Green’s function for differential operator

Assume we are given $g\in {𝒞}^0([0,T])$ and we want to find $f\in {𝒞}^1([0,T])$ such that

(1)

Expression (1) is an example of initial value problem for an ordinary differential equation. Let usshow, 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. $ℱ({\mathrm{\Omega }}_x)=\{f\in {𝒞}^1([0,T])|f(0)=0\}$
   $𝒢({\mathrm{\Omega }}_y)={𝒞}^0([0,T])$.

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:

$$f(t)={\delta }_t(A^{-1}g)=\underset{0}{\overset{t}{\int }}g(t^{\prime })𝑑t^{\prime }=\underset{0}{\overset{T}{\int }}G(t,t^{\prime })g(t^{\prime })𝑑t^{\prime },$$

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

(3)

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

Its graph is presented in Figure 1.

Figure 1: The Green’s function for the problem (1).