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

970 questions
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Integral equation $\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}}f(R \cos(x)) d x = 1$

Can we prove that there does not exist a function $f$, which satisfies this equation for all $R>0$: $$\int_0^{2 \pi} \frac{1}{\sqrt{1+R^2 \sin^2(x)}} f(R \cos(x))\, dx= 1.$$
Marc Palm
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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 right side, but it got my no where.
Daniel Katzan
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Solution of Abel type integral equation

I would like to know when (for what functions $f$) and how I can find integrable solution of equation \begin{align} f(x)=\int_x^{\infty}\frac{u(y)}{\sqrt{y-x}} \ dy, \end{align} where $u$ is unknown function? Thanks.
patric
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Find continuous $f$ with period $1$ such that $f(x) =\int_0^1 f(x-t)f(t) dt$

The problem is to find all $f : \mathbb{R} \to \mathbb{C}$ that is continuous and has a period of $1$ (not necessarily smallest period) such that the following equality holds: $$f(x) = \int_0^1 f(x-t)f(t) dt \quad \forall x \in \mathbb{R}$$ I…
user99185
  • 287
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Fredholm integral equation

How can I solve the following fredholm integral equation $$ψ(x)=x+λ\int_{0}^{2π}|x-t|ψ(t)dt$$ The kernel contains absolute value
Syria math
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Uniqueness of a positive solution to an integral equation

This is a followup to another question I asked recently. This is a slight modification to that question. In the fluid mechanics of pipe flow, it is sometimes stated that the velocity profile $u(r)$ which corresponds to a kinetic energy coefficient…
AtomicAcorn
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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. However, it involves the mag-square of the function to…
SDiv
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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 use: $$ S(x) = \int_a^x f(t) dt $$ in some way.
iveqy
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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?
Lugi
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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.
Jens Wagemaker
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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. Appreciate any assistance. Thanks, Ido.
yoki
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Solution of Volterra convolution-type integral equation

The convolution-type Volterra integral equation of the first kind $f(t) = \int_a^t k(t-t')\,x(t')\,dt' \qquad t\in [0,\infty]$ can be solved (at least formally) by applying the Laplace transformation $L$ to obtain $F(s) = K(s)X(s)$ (where the…
D_J_S
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Integral equations that can be solved elementary

Solve the following integral equations: $$ \int_0^xu(y)\, dy=\frac{1}{3}xu(x) \tag 1 \label 1 $$ and $$ \int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1. \tag 2 \label 2 $$ Concerning $\eqref 1$, I read that it can be solved by differentiation.…
user34632
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An analytical solution of the integral equation $ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz $

How to solve for the f(x) in the integral equation? $$ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz \\ \alpha \in (1,2), C_2\text{ is a positive constant.} \\ \text{What is the definition of the function } f(x)…
YuanLan
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Understanding Method of Successive Approximation for finding the unique solution to Volterra Integral Equation

Consider this non-homogeneous Volterra Integral Equation of second kind, $$u(x)=f(x)+\int_a^{x}K(x,\xi)\,u(\xi)\,d\xi$$ where, $f(x)$ and $K(x,\xi)$ are non-zero real valued continuous functions defined on $I=[a,b]$ and $I\times I$ respectively and…
mat09
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