Obtaining the Green Function for Euler Beam with Specific Boundary Conditions

[muzialis]

Hello there,

I would like to obtain the Green function for the operator F, F [u(x)] = u '''', and the boundary conditions u(0) = u'(0) = u (1) = u' (1) = 0.


I am looking for a function G ( x, s ) such that G'''' (x,s) = delta (x-s) (the apecis referring to differentiation w.r.t. x, and delta referring to Dirac's function).

All I managed to do is the following:

1) solve the ode for x other than s, u = a +b*x + c * x^2 +d * x^3
2) considering the first two B.C. for x < s, and the other two for x > s I get some conditions for the constants in terms of linear equations
3) an additional equation could be written to impose continuity at s

But then? What other conditions are avaialbe to determine the missing constants?

Thanks as usual

Muzialis

 

[Mute]

Hello there,

I would like to obtain the Green function for the operator F, F [u(x)] = u '''', and the boundary conditions u(0) = u'(0) = u (1) = u' (1) = 0.I am looking for a function G ( x, s ) such that G'''' (x,s) = delta (x-s) (the apecis referring to differentiation w.r.t. x, and delta referring to Dirac's function).

All I managed to do is the following:

1) solve the ode for x other than s, u = a +b*x + c * x^2 +d * x^3
2) considering the first two B.C. for x < s, and the other two for x > s I get some conditions for the constants in terms of linear equations
3) an additional equation could be written to impose continuity at s

But then? What other conditions are avaialbe to determine the missing constants?

Thanks as usual

Muzialis

You can generate another condition by integrating your differential equation for G(x,s) over a small window around x = s. This will give you a condition that relates the third derivatives of G(x,s) on each side of the boundary x = s.

If I recall correctly, the lower order derivatives should still be continuous; only the highest derivative is discontinuous (as you show by integrating the differential equation to get the condition mentioned above). This will provide the last two equations you need to solve for all of the coefficients.

Also, just to make sure you are aware, you have 8 constants: the a,b,c and d are different on each side of x = s.

 

[muzialis]

Mute,


I was aware of the 2 * 4 = 8 constants (thanks anyhow a lot). To recap, of the initial 8 constants only 4 need determining, as the others are taken care of by the boundary conditions for x > s and x < s.
Continuity at x = s provides another equation, and we arrive at your suggestion.
Assuming I have understood, the integration of the DE between say (s-c) and (s+c) yields that the jump in the third derivative of G equals 1.
This was extremely welcome but I still need two more conditions, assuming I have understood your help.

I am trying to work if the solution to my problem is to find similar jump conditions for the second and first derivatives.

Many thanks

 

[Mute]

Mute,I was aware of the 2 * 4 = 8 constants (thanks anyhow a lot). To recap, of the initial 8 constants only 4 need determining, as the others are taken care of by the boundary conditions for x > s and x < s.
Continuity at x = s provides another equation, and we arrive at your suggestion.
Assuming I have understood, the integration of the DE between say (s-c) and (s+c) yields that the jump in the third derivative of G equals 1.
This was extremely welcome but I still need two more conditions, assuming I have understood your help.

I am trying to work if the solution to my problem is to find similar jump conditions for the second and first derivatives.

Many thanks

Perhaps you were writing a reply while I was editing my previous post, but I added that the other derivatives should be continuous across x=s, giving you the last two equations that you need.

Perhaps I should give some hint as to how you can convince yourself that should be the case: the integral of a delta function is a step function:

$$\int_{-\infty}^x dx'~\delta(x'-s) = \Theta(x-s).$$

This is a discontinuous function. If you integrate the step function, is the result continuous or discontinuous? Why? Along this line of thought you should be able to figure out that this implies that only the highest derivative of your Green's function has a discontinuity.

 

[muzialis]

Thta was indeed the case Mute, thank you very much your first hint was sufficient, I have got my solution, much appreciated!