# 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

 $\left\{\begin{array}[]{rcl}f^{\prime}(t)&=&g(t)\\ f(0)&=&0\end{array}\right.$ (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. 1.

$\Omega_{x}=\Omega_{y}=[0,T]$.

2. 2.

$\mathcal{F}(\Omega_{x})=\{f\in\mathcal{C}^{1}([0,T])\,|\,f(0)=0\}$
$\mathcal{G}(\Omega_{y})=\mathcal{C}^{0}([0,T])$.

3. 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:

 $f(t)=\delta_{t}(A^{-1}g)=\int\limits_{0}^{t}g(t^{\prime})\,dt^{\prime}=\int% \limits_{0}^{T}G(t,t^{\prime})g(t^{\prime})\,dt^{\prime},$

where $G(t,t^{\prime})$ has the following form:

 $G(t,t^{\prime})=\left\{\begin{array}[]{rl}1,&0\leq t\leq t^{\prime}\\ 0,&t^{\prime} (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 operator  GreensFunctionForDifferentialOperator 2013-03-22 14:43:39 2013-03-22 14:43:39 mathforever (4370) mathforever (4370) 7 mathforever (4370) Example msc 34A99 msc 34A30