A somewhat conceptual question about Green's functions

In summary, the particular solution to a diff eq. that has homogeneous boundary conditions can be turned into a general solution if you take care of the inhomogeneity with a Green's function.
  • #1
BiGyElLoWhAt
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I just did a problem for a final that required us to use a green's function to solve a diff eq. y'' +y/4 = sin(2x)

I went through and solved it and got a really nasty looking thing, but I checked it in wolfram and it works out. Now, my question is this:

After I got the solution from my greens functions, I went through and tried to add in what would be the homogeneous solution, Asin(x/2) + Bcos(x/2) and apply the boundary conditions again to solve for A and B. Both came out to be zero. Is this always the case? Is my integrand of G*f the general solution and not just the particular solution? Was this just a coincidence? I have a few more problems I need to do, most of which are green's functions, and this would be handy to know. If it is the general solution, does it have to do with the fact that I used the homogeneous solution to obtain my Green's solution? (I'm not sure what you want to call it)
Thanks
 
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  • #2
BiGyElLoWhAt said:
After I got the solution from my greens functions, I went through and tried to add in what would be the homogeneous solution, Asin(x/2) + Bcos(x/2) and apply the boundary conditions again to solve for A and B. Both came out to be zero. Is this always the case?
If you have a homogeneous linear differential equation with homogeneous boundary conditions, then the solution is trivial. The point of the Green's function method is to take care of the inhomogeneities.
 
  • #3
The diff eq wasn't homogeneous. It was equal to sin (2x). It was linear, and I had homogeneous boundary conditions, though. Y (0)=y (pi)=0
 
  • #4
BiGyElLoWhAt said:
The diff eq wasn't homogeneous. It was equal to sin (2x). It was linear, and I had homogeneous boundary conditions, though. Y (0)=y (pi)=0
Yes, but you took care of the inhomogeneity with the Green's function. You cannot expect to add a homogeneous solution to this if you have homogeneous boundary conditions because your Green's function is set up to handle the homogeneous boundary conditions.
 
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  • #5
So is it reasonable to say that I turned a homogeneous solution into a general one? As in the homogeneous is buried within the resultant solution? I've been struggling to get the grand scheme of the green's function and it's solution, so to speak.
 
  • #6
BiGyElLoWhAt said:
So is it reasonable to say that I turned a homogeneous solution into a general one?
Not really. Rather, you found the solution satisfying the boundary conditions without ever having to worry about splitting it into a homogeneous and a particular part. That distinction is arbitrary anyway as there is not one particular solution, but all particular solutions are related in that their difference is a homogeneous solution.

Instead, you design your Green's function in such a way that it will let you help the particular solution that already satisfies homogeneous boundary conditions. If your problem has homogeneous boundary conditions, then you therefore never need to worry about finding a homogeneous solution.

I think it is a common thing among students to struggle with Green's function, yet you have likely been using them since high school for computing the electric/gravitational potential of a collection of point particles - just that you never called them Green's functions.
 
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Hmmm. Ok, I think I understand now. Thank you.
 

FAQ: A somewhat conceptual question about Green's functions

1. What is a Green's function?

A Green's function is a mathematical tool used to solve differential equations. It represents the response of a system to a delta function, or impulse, input. It can also be thought of as the inverse of the differential operator in the equation.

2. How are Green's functions used in science?

Green's functions are used in various fields of science, such as physics, engineering, and mathematics, to solve differential equations and model systems. They are particularly useful for systems with complicated boundary conditions or non-homogeneous equations.

3. Can Green's functions be applied to any type of differential equation?

Green's functions can be applied to linear differential equations, which are equations that can be written in the form of a linear combination of functions and their derivatives. Non-linear equations may also be solved using Green's functions, but it can be more challenging.

4. Are there different types of Green's functions?

Yes, there are different types of Green's functions depending on the type of differential equation being solved. Some examples include the Dirac delta function, Heaviside step function, and Bessel function. Each type of Green's function is specific to a particular type of equation.

5. How are Green's functions related to boundary value problems?

Green's functions are often used to solve boundary value problems, which involve finding a solution to a differential equation that satisfies certain conditions at the boundaries of the system. Green's functions provide a way to express the solution in terms of the boundary conditions and the input function.

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