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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non-trivial solution of an integral equation

The values of $\lambda$ for which the following equation has a non-trivial solution $\phi(x)=\lambda\int_0^{\pi} K(x,t)\phi(t)dt$ where $0\leq x\leq \pi$ and $K(x,t)= \begin{cases} \sin x\cos t & 0\leq x\leq t \\ \cos x\sin t & t\leq…
am_11235...
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Integral equation equal to a constant from aerodynamics application

I am stuck on part of my aerodynamics textbook which involves an integral equation. I would like some help understanding how to solve the equation. Problem Statement Find $\gamma(\theta)$ such that $\gamma(\pi)=0$…
EssentialAnonymity
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Resolvent kernel of Fredholm integral equation.

For the linear integral equation $ y(x)=x+\int_{0}^{1/2} y(t) dt$. Find Resolvent kernel $R(x,t,1)$. I tried to find resolvent kernel of Volterra integral equation by taking kernel as 1.Then I got $R(x,t,1)=e^{(x-t)}$.But I don't know…
ASHWINI SANKHE
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Solve $f(x) = \lambda\int_0^\pi(x+y)f(y)\mathrm dy$

Solve the integral equation $$ f(x) = \lambda\int_0^\pi(x+y)f(y)\mathrm dy $$ with $f(x)$ integrable in [0,π]. From my comment below: ...I think that a non trivial solution is $f(x)=Ax+B$ with $A$ and $B$ constant, but i can't check it. (see the…
Alex
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Solving an equation with an integral

I need to solve the following equation for $v(x)$: $$\int_0^tv(x)(x+1)dx=f(t)$$ I am given the function $f(t)$. I've done this so far: If we derive both sides by $t$, we get $v(t)(t+1)=f'(t)$ and $\bar{v}(t)=\frac{f'(t)}{t+1}$. The problem is that I…
Hasanhasan Hasan
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Solving A Volterra Integral Equation

So I am working on solving the following Volterra integral equation... $$f(t)=te^t+\int_0^t\tau f(t-\tau)d\tau$$ I take the Laplace transform of it and then solve for $F(s)$ and arrive at... $$F(s)=\frac{s^2}{(s-1)^2(s^2-1)}$$ Now I obviously need…
Jake Young
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Solution to Integral Equation (Fredholm Integral Equation)

I have the following equation and want to find $f(x)$ $f(x)=x $ + $\int_{0}^{1} (xy^2 + yx^2)$ $f(y)dy$ When i tried to get a solution from wolfram alpha, it gave me an answer but says it is solving a Fredholm Integral Equation. I am a high schooler…
Abhijith S. Raj
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Help solving this integral equation

I've got the following relation: For any $t_m \in (0, t_f)$, $$I(k | t_0,t_f,x_0) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(k_m | t_0, t_m, x_0) I(k-k_m|t_m,t_f,x_m) dk_m dx_m $$ I want to solve for $I$. I have reason to believe that a…
GMB
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Which of the following satisfies the equation $\phi(x)=f(x)+\int_{0}^{x}\sin(x-t)\phi(t)dt$

Let $\phi$ satisfy $$\phi(x)=f(x)+\int_{0}^{x}\sin(x-t)\phi(t)dt.$$ Then $\phi$ is given by…
Kushal Bhuyan
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How to solve this integral equation: $\log (x-1)=\int_1^{\infty } \log \left(1-f(t)^{-x}\right) \, dt$?

Find f(t) such that: $\log (x-1)=\int_1^{\infty } \log \left(1-f(t)^{-x}\right) \, dt$ I am not familiar with solving integral equations. I was thinking of expressing logarithm inside integral with series, then moving integral inside this sum, but…
azerbajdzan
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In what class do you learn how to solve integral equations?

I was interested in knowing in what class does one learn Fredholm theory (integral equations). Any textbook recommendations for learning how to solve integral equations?
user401426
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Fredholm equation with symmetric kernel

I have the following equation : $$ \phi(x) = \frac{x}{2} + \frac{\pi^2}{4}\int_{0}^{1}K(x,t)\phi(t)dt $$ where $$ K(x,t)= \left\{ \begin{array}{ll} \frac{x(2-t)}{2} & \mbox{if } 0 \leq x \leq t \\ \frac{t(2-x)}{2} & \mbox{if } t \leq x…
Michael
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Convert IVP to an equivalent Volterra integral equation

Convert the following initial value problem to an equivalent Volterra integral equation: $ \begin{cases} u'' -u' \sin x + \Bbb e ^x u= x \\ u(0)=1\\ u'(0)=-1\\ \end{cases} $ I have solved it and I have got the solution $$ x-…
Sara
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Solve integral equation that involves hyperbolic cosine

I'm trying to find a weight function w(x) that makes this integral 0 $\int_0^1 w(x) \cosh((\alpha_n+i\omega_n)x) \cosh((\alpha_n-i\omega_n)x)=0$, where where $\omega_n=(2n+1)\frac{\pi}{2}$ and…
mdornfe1
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Transforming the integral equation $u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b$ into its equivalent differential equation

Let $u \in C^2[0, 1]$ satisfy for some $ \lambda \neq 0$ and $a \neq 0,$ $$u(x) + \frac{\lambda}{2}\int_{0}^{1}|x - s|u(s)ds = ax + b.$$ Then show that u also satisfies $\frac{d^2u}{dx^2} + \lambda u = 0$ I differentiate the integral equation twice…
user242028
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