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
0
votes
0 answers

solution of a Fredholm equation of first kind

I was recently studying about solution of the following homogeneous Fredholm equation of first kind $$x=\lambda\int_0^1e^{x-t}y(t)dt$$ if there's any . But the method of regularization (since the kernel is separable) and homotopic perturbation…
am_11235...
  • 2,142
0
votes
2 answers

Solution of the Integral equation $ y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt $

This question is from a mathematics competition question paper. We are given the integral equation $$ y(x)= f(x) + \int_{0}^x \sin(x-t)y(t) dt $$ Then $y$ is given by: $y(x) = f(x) + \int_{0}^x (x-t)f(t) dt$ $y(x) = f(x) - \int_{0}^x (x-t)f(t)…
Kashif
  • 710
0
votes
1 answer

Unicity (or not) of the solution of an integral equation

Given the integral equation: $$\int_0^a f(x)\left[ \frac{d^2}{dx^2}f(x) \right]dx=a$$ with the condition: $$\lim_{x\to\infty}f(x)=0$$ how can I find its solution? Is the solution (if any) the only one possible?
Riccardo.Alestra
  • 10,546
0
votes
3 answers

Different formulations for multiple integral equation

On several papers, I found the following model for a multiple integral equation: $$g(s)=\int\limits_{\Omega} h(s,t)f(t)\,\mathrm{d}t$$ where $s,t \in \mathbb{R}^3$, and $\Omega \subseteq \mathbb{R}^3$. I would like to know wether it is possible to…
no_name
  • 445
  • 1
  • 5
  • 10
0
votes
1 answer

Determining the number of solutions of the non homogeneous integral equation.

This is a question from a competitive exam. We are given the integral equation: $$ \phi (x) = \cos(7x) + \lambda \int_{0}^{\pi} \left[ \cos(x)\cos(t) - 2\sin(x)\sin(t) \right]\phi(t) dt $$ and are asked the number of solutions of the equation…
Kashif
  • 710
0
votes
1 answer

Solution of the Integral equation $y(x) = 1 - 2x -4x^2 +\int_{0}^x \left[ 3 + 6(x-t)-4(x-t)^2 \right]y(t)dt$

I want to find the solution for the integral equation: $$y(x) = 1 - 2x -4x^2 +\int_{0}^x \left[ 3 + 6(x-t)-4(x-t)^2 \right]y(t)dt $$ I tried finding the resolvent kernel of the equation and it is getting more complicated at each new iteration. Is…
Kashif
  • 710
0
votes
3 answers

Integration of $\int_{0}^{\infty}{\frac{1}{(x^2+1)(x^{2019}+1)}dx}$

The original problem is: The proof of the equation show above is what I want. I found that the first integral equals $\pi/4$. The changes of 2019 from 1 to any real numbers bigger than 1 cause no change of the final integration.
Frank
  • 65
0
votes
1 answer

Solving a non-linear integral equation

Is there any way to solve an integral equation of this type, $s(1-b)f(s)=\int^s_{sb}f(x)f(x-sb)dx$ Any help would be highly appreciated.
Sayan
  • 121
0
votes
1 answer

Line integral with given midpoint and radius of a circle. Daugmans algorithm

I am trying to realize the daugman algorithm in java code. While reviewing the formula I found a line integral with a mid point and a radius as parameters. Now I am trying to understand how to calculate this integral. $$ \oint\limits_{x0, y0, r}^{} …
Alex
  • 1
0
votes
1 answer

Eigenvalues of the integral equation $\int_{-T}^{+T}\exp(-\alpha|t-u|)\phi(u)du=\lambda\phi(t)$

I've been going through 'Detection, Estimation, and Modulation Theory 2e' by Van Trees et al and am having trouble with Problem 6.4.1: Consider the integral…
Ruvi Lecamwasam
  • 2,002
0
votes
0 answers

Is it possible to find a solution to this equation, reducing it to the Fredholm equation?

To continue the…
dtn
  • 731
  • 1
  • 6
  • 19
0
votes
0 answers

Integral equation $\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$

In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ where $n>0$ and $\binom a b$ is the binomial…
Hans
  • 9,804
0
votes
1 answer

How to solve the following integral equation

Is there any method to solve the following integral equation, either analytically or numerically: $$A(t) cos(\omega t) + \int_0^t \omega A(\tau) sin(\omega \tau) d\tau = f(t)$$ Where: $$A(t): unknown\ function\ which\ must\ be\ found$$ $$\omega:…
Pirooz
  • 7
0
votes
0 answers

solving an Fredholm integral equation with symmetric kernel

Solve the following Fredholm integral equation with symmetric kernel : $$y(x)=\lambda\int_0^1 tK(x,t)y(t)dt$$ where $$K(x,t)=\begin{cases} \frac{x}{2t}(1-t^2) &\text{if} \ \ \ \ \ 0\leq x
am_11235...
  • 2,142
0
votes
1 answer

a problem on integral equation having no eigen value

show that the integral equation $$\phi(x)- \lambda\int^{\pi}_0 \sin x \sin 2t\phi(t)\,dt=0 , 0 \leq x \leq \pi$$ has no eigenvalue. can anyone help how can I able to solve this problem please.thanks for your time.
ketu
  • 1
1 2 3
…
9 10