Green’s function for differential operator
Assume we are given $g\in {\mathcal{C}}^{0}([0,T])$ and we want to find $f\in {\mathcal{C}}^{1}([0,T])$ such that
$$\{\begin{array}{ccc}\hfill {f}^{\prime}(t)& \hfill =\hfill & g(t)\hfill \\ \hfill f(0)& \hfill =\hfill & 0\hfill \end{array}$$  (1) 
Expression (1) is an example of initial value problem^{} for an ordinary differential equation^{}. Let us show, that (1) can be put into the framework of the definition for Green’s function.

1.
${\mathrm{\Omega}}_{x}={\mathrm{\Omega}}_{y}=[0,T]$.

2.
$\mathcal{F}({\mathrm{\Omega}}_{x})=\{f\in {\mathcal{C}}^{1}([0,T])f(0)=0\}$
$\mathcal{G}({\mathrm{\Omega}}_{y})={\mathcal{C}}^{0}([0,T])$. 
3.
$Af={f}^{\prime}$
Thus (1) can be written as an operator equation
$$Af=g.$$  (2) 
To find the Green’s function for (2) we proceed as follows:
$$f(t)={\delta}_{t}({A}^{1}g)=\underset{0}{\overset{t}{\int}}g({t}^{\prime})\mathit{d}{t}^{\prime}=\underset{0}{\overset{T}{\int}}G(t,{t}^{\prime})g({t}^{\prime})\mathit{d}{t}^{\prime},$$ 
where $G(t,{t}^{\prime})$ has the following form:
$$  (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.
Title  Green’s function for differential operator^{} 

Canonical name  GreensFunctionForDifferentialOperator 
Date of creation  20130322 14:43:39 
Last modified on  20130322 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 