Difficulties understanding Green's Functions

[harsh]

Hello people.
I am trying to understand how the Green's functions work, and how to come up with one for a given differential equation. Now, I need to write down the Green's function for 3 different types of differential equations. They are simply, underdamped, critically damped, and overdamped oscillators. I know that the Green's functions are easy to come up with for above mentioned cases if you know the homogenous solution (just replace t with (t - t')), but I am not exactly sure how the heavyside (step-function) gets involved with this. Moreover, I know that the Dirac-delta function is very useful in solving the nasty integrals, but I don't really understand how we are using them. I basically need help in coming up with Green's functions for 3 different kinds of oscillators. Any help on the step-function and the impulse function (as forcing functions), would also be greatly appreciated. Thanks in advance for any help.

- harsh

 

[Dr Transport]

First of all, the three cases you list are for different values of the damping constant for an oscillator differential equation.

To find the correct Greens' function, you need to do two things
1. Solve the differential equation

[tex] \frac{d^{2}}{dx^{2}} G(x) - \gamma\frac{d}{dx} G(x) - k{^2} G(x) = -\delta{x} [/tex]

subject to the bondary conditions that you are given. The other equation relates continuity of the derivative of the Greens' function at [tex] x = 0 [/tex]. Now for the case of the undamped harmonic oscillator, this is

[tex] \frac{d}{dx}G(0^{-}) - \frac{d}{dx}G(0^{+}) = -1 [/tex] for the damped oscillator will be different, but you get the idea. Then you can get the Greens' function, you can not just find the solution and put [tex] t - t' [/tex] in and call it quits. The solution is more difficult than that. If you want time dependence, you follow what I have quickly outlined and work in the time the difference is that you need two [tex] \delta [/tex] funtions, one in space and one in time.

Have fun and play around with it and I'll do the same.

dt

 

[GSXR750]

Hello Harsh, understanding Green's functions can definitely be a difficult task. It's great that you are seeking help in understanding them better. Let me try to break down the concept and hopefully it will become clearer for you.

Firstly, the Green's function is a mathematical tool that helps us solve differential equations. It's essentially a way to represent the response of a system to a given forcing function. In other words, it tells us how the system will behave when a particular force is applied to it.

Now, to come up with the Green's function for a given differential equation, we need to first understand the homogenous solution. This is the solution to the differential equation when the forcing function is equal to zero. Once we have the homogenous solution, we can use it to come up with the Green's function by replacing t with (t-t'). This is because the Green's function represents the response of the system at time t' when a force is applied at time t.

The heavyside function and the Dirac-delta function come into play when we have a forcing function in the differential equation. The heavyside function, also known as the step-function, is used to represent a sudden change in the forcing function. For example, if the forcing function suddenly changes from zero to a constant value, we can use the heavyside function to represent this change.

The Dirac-delta function, on the other hand, is used to represent an impulsive force. This means that the force is applied for an infinitesimally short amount of time. In this case, the Dirac-delta function helps us to solve the integral involved in finding the Green's function.

To come up with the Green's function for the three types of oscillators you mentioned, you will need to use the homogenous solution and apply the appropriate forcing function (step-function or Dirac-delta function) to it. I would suggest looking for some examples or tutorials online to get a better understanding of the process.

I hope this helps and good luck with your studies!