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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Fredholm integral equation --- almost of the first kind

I have an equation on $f(x)$ which has the form $$\int_a^b K(x, y)f(y)dy = g(x) + f'(x=b)\times h(x).$$ The value of $f'(x=b)$ (first derivative of $f$, evaluated at $b$) is finite and given by a boundary condition on an antiderivative of $f$ (that…
Aubergine
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What is the solution of this Fredholm integral equation?

Which of the following functions is the solution of the Fredholm integral equation: $$u(x)+\frac{1}{2}\int_0^1e^{x-t}u(t)dt=2xe^x$$ (A) $u(x)=e^x\big(2x-\frac{2}{3}\big)$ (B) $u(x)=e^x(x^2-x)$ (C) $u(x)=e^x(x-3)$ (D) $u(x)=e^x(3x^2-3x-1)$ My…
F. A. Mala
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How can I solve for $f(x)$ and $g(x)$?

How can I solve for $f(x)$ and $g(x)$: $$e^{-x^2}=\int_{-\infty}^\infty g(t)e^{-f(t)(x-t)^2}\text{d}t\;\;\;\;?$$ Not necessarily an elementary solution, a numerical one would also suffice. I am trying to express the Gaussian as a sum of infinitely…
artmyb
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Have you ever seen this kind of "Logarithmic" Volterra integral equations?

I am currently trying to solve a set of integral equations of the form \begin{equation} \ln \int^1_0 f(s, t) \ \text{d}s = b + \int_0^1 w(t, s) \ \ln f(t, s) \ \text{d}s, \end{equation} where $b \in \mathbb{R}$, for $f$ in the set of 1-periodic…
NoFishLikeIan
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Separable equation with $u$-substitution

I'm trying to solve this separable equation. Answer: x^3-y^3=c (Apparantly my calculus book only have answer for every second assignment) $$\frac{dy}{dx}=\frac{3y-1}{x}\\ \frac{dy}{dx}=\frac{3y-1}{x} \Rightarrow…
user9060784
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Integral equation problem

Consider the integral equation $\phi(x)-\frac{e}{2} \int_{-1}^{1} x e^{t} \phi(t) d t=f(x) .$ Then there exists a continuous function $f:[-1,1] \rightarrow(0, \infty)$ for which solution exists there exists a continuous function $ f:[-1,1]…
ਮੈਥ
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Solution of the following integral Equation $\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$

Consider that the following equation is solvable then analyze with respect to $\lambda$ $$\varphi(x) - \lambda\int\limits_{-1}^1 x e^t\varphi(t) \: dt=x$$ Can someone tell me how can I solve it ?
Hitman
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solve the following integration equation (integral equation/variation of calculus)

I'm still rusty on integral equations. I need to solve the following $$ f(x) = e^{-{\left | x \right |}} + \lambda \int_{-\infty}^{\infty} e^{-{\left | x - y \right |}} f(y)dy $$ where $f(x)$ is to remain finite for $x \rightarrow \pm \infty $ I'm…
Gerg
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How to find Resolvent Kernel?

Find the resolvent kernel associated with the kernel $K(x,t) = |x-t|$ in the interval $(0,1)$ ? I tried solving it but I am stuck at the part where we have to split the integral.
Nihir Agarwal
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How to solve the following integral equation?

$g(s) = \frac{\lambda A}{1-e^{-\lambda T}}s$ for $s\in [0, T]$. $g(s) = \int_{0}^{T} \lambda e^{-\lambda t} g(s-t) dt + A$. for $s \ge T$. $T, A, \lambda$ are constant. I want to get the closed-form of $g$ for $s > T$.
ftor
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Resolvent Kernel for $\sinh(x-t)$

Just starting my study of Integral Equations on Rahman's book, I have studied how to find the resolvent kernel $R(x,t;\lambda)$ of the integral equation since its kernel $K(x,t)$. The expression for the resolvent kernel is …
mini1998
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Help with unsolved indefinite integral ! $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$

I really need to solve this indefinite integral: $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ It seems much more complicated than it looks. I have found a integral table with a integral resembles my integral, and the…
Romildo Junior Carla Beatriz
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solving the Volterra integral equation $Y(t)=F(t)+\int_0^t (t^2-x^2)Y(x)dx$

Solve the inhomogeneous Volterra integral equation and hence find its resolvent kernel : $$Y(t)=F(t)+\int_0^t (t^2-x^2)Y(x)dx$$ We know that if the kernel $K(t,x)$ is of the form $K(t-x)$, then it can be solved easily by using Laplace transform.…
am_11235...
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an integral equation with modulus kernel

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 $u$ also satisfies $(a)$ $\frac{d^2u}{dx^2}+\lambda u=0$ $(b)$ $\frac{d^2u}{dx^2}-\lambda u=0$ $(c)$…
am_11235...
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Volterra Equations of the First Kind

Recently, I managed to find a digital copy of "Handbook of Integral Equations" by Andrei D. Polyanin and Alexander V. Manzhirov (1998 edition). Linear integral equation of the first kind have the form: $$ \int_a^x K(x,t) y(t) dt = f(x) $$ where…
PackSciences
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