## You are here

HomeGreen's function for differential operator

## Primary tabs

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

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.

## Mathematics Subject Classification

34A99*no label found*34A30

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: numerical method (implicit) for nonlinear pde by roozbe

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella