Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integral-equations]

This tag is about questions regarding the integral equations. 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.

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.

A general integral equation for an unknown function $y(x)$ can be written as $$f(x) = a(x)y(x) +\int^b_a k(x,t)y(t)dt$$ where $~f(x),a(x)~$ and $~k(x,t)~$ are given functions (the function $~f(x)~$ corresponds to an external force).

The function $k(x,t)$ is called the kernel.

Classification : There are different types of integral equations. We can classify a given equation in the following three ways.

  • The equation is said to be of the Integral Equations of First kind if the unknown function only appears under the integral sign, i.e. if $a(x) ≡ 0$, and otherwise of the Integral Equations of Second kind.

  • The equation is said to be a Fredholm Integral Equations if the integration limits $~a~$ and $~b~$ are constants, and a Volterra Integral Equations if $~a~$ and $~b~$ are functions of $x$.

  • The equation are said to be Homogeneous Integral Equations if $f(x) ≡ 0$ otherwise Inhomogeneous Integral Equations.

Applications: Integral equations arise in many scientific and engineering problems. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations. The potential theory contributed more than any field to give rise to integral equations. Mathematical physics models, such as diffraction problems, scattering in quantum mechanics, conformal mapping, and water waves also contributed to the creation of integral equations.

References:

"Handbook of Mathematics" by I.N. Bronshtein · K.A. Semendyayev · G.Musiol · H.Muehlig

"https://en.wikipedia.org/wiki/Integral_equation"

"Integral Equations" by Francesco Tricomi

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Homogeneous integral equation $\xi (t) - \int_0^{\pi} \sin(x+t) \xi (x) dx = 0$

I have been given the homogeneous equation: $$\xi (t) - \int_0^{\pi} \sin(x+t) \xi (x) dx = 0$$ and I have to show that the only solution is the trivial solution. This is the adjoint homogeneous equation of a Fredholm problem I have been given,…
Henrymerrild
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the uniqueness of the integral equation

If $x(0)=\int_{0}^{T} A(t)y(t)dt-\int_{t=0}^{T}\int_{s=0}^{t} B(t, s)f(s, x(s))\hspace{0.1 cm}ds \hspace{0.1 cm}dt,$ where $A(t),$ $B(t, s)$ are the known $n \times m$ and $n \times n$ matrices respectively, $y(t)$ and $f(s, x(s))$ are the vectors…
thomus
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Solve $ \int_0^\infty e^{-\frac{c}{2x^2}} g(x) dx =\sqrt{\frac{\pi}{2}} e^{-\frac{\sqrt{c}}{2}} $

How to find a positive function $g(x)$ such that \begin{align} \int_0^\infty e^{-\frac{c}{2x^2}} g(x) dx =\sqrt{\frac{\pi}{2}} e^{-\frac{\sqrt{c}}{2}} \end{align} I know that the solution is $e^{-\frac{x^2}{2}}$ but not sure how to search for such…
Boby
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Voltera integral equation of first kind.

The curve $y=y(x)$ passing through the point $(\sqrt{3},1)$ and defined by the following property $$\int_{0}^{y}\frac{f(v)dv}{\sqrt{y-v}}=4\sqrt{y}$$ where $f(y)=\sqrt{1+\frac{1}{y^{'2}}}$ is a part of a $1.$ Straight line. $2.$ Circle. $3.$…
neelkanth
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Need help with a differential equation -like problem.

$\forall y \in \mathbb{R}, \int_{-\infty}^{\infty} f(x)f(x-y)dx=f(y)$ I also know that $\int_{-\infty}^\infty f(x) dx$ converges and that $f$ is symmetric about the origin. What does $f$ look like? Is it possible to identify a parametric set of…
GMB
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Integral Hammerstein-like equation solution

In signal processing theory, I found this integral equation which I suppose to be of Hammerstein type: $$u(t)-\int_0^1\frac{\cos(\omega t+\phi)}{u(\phi)}d\phi=0$$ I didn't find anything in literature apart…
Riccardo.Alestra
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For what value(/s) of $\lambda$ , solution of the following Integral Equation does not exist?

For what value(/s) of $\lambda$ , solution of the following Integral Equation does not exist ?$$y(x)=1+\lambda\int_0^1(1-3xt)y(t)\,dt$$ Let , $$y(x)=1+\lambda C_1-3\lambda xC_2$$where , $$C_1=\int_0^1(1+\lambda C_1-3\lambda tC_2)\,dt=1+\lambda…
Empty
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Fredholm integral equations

I'm having problem obtaining the solution of the homogeneous Fredholm integral equation of the 2nd kind with a separable kernel. I always get a zero if I use the normal method I was taught for the non homogeneous type. I have an example: y(x) =…
HenryMath
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Some linear integral equation

Please help me with the following problem: Let $\gamma\in (0,1)$ and $a<0
Darek
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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 how to solve it?
cherep
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If $\phi(x)=\lambda\int_0 ^1 e^{x+t}\phi(t)dt$, then for what value of $\lambda$ does there exist a non-trivial solution?

For the homogeneous Fredholm integral equation $$\phi(x)=\lambda\int_0 ^1 e^{x+t}\phi(t)dt$$ For what value of $\lambda$ does there exist a non-trivial solution for $\phi(x)$?
F. A. Mala
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Can there exists a solution to the following integral equations?

I have come across the following two integral equations: \begin{align} a&=f(1)+f(2)+\int_{0}^{3}t f(t)\mathrm{d}t\\ b&=\int_{0}^{3}t^2f(t)\mathrm{d}t. \end{align} Now, my question is can there exists at least one piecewise continuous function $f(t)$…
thomus
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