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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29 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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53 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
33 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
28 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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35 views

Solution for a Fredholm integral equation

I have an integral equation: $\int_0^T( t^\alpha + s^\alpha -|t-s|^\alpha) \phi(s)ds=\lambda\phi(t)$ for $\alpha\in(0,2)$. I think this is a Fredholm equation but I am not sure how to solve it. ...
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67 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 ...
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20 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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34 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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346 views

Integral equation solution (Fredholm, second type)

There is an equation $$ w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy $$ where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. ...
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2answers
80 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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53 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\\ ...
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1answer
41 views

Understanding solution to first-order differential equation written in integral form, $y(x)=\int_0^xy(t)dt + x + 1$

I have a differential equation and I'm trying to understand the solution printed in the back of the book. I will specify what part I'm struggling to understand: Problem statement: Solve the following ...
2
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1answer
15 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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18 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, ...
2
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3answers
67 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 ...
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1answer
2k views

How to solve partial integro-differential equation?

Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$: $\partial_t u = \partial_{xx} u + f(u,\lambda)$ $\lambda = B\left(u_0 - \int_{x=0}^L ...
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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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39 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 ...
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33 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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1answer
27 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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35 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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35 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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32 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 $h$ ...
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2answers
46 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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2answers
89 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 ...
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43 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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20 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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52 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
37 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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1answer
97 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 ...
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1answer
153 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) + ...
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23 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 ...
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60 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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26 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 ...
3
votes
1answer
91 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 ...
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2answers
59 views

Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
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22 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 ...
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36 views

Integral Equation of convolution type

given is the following integral equation: All variables and functions are given, except for n(x). I need to find n(x). Does anybody have an idea how to approach this problem? Many thanks in ...
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1answer
72 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.
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355 views

Find a continuous function $f$ that satisfies…

Find a continuous function $f$ that satisfies $$ f(x) = 1 + \frac{1}{x}\int_1^x f(t) \ dt $$ Note: I tried differentiating with respect to $x$ to get an ODE but you get one that contains integrals - ...
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2answers
67 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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3answers
138 views

Solving a non-linear integro-differential equation

I am trying to solve the following equation $$ f^2(x) - g^2(x) = \alpha\int_0^x f(u) (x-u)du $$ For $\alpha=0$ we get $f=g$. I would like to see how the solution moves away from $g$ when I increase ...
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0answers
40 views

Integral equation with convolution

I need to solve the following integral equation for $\phi(t)$: $$ \ln \phi(t) - c_2\int\limits_{-\infty}^{\infty} k(t-\tau) \, g(\tau,\phi(\tau)) \, d\tau = c_1 $$ On the web I found a solution for ...
2
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1answer
48 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 ...
3
votes
1answer
323 views

Is there a solution to this integral equation?

The problem is related to this question: How to find eigenfunctions of a linear operator (follow-up question) I posted earlier. Suppose I want to solve the following integral equation: $$\int_0^1 ...
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4answers
55 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
37 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 ...
1
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0answers
29 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 ...
1
vote
1answer
125 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') = ...