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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Existence and uniqueness of Volterra integral equations of the first kind with vanished kernel

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...
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2answers
52 views

Finding function $f(x)$

How do we find the function(s) $f(x)$ given that $$f(x)=\int_{0}^{x} te^tf(x-t) \ \mathrm{d}t$$ My Try : I first used the property $\int_{0}^{a}g(x) \ \mathrm{d}x=\int_{0}^{a}g(a-x) \ \mathrm{d}x$ ...
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25 views

I need a general solution of the following PDE:

$$\frac{\partial}{\partial t} F(t, x) = xF(t,g(tx)), F(0,x)=1=F(t,0) $$ where $g(x)$ is given. In fact, I need the case $g(x)=x.$ This PDE comes from an integral equation $$ F(t,x)=1+x \int_0^t F(s, ...
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1answer
18 views

Is it possible to find a solution to this integral equation?

I have an integral equation of the following form: $y(t)=\lambda x(t) + x(t)\int_{-\infty}^{\infty}K(t,s)x(s)ds$ I haven't been able to find any discussion online of integral equations with the ...
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60 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the simple and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
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0answers
34 views

Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
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1answer
30 views

How to find the solution of integer equation group

I have the following problem: to find the general item of the following equation: let $a_1=b_1=1$, $$a_{n+1}=6a_n+2b_n, b_{n+1}=3a_n+2b_n$$ for any $n\geq 1$, find $a_n=?, b_n=?$
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31 views

finding solution to a partial integro differential equation

I want to find a function (or a set of functions) such that $u(x,t)$ satisfies the following partial integro-differential equation with singular kernel \begin{eqnarray} &&u_x(0,t) = \int_0^t ...
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1answer
55 views

How to solve $xy=2\int_1^xy(t)dt+5$?

Could you please give me some hint how to solve this equation: $xy=2\int_1^xy(t)dt+5$. It is not known whether $y(x)$ is continuous or not, so I could not use Fundamental Theorem of Calculus for ...
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29 views

FM signals and non-trivial solutions to a homogeneous Fredholm integral equation of the first kind

I am looking for any non-trivial solution to the following integral equation. That is, find any function $q(\theta)\neq 0$ which satisfies the following equation: $$\int_0^a ...
3
votes
2answers
79 views

Volterra equation for a Bessel type IVP that appears in inverse scattering

I have the following i.v.p. (Colton Kress-Inverse acoustic and electromagnetic scattering theory, Springer) $$y''(r)+(k^2n(r)-\frac{l(l+1)}{r^2})y(r)=0$$ $$y(0)=0, y'(0)=1$$ using the Liouville ...
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0answers
30 views

Integral equation involving magnitude/modulus squared

I wish to solve the following integral equation that has popped up in my studies of focused light. If you notice, it looks almost like a homogenous Fredholm integral equation of the first kind. ...
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1answer
30 views

Define all functions using the main statement

Define all functions that are continious and fullfill the equation $$ f(x) = -1 + \int_0^{x^2} \frac{(f(\sqrt{t})^2 \sin t}{\cos^2t} dt$$ I'm completely lost on this one. I think that you should ...
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2answers
42 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
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1answer
45 views

Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
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31 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
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0answers
40 views

Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...
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1answer
61 views

Existence of integral equation solution

I am trying to prove a differentiable solution in some open interval about the origin for the equation: $$u(x) + u(x)^2 + \int_0^x (1+\cos(x+u(y))) dy = 0$$ I have been trying to prove it as a ...
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2answers
87 views

A Matrix Integral Equation

We have an integral equation on matrix. ${\Im(t)}=\Im(0)+\int_{0}^{t} \Im(s)[K(s)]_{ \times }ds \tag 1$ $[\hspace{.2cm} ]_{\times}$ is skew symmetric matrix with diagonals zero and is non ...
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65 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
2
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1answer
19 views

Solution sets/ existence and uniqueness of solutions to $Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x)$

Given $$ Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x) $$ A) For what values of $\lambda$ does there exist a unique solution for all $f\in L^2(0,1)$? B) Find the solution set ...
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26 views

Condition of existence and uniqueness of solution for abel integral equation

It is well known that Abel integral equation has a unique continuous solution. For example, $$ f(t)=\int_0^t\frac{g(s)}{(s-t)^{\alpha}}ds , 0<\alpha<1 $$ where $f(t)$ is known. Specifically, ...
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74 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
2
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2answers
40 views

Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
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40 views

An integral equation of second type.

Let $A,B$ be real numbers smaller in absolute value than $1000$. Consider the integral equation $$ f ( x ) = A + B\int_1 ^{\sqrt x} f \left( \frac{x-1}{t} \right) \,\mathrm{d} t $$ where the ...
2
votes
1answer
28 views

Taking Fourier transform of integral-differential equation

If $u$ is a solution of the equation $$\frac{\partial}{\partial t} u(x,t) + \int_{-\infty}^{\infty} \text{sinc}(x-y) \cdot \frac{\partial^{2}}{\partial y^{2}} u(y,t) \ dy = 0,$$ with initial condition ...
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63 views

existence and uniqueness 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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39 views

Contraction mapping principle application

I'm to prove that the following equation has a unique solution: $$f(x) = \int_0^1 e^{-sx} \cos(\alpha f(s)) ds.$$ (Here, $\alpha \in (0,1)$.) The form of the exercise screams to apply the ...
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1answer
42 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
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38 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
2
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2answers
49 views

Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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0answers
44 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
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2answers
93 views

Integral equation/ODE

I have to find all the functions $f(x)$ such that $$f(x)=xe^{(1-x^{2})/2}-xe^{-x^{2}/2}\int_{1}^{x}t^{-2}e^{t^{2}/2}dt$$ which satisfies $$f(x)=1-x\int_{1}^{x}f(t)dt$$ I tried to equal both, but when ...
3
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0answers
27 views

Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
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0answers
22 views

Solving a nonlinear volterra integral equation with two integrals each with a non separable kernel

I am trying to solve the nonlinear volterra integral equation ...
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4answers
61 views

Solve $y = 2 + \int^x_2 [t - ty(t) \,\, dt]$

While working on some differential equation problems, I got one of the following problems: $$y = 2 + \int^x_2 [t - ty(t) \,\, dt]$$ I have no idea what an integral equation is however, the hint ...
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3answers
40 views

Solving integral equation using Laplace transform

$x(t)+\int_0^t(t-\tau)x(\tau)=t^2$ Is $x(\tau)$ the equivalent of $d\tau$? How do I solve this particular equation?
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0answers
28 views

Writing $y'' + P(x)y' + Q(x)y = g(x) $ as a Fredholm Integral Equation

Is it possible to convert a general linear second order boundary value ode $$y'' + P(x)y' + Q(x)y = g(x), y(a) = y_a, y(b) = y_b$$ to a Fredholm integral equation, explicitly determining the Kernel ...
3
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0answers
62 views

Finding the integral of $\cos\theta \cdot dt$ in terms of the integral of $\sin\theta \cdot dt$

I have an integral as follows: $$\int_0^T \cos\theta\cdot dt = xT$$ where $\theta$ is a function of $t$ I also have, $$\int_0^T \sin\theta\cdot dt = y$$ I want to solve for $T$. If the ...
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37 views

A solid rocket model: a differential equations set with ending time unknown

I am modelling a rocket model. Consider a solid rocket motor, (let us for sake of simplicity assume that the propellant is distributed in the case with a cylindrical shape: see shape in fig.1 of the ...
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0answers
24 views

asymptotics of the solution of an integral equation

Suppose we are given the integral equation $$ u(x;a) =v(x)+\int_0^a K(x,y)\,u(y;a)\,dy, $$ where $K(x,y)$ and $v(x)$ are known functions, and $a>0$ is a constant. What I am interested in is the ...
2
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2answers
70 views

solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$ |f(t,x)|\leq \phi(t)|x|. $$ Suppose ...
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1answer
78 views

How can i solve the integral equation

How can i solve the integral equation $$z(t) =\int_0^t z(q)(1-(t-q))\,dq+C?$$ Solving for the function $z$. I have access to Mathematica.
2
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1answer
54 views

Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
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4answers
57 views

A little doubt with integral equation

I have the next equation: $$\int_{0}^{t}h(\tau)e^{-(t-\tau)}\mathrm{d}\tau=10e^{-t}\cos(4t) \tag{1}$$ Derivating both sides, I get: $$h(t)e^{-(t-t)}=h(t)=10[(-1)e^{-t}\cos(4t)+e^{-t}(-4)\sin(4t)] ...
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0answers
44 views

Integral equation solution in power series

Given the integral equation $$ g(x)= \int_{-\infty}^{\infty}K(x-y)f(y) \, dy$$ for a known function $ g(x) $ and kernel $ K(x)$. Of course I know this is a Wieener-Hopf integral equation but I ...
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0answers
31 views

Does this integral equation have a name, and is there a reference that deals with it?

The equation is $$S(z,t)=\int_a^b \int_c^d f(z,u) g(t,s) S(u,s) \, du \, ds$$ where $S$ is the unknown function and $f,\ g$ are fixed from the outset. I can approximate solutions in some special ...
3
votes
1answer
97 views

Showing that a sequence of Picard iterates converges

I have a sequence of functions: $$y_{n}(x) = 1 + \int \limits_0^x 1 + t^2 + y_{n-1}^2(t)\,\mathrm dt$$ With $y_0 = 1$. I'm trying to show that this converges in a box $-1 \le x \le 1$ and $-10 \le ...
1
vote
1answer
137 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
0
votes
0answers
13 views

inequality for compact operator

Let $K(x)$, $x\ge0$ be a nonnegative-valued continuous function with support $(0,\infty)$ and such that $\int_0^\infty K(x)\,dx=1$. Let $\mathcal{K}$ be an integral operator given by $$ ...