(adsbygoogle = window.adsbygoogle || []).push({}); Questions about Green's functions for ODEs, jump conditions

I'm having a hard time understanding Green's functions which have been introduced quite early on in the course, and which I think hasn't been well motivated. I can't find any other resource which explains this at this level (have only covered most of 1st order linear DEs before this) so the only thing I have to go by are the lecture slides.

The (one-sided) Green's function is been introduced in the notes as the solution to the DE:

[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex]\tau > t_0[/itex]

the solution being

[itex]y(t)=e^{-a(t-\tau)}[/itex] for [itex] t_0 < \tau < t[/itex]

[itex]y(t)=0[/itex] for [itex] t_0 \leq t < \tau[/itex]

and this is referred to as the one-sided Green's function.

Two methods for calculating the Green's function are described. First is to obtain the integral solution to the initial value problem

[itex]y'+ay=g(t), y(t_0)=0[/itex] and just 'read off' the Green's function...but this apparently is not instructive since it gives no insights into what it means.

Second method is to calculate it for

[itex]y'+ay=\delta(t-\tau)[/itex] with [itex]y(t_0)=0[/itex] and [itex] \tau>t_0[/itex] in the two domains [itex]t_0 \leq t < \tau[/itex] and [itex]t > \tau[/itex] with thejump condition[itex][y(t)]^{t=t^+}_{t=t^-}=1[/itex] which supposedly will give the Green's function.

---

Firstly, what is the point of the Green's function?

What exactly is a 'jump condition' and how do I work with it?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Green's functions for ODEs, jump conditions

Loading...

Similar Threads - Green's functions ODEs | Date |
---|---|

A Green's Function ODE BVP | Jan 3, 2018 |

Solve second order ode with Green function | Jul 6, 2013 |

Solve second order ode with Green's functions | Jun 23, 2013 |

Green's function for third order ODE | Mar 2, 2012 |

Need to find Green Function to solve ODE | Oct 8, 2006 |

**Physics Forums - The Fusion of Science and Community**