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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Existence and uniqueness of an integral equation

Does this equation $$f(u)=1+\frac{1}{\pi}\int_{-\infty}^{\infty} \frac{1}{(u-v)^2+1}f(v)dv$$ has a bounded continuous solution? Is this solution unique? Here $f$ is defined over $\mathbb{R}$ and bounded. Here are my attempts: I tried the…
Golbez
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Solving integral equation

Solve the following integral equation: $${u(x)}={x}+{e^{x}}+\int_{0}^{1}\left({5}{x}^{2}{t}^{2} -{3}{t}^{2}+{t}\right) {u(t)}dt. $$
user90321
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Let $f(x)=\sin(x)-\int_{0}^{x}{(x-u)f(u)du}$ where $f(x)$ is continuous. Find $f(x)$.

Let $$f(x)=\sin(x)-\int_{0}^{x}{(x-u)f(u)du}$$ where $f(x)$ is continuous. Find $f(x)$. Initially, I use FTC and obtain $f(x)=\sin(x)$ but in the question didn't mention $f$ is differentiable. Then I get stuck for half an hour. Can anyone guide me…
Idonknow
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It is possible to solve this integral equation?

I have tried (hard) to solve the following integral equation, but without success: $$\frac{1}{2{\pi}i}\int_{-\infty}^\infty{\frac{\sigma(s)}{s-(2hi+x)}ds}=\frac{1}{(x+hi)^2}$$ where $h$ and $x$ are real, $h>0$ and $x\in[-\infty,\infty]$. At first…
jesusvaleo
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Fredholm integral equation of the first kind

Can we solve the following specific integral equation: $$ \int_0^1v^n(1-v)^{x-1}K(v)dv=f(x) ,x\in[0,1) $$ If it is solvable, I wonder whether its solution can be represented in a closed form.
Mathwinnie
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Find $y(1)$ if $y(x)=x^3+\int_0^x \sin(x-t)y(t)dt$

Consider the integral equation $y(x)=x^3+\int_0^x \sin(x-t)y(t)dt, x\in[0,\pi].$ Then the value of y(1) is $1. \;\;19/20 \\ 2. \;\;1\\ 3. \;\;17/20\\ 4. \;\;21/20 $ My Attempt: $y(x)=x^3+\int_0^x \sin(x-t) \ y(t)dt \\ \Rightarrow…
Priyanka
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What is the value of C in this integral equation?

Consider the following integral equation: $$\int_0^1 С(yW(x))W^3(x)\,dx=F(y),$$ $$ W(x)=0.5( \cosh(kx)-\cos(kx)-A( \sinh(kx)-\sin(kx))) $$ $F$ is known. $A= \frac{ \cosh(k)+\cos(k)}{ \sinh(k)+\sin(k)}$. What is the value of $C$?
Айрат Камалутдинов
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Solving Volterra integral equation

I would like to solve $4u(t)+\int_0^t\sin(t-s)u(s)ds=5t, \ t\geqslant 0$. Any ideas on how to approach this equation?
user339135
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convert differential equation to Integral equation

$$ y''(x) + y(x) = x$$ with b.v conditions $$ y(0) = 1, y'(1) = 0 $$ Integrating $$ y'(x) - y'(0) + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$ $ let y'(0) = c_1 $ $$ y'(x) - c_1 + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$ $$ y'(x) = c_1 - \int…
Ali Jan
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Solutions of an integral equation

Given the integral equation: $$\sqrt{f(x)}\int_{0}^{x}f(\tau)d\tau=g(x)$$ with g(x) known function, in what cases and how is it possible to solve it? Thanks
Riccardo.Alestra
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Solving a Volterra Integral Equation of the 2nd Kind

Can anyone help me in finding a closed-form solution to the integral equation $$x\left(t\right)=1-\lambda \int _{0}^{t}e^{-\alpha \left(t-\tau \right)} \cos ^{2} \left(k\tau \right) x\left(\tau \right)d\tau$$ Thanks a lot.
Xiaojing Huang
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Volterra integral equation of second type

Solve the Volterra integral equation of second kind : $$ y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds $$ I know two methods for such integral equations: Picard's method The method of finding the resolvent kernel and the Neumann…
passenger
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Homogeneous Fredholm Integral Equation

I'm having problem obtaining the solution of the homogeneous Fredholm Integral Equation of the 2nd kind, with separable kernel. I always get a zero if I use the normal method i was taught for the nonhomogeneous type. I have an example: $$y(x) =…
HenryMath
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Solution of inhomogeneous Fredholm integral equation of the first kind with symmetric rational kernel

Please explain how to solve this inhomogeneous Fredholm integral equation of the first kind: $$f(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{g(y)}{x+y}dy$$
iskar
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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: $$\int_0^a \sin(x\cos\theta)q(\theta)d\theta=0,$$ where $0
SDiv
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