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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Is there a solution to the integral equation $ H(y)=\int_0^\infty G(\frac{y-\phi_2(v)}{\phi_1(v)}) \exp(-v) ~\mathrm{d}v $?

Consider the following equation $$ H(y) = \int_{0}^{\infty} G\left(\frac{y-\phi_2(v)}{\phi_1(v)} \right) \exp(-v) ~\mathrm{d}v $$ where $\phi_i$ are well-behaved differentiable functions on the region of integration and G non-decreasing on the real…
Archimedes
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Solving integral equation $a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$

I've got this nasty-looking integral equation involving taking two minimums: $$a+c\min(b,x)=\int_{-\infty}^{\infty}f(x-t)\left(a+d\delta(t-(x-b))+c\min(b,t)\right)dt$$ where $\delta(\cdot)$ is the Dirac delta function and $a$, $b$, $c$, and $d$ are…
M.B.M.
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How does one solve this integral equation $1+ax=\int_{-\infty}^xf(x-t)dt$

I've run into having to solve this equation for $f(x)$: $$1+ax=\int_{-\infty}^xf(x-t)dt$$ Unfortunately, I am not familiar with solving integral equations. Can anyone help? Is is even soluble? Edit: Fixed a typo in the upper limit in the integral.
M.B.M.
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Proof of the uniqueness of the solution to an integral equation

In the fluid mechanics of pipe flow, it is sometimes stated that the velocity profile $u(r)$ which corresponds to a kinetic energy coefficient of 1 is always uniform, so $u(r) =$ some constant. Munson's fluids textbook states this (5th ed., p.…
AtomicAcorn
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How to solve the integral equation?

How to solve the integral equation $$ \int_{-20}^{x} \left| \left| \left| \left| \left| \left| \left| \left| t \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 \right| -1 \right| \,{\rm d}t={\frac {4027}{2}}?$$
Markiyan Hirnyk
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A little doubt with integral equation

I have the next equation: $$\int_{0}^{t}h(\tau)e^{-(t-\tau)}\mathrm{d}\tau=10e^{-t}\cos(4t) \tag{1}$$ Derivating both sides, I get: $$h(t)e^{-(t-t)}=h(t)=10[(-1)e^{-t}\cos(4t)+e^{-t}(-4)\sin(4t)] \tag{2}$$ However, if first simplify (1) and then…
Brallan Aguilar
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Integral equation solution in power series

Given the integral equation $$ g(x)= \int_{-\infty}^{\infty}K(x-y)f(y) \, dy$$ for a known function $ g(x) $ and kernel $ K(x)$. Of course I know this is a Wieener-Hopf integral equation but I would like to know another formula, if possible in power…
Jose Garcia
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Integral equation and constant rules

I have an integral equation of the form: $$f(x)=3+4\int_a^bf(t)~dt$$ How can I put the constants inside the integral to get something where I can apply the fundamental theorem of calculus?
solsol
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Clarification of intergal equations.

I feel like asking a question that show how long I have to go. Clarification. In differential equations, I start with a rate of change and find the indefinite integral to find the function. In integral equations I start with an integral of some…
Chris
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Finding a nonzero continuous function that satisfies this integral equation, but not unique?

If $h(x)$ is such that it satisfies $$ h(x) = \lambda \int_a^b K(x,y) h(y)\, dy $$ Then for $\phi(x)$ a solution of $$ \phi(x) = f(x) + \lambda \int_a^b K(x,y)\phi(y)\, dy $$ it is true that $\phi + h$ is also a solution. But I am confused as to…
Moderat
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Non-local boundary condition and integral equations

I'm solving initial value problem with non-local boundary condition $u(0,t)=\int_{0}^{l}\beta(s)u(s,t)ds = \gamma(t)$. I have already found function u for to cases $xt$. But I have some troubles with finding general solution to function…
Kyle Crane
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Inverse Integral problem: substituting infinite upper bound with constant

In the handbook of integral equations, there are provided solutions for two integrals involving functions $f(x)$ and $y(t)$: Equation 1: Given $$f(x) = \int_{a}^{x} \frac{y(t) \ dt}{\sqrt{x^2 - t^2}},$$ the solution is $$y(t) = \frac{2}{\pi}…
David khoder
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Solution of the Volterra integral equation of the 2nd kind

Please tell me where I made a mistake? Or maybe I used the wrong method to solve it? Link to my attempted solution: https://ru.overleaf.com/read/xcxsthdjpnmx#17beab For the successive approximation method, I did the calculations in Wolfram…
Mark
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Fredholm integral equation of the second kind with constant kernel

I'm trying to read Kress' Linear integral equations, and I'm stuck at the first example. There must be something obvious I'm missing, and to that end, should I read something before this text? $f(x)=\phi(x)-\int_a^b K(x,y)\phi(y)dy,\quad…
LiquidMikerrs
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eigenvalues and eigenfunctions for a nondegenerate $L^2$ kernel

Consider the symmetric $L^2$ kernel $K(x,t)=\log(1-\cos(x-t))$ for $0\leq x,t\leq 2\pi$ . Find the eigenvalues and corresponding eigenfunctions of $K(x,t)$ By general procedure we consider a Fredholm equation…
am_11235...
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