## 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: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag