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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Solving $g(x)=\int_{3}^{x} g(t) dt$

The question is what set of continuous functions solves the problem $g(x)=\int_{3}^{x} g(t) dt$. My answer so far: g(3)=0, g'(x)=g(x)-g(3) therefore g(x)=g'(x)=$ce^x$. Obviously $ce^x=ce^x-ce^3$ doesn't really work out. Where's my mistake? And…
red
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Solve: $\int_{0}^{2\pi}g \psi e^{i n \theta}\,\text{d}\theta = n/(n-i\alpha) \int_{0}^{2\pi}\psi e^{i n \theta}\,\text{d}\theta$

For $\alpha>0$, I want to find a $g(\alpha, \theta)$ such that $$ \int_{0}^{2\pi}g(\alpha, \theta)\psi(\theta)e^{i n \theta}\,\text{d}\theta = \frac{n}{n-i\alpha} \int_{0}^{2\pi}\psi(\theta)e^{i n \theta}\,\text{d}\theta $$ for some $\psi(\theta)$.…
Mr. G
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Solving an Integral Equation

Here is the Question; Solve the integral equation, $$\int_0^tY(u)Y(t-u)du = \frac12 (\sin t-t\cos t)$$ Really not sure how to go about this, took the Laplace transform of the right side…
slow_Math
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Unique Solution to a Set of Bivariate Integral Equation?

Suppose we have the following two integral equations: $$ f_1(x) = \int K(x,t) \varphi(x,t) dt, \quad f_2(x) = \int K(h(x),t) \varphi(x,t) dt, $$ where $f_1,f_2,K,h$ are known functions, $h$ is strictly increasing, and $\varphi$ is unknown. Both $x$…
WaitedLeastSquare
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What is the meaning of the definition below? Taken from a 1909 book on Integral Equations.

Definition: We say that the discontinuities of a function of $(x, y)$ are regularly distributed in $S$ or in $T$ if they all lie on a finite number of curves with continuously turning tangents, no one of which is met by a line parallel to the axis…
user79213
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A problem related to an integral equation

I am stuck on the following problem that is as follows: The integral equation $\quad \varphi(x)-\lambda \displaystyle\int_{-1}^{1}\cos[\pi(x-t)]\varphi(t) dt= g(x)$ has 1.a unique solution for $ \lambda \ne 1$ when $g(x)=x$ 2.no solution for…
learner
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Resolvent Kernel of Fredholm integral equation $R(x,t;\lambda)$ is bijective in $\lambda$

We have the integral Equation: $$y(x)=f(x)+λ\int_{a}^{b} K(x,t)y(t) dt ,x\in [a,b],K\in L^{2}([a,b]\times[a,b]),f\in L^{2}([a,b])$$ Prove that $R(x,t;\lambda)$ is a one-to-one function in $\lambda$. I have tried to solve this, but I got stuck. I…
Θάνος Κ.
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Need help solving an Integral Equation

Need help solving: $$ f(x) = x + \lambda \int_{0}^{1}y(x+y)f(y)dy $$ keeping terms through $\lambda^{2}$, (a) by using the Fredholm method (b) by using the Neumann method
Gerg
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Solution of an integral equation $\phi(x)+\int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $

Solve the following integral equation: $\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $ I need to solve the integral equation above. Can anyone help me please?
sumon
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Fredholm Integral Equation of First Kind with finite limits, 1-D

$$ \int_{-1}^{1} \frac{f(\xi)}{h^2 + (x-\xi)^2} d\xi = 1 $$ for $x \in [1,1]$ and $h < 1$ I went through all the relevant examples in Andrei D. Polyanin Alexander V. Manzhirov's 'Handbook of Integral Equations', but couldn't figure out a way to…
boidy
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How to solve this integral Fredholm equation?

Find the solutions of the integral equation $y(x)=1+3\int_0^1 K(x,t)y(t)dt$, where $K(x,t)=\begin{cases} \cosh x \cdot \sinh t, \text{if}\;\; 0 \leq x \leq t\\ \cosh t\cdot \sinh x, \text{if}\;\; t \leq x \leq 1 \end{cases}$. I know how to solve…
Pinkimani Goswami
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Solve an integral equation in a shorter time

Consider the integral equation $y(x)=x^3+\int _0^x\sin (x-t)y(t) dt$ Find the value of $y(1)$. This is a Volterra I.E.I know that in order to solve it I will have to use the technique of Resolvent Kernel but that is very lengthy and it will…
Learnmore
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General question about solving equations involving a definite integral

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\mathbb{C}$ is the unknown variable, and the…
pshmath0
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Asking solutions for the integral equations

This is from Berkeley Problems in Mathematics, Spring 86. It asks for $\lambda\in \mathbb{R}$, find all solutions of the following two equations: $$\phi(x)=e^{x}+\lambda\int^{x}_{0}e^{x-y}\phi(y)dy;…
Bombyx mori
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Eigenvalues and eigenfunction of integral operator

Suppose we are given an integral operator $ g(x)=f(x)+ \lambda \int_{0}^{\infty}K(x,t)f(t)dt $ with the kernel $ K(x,t)=K(t,x)$. According Hilbert-Schmidt theory then, the function can be obtained as $ f(x)= \sum_{n=0}^{\infty}\frac{c_{n}}{\lambda…
Jose Garcia
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