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 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. $ℱ({\mathrm{\Omega }}_x)=\{f\in {𝒞}^1([0,T])|f(0)=0\}$
   $𝒢({\mathrm{\Omega }}_y)={𝒞}^0([0,T])$.

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) 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
Canonical name	GreensFunctionForDifferentialOperator
Date of creation	2013-03-22 14:43:39
Last modified on	2013-03-22 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