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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

1008 questions
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A very odd resolution to an integral equation

Here is something I've found on the internet $$\begin{aligned} f-\int f&=1\\ \left(1-\int\right)f&=1\\ f&=\left(\frac1{1-\int}\right)1\\ &=\left(1+\int+\int\int+\dots\right)1\\ &=1+\int1+\int\int1+\dots\\ &= 1+x+\frac{x^2}2+\dots\\ &=…
Alma Arjuna
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2 answers

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
Victor
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Do you lose solutions when differentiating to solve an integral equation?

The question is more general but here's the problem that motivated it: I want to find all solutions to the integral equation $$f(x) + \int_0^x (x-y)f(y)dy = x^3.$$ Differentiating twice with respect to x this yield the second order differential…
Francisco José Letterio
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33
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Spectrum of Indefinite Integral Operators

I've considered the following spectral problems for a long time, I did not know how to tackle them. Maybe they needs some skills with inequalities. For the first, suppose $T:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $$Tf(x)=\int_{0}^{x} \!…
yaoxiao
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Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of what's the reason for caring whether an operator is…
csss
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Find $f$ such that $f \star f(x) = \frac{1}{1-x}$.

I'm looking for a measurable function $f$ defined on $]0,1[$ such that : $$f \star f(x) = \int_{0}^1 f(x-y) f(y) \ \mathrm{d}y = \frac{1}{1-x}$$ for (almost) any $x \in ]0,1[$. Is it possible to find or construct such a function f ? Eventually, we…
Velobos
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Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) \, dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) \,dt.$$ The sources I've checked* seem to say that the process is fairly involved. Has anything been published on this…
Antonio Vargas
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3 answers

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such functions $f$. So far, I've managed to show that…
Sntn
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2 answers

if $f(x)-a\int_x^{x+1}f(t)~dt$ is constant, then $f(x)$ is constant or $f(x)=Ae^{bx}+B$

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant, show that $f(x)$ is constant; or $$f(x)=Ae^{bx}+B$$ where $A\ge 0,|B|\le A$ and $A,B$ are constant, and…
math110
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Why solving a differentiated integral equation might eventually lead to erroneous solutions of the original problem?

Consider the following integral equation araising in a mathematical-physical problem: $$ \int_0^r f(t) \arcsin \left( \frac{t}{r} \right) \, \mathrm{d}t + \frac{\pi}{2} \int_r^R f(t) \, \mathrm{d} t = r \, \qquad (0
Eulerian
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11
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2 answers

Numerical solution of an integral equation

I have problems with a solution of an integral equation in MATLAB: all conditions are double-checked, but the answer is incorrect. Let me state the equation: $$ x(s) = g(s)+\int\limits_0^1x(t)f(t-s)\,dt\quad(1) $$ where $$ f(t) =…
SBF
  • 36,568
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4 answers

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 - likely difficult to solve.
Ryan
  • 850
11
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1 answer

Generalizing Archimedes' "The Quadrature of the Parabola"

In the third century BC Archimedes discovered that The area enclosed by a parabola and a line (left figure) is 4/3 that of a related inscribed triangle (right figure). Consequentially, the area enclosed by a parabola and a line is 2/3 that of a…
user1337
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10
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0 answers

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means convolution as in $(f*g)(x)=\int\limits_{\mathbb…
Tobias Kienzler
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10
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1 answer

How do I solve this integral equation?

Case 1: We will look at an easier problem first. Let $|\alpha|, |\beta| \leq \alpha_c, \alpha_c \leq \pi$. I want to solve for $\rho(\beta)$ in the following equation, where $P$ denotes the principal value of the…
user307856
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