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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44 views

Solve the integral equation with symmetric kernel

I have the following integral equation with symmetric kernel $$g(x)=\cos \pi x +\lambda \int_{0}^{1} k(x,t)g(t)\,dt $$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= \begin{cases} (x+1)t, ...
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0answers
34 views

Solving a Fredholm equation using Neumann series technique

It is a very simple kind of Fredholm equation: $$f(x)=x+\int_0^1(1+xt)f(t)\,dt.$$ I solved it and I know that the answer is $f(x)=-2$. But how can i solve this equation by the Neumann series ...
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0answers
17 views

A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$ 0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt $$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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1answer
25 views

Optimal Control of Integral Equations of the first kind

I have managed to find a couple of papers which deal with the optimal control of systems governed by integral equations of the second kind (e.g. here: ...
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0answers
20 views

Solving a weighed integral equation

I am struggling in solving a specific type of equation which simply pops out when we're dealing with weighed functions. I think the general context is worth mentioning. Let's say we have a discrete ...
2
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0answers
51 views

Optimization Problem Involving an Integral Equation

I have an integral equation for a probability current $j(t)$ given by: $$ \large\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} ...
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0answers
57 views

How to solve a homogeneous Fredholm integral equation of the second kind with a symmetric non-seperable kernel?

I have the equation \begin{eqnarray} \lambda L(p)=\int dq\,K(p,q)L(q) \end{eqnarray} Where $L$ is an unknown function, $\lambda$ is some constant, and $K$ is a known function. This is a homogeneous ...
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4answers
112 views

What is the function that satisfies $\int_0^x f(t) dt=constant$ [closed]

$$\int_0^x f(t) dt=constant$$ What is the function that satisfies this condition ? Thank you!
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0answers
47 views

The resolvent kernel

Good morning. How can we prove that $$ \Gamma (s,t, \Lambda ) - \Gamma (s,t, M ) = \Lambda - M \times \int { \Gamma(s,t, \Lambda ) \times \Gamma (s,t,M) \, dx }$$ I tried to use the formula of ...
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votes
0answers
21 views

Integral parameter equation

How to solve the following equation, $$\int_{-i\infty}^{+i\infty} e^{st}\frac{\Sigma_{k=0}^{m}b_ks^k}{\Sigma_{k=0}^{n}a_ks^k} ds=0$$ to obtain $t$ where, $m\le n$ and $a_i$, $b_i$ are the given ...
1
vote
1answer
73 views

How to find the characteristic number of a given integral equation?

How to find the characteristic number of the following integral equation? $$y(x)= \lambda \int_{0}^{1} (3x-2)ty(t)dt$$
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3answers
82 views

Solution of the integral equation $y(x)+\int_{0}^{x}(x-s)y(s)ds=x^3/6$

So the question is to solve the integral equation to find out $y(x)$. $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\dfrac{x^3}{6}$$ So I find out the Resolvent kernel for $(x-s)$ and got $\sin{(x-s)}$, so I ...
2
votes
2answers
86 views

derivative over nested integrals

I have the following problem. First an example for two-variable functions Let $A=A(t,\tau)$ and $B=B(t,\tau)$ If I want to compute the following $\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t ...
1
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0answers
30 views

Function equation with sth like self-convolution

I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$. Or a more simple question may be if we assume ...
4
votes
2answers
63 views

Check whether a functional has an extremal or NOT

Find the extremal of the functional $$J(y)=\int_a^b F(x,y,y')\,dx$$where , $F(x,y,y')=y'+y$ , for admissible functions $y$. From Euler-Lagrange equation , $\displaystyle ...
1
vote
1answer
55 views

Which method solves this integral equation? $\int_{-1}^{1}w(x)\,e^{tx}\,dx=6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$

Today I encountered this integral equation wrt. $w(x)$: $$\int_{-1}^1 w(x)\ e^{t x}dx = 6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$$ I never solved such equations, and when I tried to ...
0
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0answers
38 views

Second kind Volterra integral equation with weakly singular.

Consider the second kind Volterra integral equation. $$ f(t)=g(t)+\int_0^t f(t) K(t,s)ds $$ where $g(t)$ is continuous and $K(t,s)$ is a weakly singular kernel, i.e. $$ K(t,s)\leq ...
2
votes
3answers
52 views

Integrate $\int\frac{x+1}{x^{4}+6x^{2}+4x}dx$

This gives a cubic polynomial in denominator and nonfactorisable It looks so simple but i just am not able to solve it
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1answer
25 views

Checking whether the function $u(x)=e^x$ solves the integral equation $u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$

The function being $u(x)=e^x$ and the integral equation is $$u(x) + \lambda \int_0^1sin(xt) u(t) dt=1$$ I can't do the integration and I'm confused about how to deal with the parameter $\lambda$ ...
2
votes
1answer
34 views

Solving a Volterra Integral Equation of the 2nd Kind

Can anyone help me in finding a closed-form solution to the integral equation $$x\left(t\right)=1-\lambda \int _{0}^{t}e^{-\alpha \left(t-\tau \right)} \cos ^{2} \left(k\tau \right) x\left(\tau ...
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0answers
12 views

when the solution of a volterra integral equation is a probability distribution?

I have a Volterra integral equation looking like this $$F(t) = \int_0^t \lambda(s)ds - \int_0^t{F(s)\lambda(t-s)ds}$$ is there any condition I can ask on $\lambda$ for the solution $F$ to be upper ...
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0answers
37 views

Love's equation $f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$

Let us consider Love's equation: $$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$ Is $f(x)$ a two times differentiable function?
2
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0answers
56 views

Cosine Fourier series solution of semi-major axis nonlinear integral equaton

Consider an integral equation $$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$ I am required to solve for $z(t)$. I approached this problem by considering the integral on right hand ...
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votes
1answer
52 views

How to find exact solution of this volterra equation?

I was working on numerical solution of this equation (by block pulse). $$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ...
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0answers
26 views

How to solve this special case of Fredholm Integral Equation of the First Kind

General form of 'Fredholm Integral Equation of the First Kind' $f(x) = \int_a^b{K(x,t)\phi(t)} dt$ Where $\phi(t)$ is the unkown My special case is $1 = \int_a^b{k(t)\phi(t)} dt$ A trivial ...
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2answers
74 views

How can I solve this integral equation by converting it to a differential equation

Let we have the following integral equation :$$y(x)=e^{-x}cos(x)-\int_{0}^{x}e^{-x+t}cos(x)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation
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2answers
92 views

Solve integral equation by converting to differential equation

One has the following integral equation: $$y(x)=\frac{1}{1+x^2}+\int_{0}^{x}\sin(x-t)y(t)dt$$ How can I solve this integral equation by converting it to a differential equation?
2
votes
2answers
69 views

prove $ \int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$

prove that for every $ p,q \gt 0$ $$ \int_0^1 (1-x^p)^\frac{1}{q} \,dx = \int_0^1 (1-x^q)^\frac{1}{p} \,dx$$ I tried to start from one side and change variables to get something similiar to the ...
2
votes
0answers
27 views

A Gronwall type inequality for Fredholm operators

I am interested in finding out the result of Gronwall type inequality for Fredholm operators. What will be the form? How one can show such inequality for fredholm operators.
5
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0answers
59 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
4
votes
3answers
65 views

Fourier-like integral equation

(This question inspired by question A specific 1st order PDE which looks almost like a linear PDE.) Solve integral equation $$ g(x)=\int\limits_{-\infty}^\infty \rho(\omega)\left[e^{i\omega x} - ...
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0answers
42 views

Numerical Solutions of Fredholm Integral Equations of the First Kind

Can anyone recommend me some papers about numerical solutions of Fredholm integral equations of the first kind? Thanks in advance
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0answers
37 views

Existence and uniqueness of a pde solution

I have the PDE system: $\frac{\delta}{\delta t}u(t,r)=-\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)$ $\frac{\delta}{\delta t}v(t,r)=\int_0^1 H(|r-r'|)v(t,r')dr'u(t,r)-v(t,r)$ $x(0,r)=\rho(r), ...
0
votes
1answer
41 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
0
votes
1answer
244 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
4
votes
1answer
77 views

Explicit solution for equation

The claim is that this equation has an explicit solution. $$\frac{\partial}{\partial t}c(x,t)=\frac{a}{\pi}\int_{\mathbb{R}}\frac{c(y,t)-c(x,t)}{(y-x)^2}dy.$$ What can one do to find this solution? ...
2
votes
0answers
44 views

Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
3
votes
1answer
71 views

transform integral to differential equations

I found a similar system of integral equations in a paper. It says that it can be solved by differentiating and then using standard techniques. My question is, how can I differentiate such a system in ...
0
votes
1answer
29 views

Singular Integral Equation

I need to find an approximate solution $u(z)$ of the following equation: $\int_{-H}^0 q(s,z)\,u(s)\,ds = -2\,\rm{i}\,\xi_0(z)$ where $q(s,z) = ...
0
votes
1answer
69 views

Solving for a function inside an integral

Is there a way to solve for $f(x)$ when $$ g(x)=\int_0^x dx' W(x,x') f(x') $$ If it weren't for the x-dependence in $W(x,x')$, I could write for example, $$ f(x)=\frac{1}{W(x)}\frac{\partial ...
1
vote
1answer
81 views

Integral Equation Unknown Limits

What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution? The simplest example is the ...
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0answers
44 views

Numerically finding eigenvalues of a Volterra operator of first kind

I'm looking for a solution to the following problem - $\int_{-\infty}^{\infty} K(x-y) f(y) = \lambda f(x)$ Consider $K(x-y) = \left\{ \begin{array}{lr} e^{-(x-y)} & : x > y \\ 0 & : x ...
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0answers
33 views

which kernel has finite rank?

in integral equations $$x(s)=y(s)+\lambda\int_a^b k(s,t)x(t) dt\\k \in L^2[a,b]$$ which one of listed kernels has finite rank ? how to show (or proof)? ...
4
votes
2answers
40 views

Need solution to Volerra integro-diff equation

I need to solve a system of Volterra integro-diff equation of form $$ y(t) = x(t) - \int_{0}^{t} k(t-\tau) y'(\tau) \;\mathrm{d}\tau $$ where kernel is of form $$ k(t-\tau) = P(t)Q(\tau) $$ Is it ...
1
vote
1answer
230 views

Convert IVP to an equivalent Volterra integral equation

Convert the following initial value problem to an equivalent Volterra integral equation: $ \begin{cases} u'' -u' \sin x + \Bbb e ^x u= x \\ u(0)=1\\ u'(0)=-1\\ \end{cases} $ I ...
3
votes
0answers
44 views

Solvability of an integral equation

Is the following integral equation solvable ? $$ F(x)-\int^{1}_{-1} K(x,y)F(y)dy=f(x) $$ Where $$K(x,y)=\frac{\sin \gamma(x-y)}{\pi(x-y)}$$ and $$f(x)=e^{i\gamma x}$$ and $\gamma$ is a parameter.
1
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0answers
28 views

Solve integral equation that involves hyperbolic cosine

I'm trying to find a weight function w(x) that makes this integral 0 $\int_0^1 w(x) \cosh((\alpha_n+i\omega_n)x) \cosh((\alpha_n-i\omega_n)x)=0$, where where $\omega_n=(2n+1)\frac{\pi}{2}$ and ...
1
vote
0answers
43 views

Integro-differential eigenvalue problem

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
1
vote
2answers
35 views

Transforming the integral equation $u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b$ into its equivalent differential equation

Let $u \in C^2[0, 1]$ satisfy for some $ \lambda \neq 0$ and $a \neq 0,$ $$u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b.$$ Then show that u also satisfies $\frac{d^2u}{dx^2} + \lambda u ...
0
votes
1answer
46 views

solve integral equation using the theory of compact operator

Find solutions of $$u(x)-\lambda\int^{2\pi}_0\sum_{j=1}^n\frac{1}{j}cos(jy)cos(jx)u(y)dy=sin^2x$$ for all values of $\lambda$. Find the resolvent kernel for this equation. (Find the least squares ...