 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
35 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} ...
1
vote
1answer
70 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
157 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
51 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
214 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
92 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
52 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
118 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
35 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
72 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 ...