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Green's function for differential operator

(Example)

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

$\displaystyle \left\{ \begin{array}{rcl} f'(t) & = & g(t) \\ f(0) & = & 0 \end{array} \right.$

(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.

$\Omega_x=\Omega_y=[0,T]$ .
$\EuScript{F}(\Omega_x)=\{ f\in\mathcal{C}^1([0,T])\,\vert\,f(0)=0 \}$
$\EuScript{G}(\Omega_y)=\mathcal{C}^0([0,T])$ .

Thus (1) can be written as an operator equation

$\displaystyle Af=g.$

(2)

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

$\displaystyle f(t)=\delta_t(A^{-1}g)=\int\limits_0^t g(t')\,dt'=\int\limits_0^T G(t,t')g(t')\,dt',$

where

has the following form:

$\displaystyle G(t,t')=\left\{ \begin{array}{rl} 1, & 0\leq t \leq t'\\ 0, & t'< t \leq T \end{array} \right.$

(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.

**Figure:** The Green's function for the problem (1).
$\includegraphics[scale=.5]{GrFn.eps}$

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AMS MSC:	34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)
	34A99 (Ordinary differential equations :: General theory :: Miscellaneous)

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