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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2answers
28 views

Integral equation involving Binomial distribution

I am trying to find the form of a function $u^{(n)}(p)$ which satisfies $\forall k \in [0,n] \int_0^1 dp\, u^{(n)}(p) \binom{n}{k} p^k(1-p)^{n-k} = \frac{1}{n+1}$. This is a private case of a more ...
1
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0answers
14 views

numerial solution to fredholm integral equation

Consider the integral equation: $$ y(x) =1+\int_0^cK(x,t)\,y(t)\,dt, $$ where $x\ge0$ and $$ K(x,t) = \frac{\partial}{\partial ...
1
vote
1answer
26 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
1
vote
1answer
25 views

Do stochastic processes form a Banach space?

I'm interested in solving a particular integral equation: $$g(X) = \int_0^1 K(X,p)f(p) \ dp$$ where $f(p)\in L^1([0,1])$ and $X$ is a stochastic process of finite length; i.e. a collection of random ...
0
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0answers
37 views

How to solve integral equations like this?

sorry for such a non-specific question and lack of research effort, but I'm new to integral equations and don't know where to start. How does one go about solving equations of the form ...
4
votes
2answers
1k views

How to solve Fredholm integral equation of the second kind? ($f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2$)

I have an equation : $\ f(x) - \lambda\int\limits_0^1 9xtf(t) dt = ax^2 - 4x^2\ $ in $\ L_2[0,1]\ $ space. And I want to understand how to solve it, not just obtain an answer.
-1
votes
1answer
17 views

Simple integral equation

There is a simple integral equation: $$\int_0^\infty p(s) ds = 1- \frac{1}{\lambda}p(0).$$ Do you know how to solve it for $p(0)$ given we know $p(s)$ for $s>0$? Or maybe some references to methods ...
12
votes
1answer
355 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 ...
2
votes
0answers
53 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
2
votes
1answer
99 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
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0answers
23 views

Smoothness of solutions to Fredholm integral equation

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel ...
1
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0answers
56 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 ...
2
votes
2answers
47 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 ...
0
votes
1answer
27 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
votes
1answer
23 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
vote
0answers
33 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 $z=0$. ...
1
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0answers
37 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: ...
0
votes
0answers
34 views

How to solve an integral equation?

Consider the following integral equation: $$\log f(x)+\frac{\displaystyle 1}{\displaystyle 2\,\pi}\displaystyle\int_{\mathbb{R}^2} f(y)\log|x-y| dy+f(x)=0\tag{*}$$ How to prove the existence of ...
1
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0answers
21 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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0answers
28 views

Regulairty of eigenfunctions of singular integral equations

Can you provide a proof or a reference, to study from, for the following problem: Let $\Gamma$ be a real analytic rectifiable closed curve in the plane, $ds$ is the arc-length , and kernel $K(z,w)$ ...
0
votes
1answer
30 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 ...
0
votes
0answers
15 views

Resolution of numerical solution of first-kind Volterra integral equation

This is a Volterra integral equation of the first-kind with a convolution-type kernel $k(t-t')$: $f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$ It is well known and expected that when ...
24
votes
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?
1
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0answers
27 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 ...
0
votes
0answers
33 views

An integral equation of the first kind with variable limits

How should one solve the following integral equation to find $F(\tau)$: $$ \int_{\tau -T}^{\tau}F(t)X(\tau -t)dt=e^{-(\frac{\tau}{T})^2} $$
0
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0answers
48 views

Integral equation problem

I has worked on this problem for a while and still stucked on it. Hopefully someone give me a hint. Consider the following integral equation, find the function $g(r)$: $$\int_0^\infty {K(s,r)g(r)dr} ...
2
votes
1answer
66 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 ...
1
vote
2answers
28 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 ...
0
votes
0answers
19 views

integral equation with beta kernel

Is there any way to solve the integral equation $$ z(a,b;x) = 1+\dfrac{(1+x)^{b}}{B(a,b)}\int_0^c\dfrac{y^{a-1}}{(1+x+y)^{a+b}}z(a,b;y)\,dy,\;\;x\ge0, $$ where $a,b,c>0$ are parameters, and ...
0
votes
1answer
55 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 ...
0
votes
0answers
20 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
43 views

Combination of Integration and Derivative

How to solve this equation: $$ \int_0^{\frac{\pi}{2}}\left(a\sin y-b\left(\frac{dy}{dx}\right)^2\right)~dy=c, $$ where $a$, $b$, and $c$ are constant. Thank you for your help.
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 ...
1
vote
1answer
50 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 ...
3
votes
4answers
138 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 ...
1
vote
0answers
24 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
137 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
59 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
2answers
50 views

Uniform convergence of matrix integral sequence

I was given recursively defined: $M_k=I+\int_{t_0}^tA(s)M_{k-1}(s)~ds$ 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 we can ...
0
votes
2answers
29 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?
1
vote
0answers
40 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
votes
0answers
64 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
votes
1answer
61 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 ...
3
votes
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 ...
0
votes
2answers
57 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
vote
0answers
35 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 ...
1
vote
2answers
76 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
63 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? ...
3
votes
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 ...
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 ...