integral equation

An integral equationMathworldPlanetmath involves an unknown function under the .  Most common of them is a linear integral equation

α(t)y(t)+abk(t,x)y(x)𝑑x=f(t), (1)

where α,k,f are given functions.  The function  ty(t)  is to be solved.

Any linear integral equation is equivalent ( to a linear differential equation; e.g. the equation  y(t)+0t(2t-2x-3)y(x)𝑑x=1+t-4sint  to the equation  y′′(t)-3y(t)+2y(t)=4sint  with the initial conditionsMathworldPlanetmathy(0)=1  and  y(0)=0.

The equation (1) is of

  • 1st kind if  α(t)0,

  • 2nd kind if α(t) is a nonzero constant,

  • 3rd kind else.

If both limits ( of integration in (1) are constant, (1) is a Fredholm equation, if one limit is variable, one has a Volterra equation.  In the case that  f(t)0,  the linear integral equation is .

Example.  Solve the Volterra equation  y(t)+0t(t-x)y(x)𝑑x=1  by using Laplace transformMathworldPlanetmath.

Using the convolution (, the equation may be written  y(t)+t*y(t)=1.  Applying to this the Laplace transform, one obtains  Y(s)+1s2Y(s)=1s,  whence  Y(s)=ss2+1.  This corresponds the function  y(t)=cost,  which is the solution. on some integral equations in EqWorld.

Title integral equation
Canonical name IntegralEquation
Date of creation 2013-03-22 18:03:57
Last modified on 2013-03-22 18:03:57
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 45D05
Classification msc 45A05
Related topic Equation
Defines linear integral equation
Defines Volterra equation