# Integral Equation

#### matematikawan

This forum discuss Differential Equations - ODE, PDE, DDE, SDE, DAE
I know the abbreviation ODE and PDE. But what are DDE and DAE ?
SDE must stand for system of DE.

Do you also discuss Integral Equation in this forum?

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#### D H

Staff Emeritus
DDE = Delay Differential Equation, for example

$$\frac {dx(t)}{dt} = f(x(t),x(t-\Delta t),x(t-2\Delta t), ...)$$

where $\Delta t$ is some finite (non-infinitesimal) sampling interval.

SDE = Stochastic Differential Equation, for example

$$\frac {dx(t)}{dt} = f(x(t)) + g(x(t))\dot(w(t))$$

where $w(t)$ is a white noise process.

DAE = Differential-Algebraic Equation, for example

$$\frac 1 2 \frac{d\boldsymbol{x}}{dt}\cdot\frac{d\boldsymbol{x}}{dt} - \frac{\mu}{(\boldsymbol{x}\cdot\boldsymbol{x})^{1/2}} - E = 0$$

or more generally,

$$f(t,x,\dot x,\ddot x,...) = 0$$

This is as good a spot as any to discuss integral equations in general.

#### matematikawan

Oop! I got it wrong for SDE. That's why I have to ask the question.

Thanks D H for the explaination.

Lo! Never been exposed to DDE, SDE, DAE before. What so interesting about those equations?

I have just started looking into integral equations and haven't really appreciate them.
As a generalization, I think we can alway eliminate any integral sign in an integral equation by differentiating the equation. My question is why bother to solve the integral equation when we can solve the corresponding differential equation? We have so many methods for solving DE.
Is it because it is much easier to solve integral equation then the corresponding DE?

#### D H

Staff Emeritus
Never been exposed to DDE, SDE, DAE before. What so interesting about those equations?
DDE and SDE are very important in the signal processing world and in physics. Diffusion, brownian motion, quantum physics, getting a spacecraft to the moon, GPS, ... The list of applications is endless.

As a generalization, I think we can alway eliminate any integral sign in an integral equation by differentiating the equation.
Sometimes, but not always. Maxwell's equations can be expressed in differential or integral form. On the other hand, if you differentiate a Fredholm or Volterra integral equation, you get another integral equation.

#### matematikawan

Sometimes, but not always. Maxwell's equations can be expressed in differential or integral form. On the other hand, if you differentiate a Fredholm or Volterra integral equation, you get another integral equation.
Sorry that I'm really new in this integral equation business. If Maxwell's equations can be expressed in both differential or integral form, then people usually solve which equations. Differential Maxwell equation or integral Maxwell equation ? (If my question make sense.)

A Fredholm or Volterra linear integral equation look something like this

$$u(x) = f(x) + \lambda \int{K(x,t) u(t)dt}$$
(the integration limits missing. How do you insert integration limit in latex?)

If the kernel is a polynomial in x then we can always convert the above integral equation into a differential equation. Am I right?
How often in applications that we met integral equation whose kernel is not a polynomial in x?

ps. Similar Threads for: Integral Equation below does not really discuss integral equation. Look more like integration problem!