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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What is the relation between discontinuities and stiffness

I am trying to understand the effect of introducing discontinuities inside ODE on system stiffness. simply speaking if we have an equation : you can have further look here https://ch.mathworks.com/help/matlab/math/ode-event-location.html [Simple…
Adnan j
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Solution of convolution Integral Equation

I have an integral equation as follows: $f(t)=\int_0^t \sqrt{\tau } g(t-\tau ) \, d\tau$ For a prescribed continuous function $f(t)$, How can I determine an analytical solution for $g(t)$? It should be noted that the derivative of $\sqrt{\tau }$ is…
Mohammad Tahaye Abadi
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reduction of IVP with variable coefficient into Volterra integral equation

Convert the initial value problem $y''+\lambda x^2y=f(x)$ with $y'(0)=y(0)=0$ into a Volterra integral equation. Here if we consider the two linearly independent solution of the above equation as $u(x)$ and $v(x)$, then we know that the reduced…
am_11235...
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Сontour integral

everyone. Can you help with the task?
Nike
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Find $\phi(\log 2)$ for the integral equation $\phi(x)=1-2x-4x^2+\int_0^x[3+6(x-t)-4(x-t)^2]\phi(t)dt$ .

If $\phi$ is the solution of the integral equation $$\phi(x)=1-2x-4x^2+\int_0^x[3+6(x-t)-4(x-t)^2]\phi(t)dt$$ Then the value of $\phi(\log 2)$ is (a). 2 (b). 4 (c). 6 (d). 8 I tried this and I get the solution is $23$ but I am not sure about it…
Gopinath
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To solve integral equation

Let $\phi : [0,\infty) \to [0,\infty)$ be a continuous function such that $\phi (0) =0$. If $(\phi (t))^{2} \leq 2 + \int _{0}^{t} \phi (s) ds \ \ \ \forall t \geq 0$ then what is $\phi ( \sqrt{2})$? I tried using derivative but it didn't help…
Naman
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Integration of Bessel function multiplied with an algebraic and trigonometric functions.

I tried my best to solve the following definite integration. Struggling a lot, ended with no luck. I know similar but a bit little simplified identity. The question is how to change this identity for my case which include two extra constants or…
Ubaid Ullah
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Solutions to $\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$

I have been looking at a problem requires the solution of an equation of the form: $$\int_{x_0}^{x_1} f(x,y)^{-n} \left(\frac{\partial}{\partial y} f(x,y)\right)^m dx = 0$$ for integer values of $m$ and $n$ greater than 1. What non-trivial functions…
Lucas
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Coupled Volterra Integral equation

Calculate $\frac{d}{dx} \int_{0}^{x} \frac{1}{\sqrt (x - t)} dt$ I want to use Leibnitz rule to calculate this value. But I am unable to find it as the second term would be undefined. Anyone can consider $t = x sin ^2 z$ and then after integrating…
Manglu
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The type of the integral equation

I'm solving the integral equation with unknown parameter in the kernel, $g(t)=\int K(t,s,\sigma) f(s) ds$, where $f(s)$ and $\sigma$ have to be defined. Without the parameter, it is the Fredholm problem of the first kind. But I could not find the…
Svyatoslav Korneev
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Eigen values and Eigen functions of the homogeneous integral euation:

let $\lambda_1, \lambda_2$ be the charactieristic numbers and $f_1, f_2 $be the corresponding eigenfunctions for the corresponding eigenfunctions for the homogeneous integral equation $\varphi(x)-\lambda\int_0^1(2xt+4x^2)\varphi(t)dt=0 \; then \\1.…
Priyanka
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Solving x from$\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$?

I found something strange when I try to solve this equatiin of $x$: $\int_0^t \frac{1}{xW(\frac{1}{xf(\tau)})}d\tau=c_0t$, where $t$ and $c_0$ are constants. $f(\tau)$ is a known polynomial function. $W(z)$ is the Lambert W function, i.e.…
Severals-user45972
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Resolvent Kernel of Volterra Equation

I need to find the resolvent kernel of $\phi(x)=f(x)+\int\limits_0^x \sin(x-t)\phi(t)\,dt$. Differentiating both sides I get $$ \phi'(x) = f'(x) + \int_0^x \cos(x-t)\phi(t)\,dt. $$ And now I am stuck. I have also tried the resolvent kernel sum…
Dharmaraj Deka
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Boundary conditions for a ODE from Volterra integral equation

I am modeling a cyclic voltammetry experiment. The mathematical model for it boils down to a Volterra integral equation of the first kind, with an unknown function $\chi(z)$ and known constants $\xi$ and $\theta$: $$\int_0^{\sigma…
svavil
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The Continuous Dependence of Solutions to Volterra Equation

Can any tell me why the solutions of the lineal Volterra integral equation of second kind, have continuous dependence? $y(t)=g(t)+\int_{0}^{t}k(t,s)y(s)ds$ k and g continuous....
David Malagón
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