Green's function for differential operator

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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.

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

$Af=f^{{\prime}}$

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}}<t\leq T\end{array}\right.

(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.

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