Green’s function for differential operator

Assume we are given g𝒞0([0,T]) and we want to find f𝒞1([0,T]) such that

{f(t)=g(t)f(0)=0 (1)

Expression (1) is an example of initial value problemMathworldPlanetmathPlanetmath for an ordinary differential equationMathworldPlanetmath. Let us show, that (1) can be put into the framework of the definition for Green’s function.

  1. 1.


  2. 2.


  3. 3.


Thus (1) can be written as an operator equation

Af=g. (2)

To find the Green’s function for (2) we proceed as follows:


where G(t,t) has the following form:

G(t,t)={1,0tt0,t<tT (3)

Thus, function (3) is the Green’s function for the operator equation (2) and then for the problem (1).

Its graph is presented in Figure 1.

Figure 1: The Green’s function for the problem (1).
Title Green’s function for differential operatorMathworldPlanetmath
Canonical name GreensFunctionForDifferentialOperator
Date of creation 2013-03-22 14:43:39
Last modified on 2013-03-22 14:43:39
Owner mathforever (4370)
Last modified by mathforever (4370)
Numerical id 7
Author mathforever (4370)
Entry type Example
Classification msc 34A99
Classification msc 34A30