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20 views

Existence and Uniqueness of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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31 views

existence of solution of volterra integral equation for the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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0answers
26 views

How to solve integral equation? [duplicate]

I have an integral equation such that $$ \int_t^T f(s)g(s,t)ds= h(t) $$ where g and h is given. we want to know function f explicitly. As i know, this type of question is about the integral equation. ...
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0answers
43 views

can we generalize the cauchy principal value and hadamard's finite part for multiple integrals $ n>1 $

I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one My question is when can we (or if negative answer why can not ) generalize the ...
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1answer
126 views

Solvability of a Fredholm system in $L^2$

Suppose $\lambda\not=0\in\mathbb{C}$. Does the following system have a non trivial solution in $L^2 [0,1]$? \begin{array} {lcl} \int_0 ^1 f(y)\log|x-y|dy=\lambda f(x) \\\int_0 ^1f(x)dx=0& ...
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1answer
179 views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
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1answer
92 views

Solvability of an integral equation

Is it possible to find $f,g\in L^2[(0,1)\times(0,1)]$ such that $$\log|x-y|=\int_0^1f(x,t)g(t,y)dt\:\:\:\forall x,y\in(0,1).$$
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59 views

Fredholm Integral Involving Stochastic Process

I wish to solve an integral equation of the form $$g(X) = c\int_0^1 K(X,t)f(t) \ dt $$ where $f\in L^1([0,1])$ and $g$ is some function on finite sequences of random variables. So, $X$ is a stochastic ...
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74 views

$L^p$-derivative of Log

Let $\gamma$ be a smooth closed planar curve. Is $\int_\gamma\big(\frac{\log|\zeta-z|-\log|\zeta-w|}{|z-w|}\big)^2ds(\zeta)$ uniformly bounded in $z,w$ as $|z-w|$ is sufficiently small?another word ...
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35 views

How to solve a multidimensional integral equation of convolution type 2 with real coefficients?

Can someone suggest suitable reference for solving a 4*4 integral equation of convolution, and whether the following equation has a closed form solution? I really appreciate your help. where y, K, ...
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92 views

A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$, $$\int_z^\infty ...
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0answers
89 views

Schwartz kernel of a pseudodifferential operator with singular symbols

If we have a smooth symbol $r(x,\xi)$ of order $d$ the corresponding pseudodifferential operator $P$ is integral operator with kernel given by $$K(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi} ...
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147 views

Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function. Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
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138 views

Solving a Volterra integral equation with both limits variable

I would like to obtain an analytical solution to the following Volterra integral equation: $\int_{\alpha(x)}^{\beta(x)}y(t)(t^2-g(x))\,\mathrm dt=0$. The functions $\alpha(x)$, $\beta(x)$ and $g(x)$ ...
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0answers
66 views

A question about the $\partial_q$

How can I prove the following equation: $\partial _q^{-1} f=\theta^{-1}(f)\partial_q^{-1}+\partial_q^{-1}\circ(\partial_q^*f)\circ \partial _q^{-1}$. Where $\partial_q^{-1}$ is the formal inverse of ...
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2answers
141 views

Double Integral of plane wave squared over a circular domain

Hello everyone! I need to calculate the following integral: $\int\int_{\sqrt{x^2+y^2}\leq a}\cos^2(v_x x+v_y y)dxdy$ First step, I convert to polar coordinates: $\int_0^a\int_0^{2\pi}\cos^2(v_x ...
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168 views

Notation for a functional L2 matrix norm

Hi, Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation: ...
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2answers
437 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...
6
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2answers
375 views

Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by $$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$ Is it possible to decompose this into two separate ...