Green's function in n-dim, but with one independent variable.

In summary, the conversation discusses the use of Green's functions in a partial differential equation for a scalar ##f##, which is independent of the variable ##y:=x^n## and only contains derivatives with respect to ##x^1,\ldots, x^{n-1}##. The question is whether it would be correct to use a Green's function in ##n-1## dimensions and ignore the ##n##'th dimension.
  • #1
center o bass
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Suppose we have some partial differential equation for a scalar ##f##
$$Df = \rho$$
taking values in ##\mathbb{R}^n##, and further suppose that the differential equation is completely independent of the variable ##y:=x^n## so that the differential operator ##D## only contains derivatives with respect to ##x^1,\ldots, x^{n-1}##, and ##f## as well as ##\rho## is also independent of ##x^n##. Would it then be correct to use a Green's function

$$D G(\vec r- \vec r') = \delta^{(n-1)}(\vec r - \vec r')$$
for ##\vec r, \vec r' \in \mathbb{R}^{n-1}## and with
$$f(\vec r) = \int_{\mathbb{R}^{n-1}} d^{n-1} \vec r' G(\vec r - \vec r') \rho(\vec r')$$?

In other words, would it be correct to just use the method of greens functions in ##n-1## dimensions, and pretend that the ##n##'th dimension is not there?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 

1. What is a Green's function in n-dim with one independent variable?

A Green's function in n-dim with one independent variable is a mathematical tool used in solving differential equations. It is a function that represents the inverse of a differential operator, and can be used to find the solutions to the differential equation by convolving it with the source term.

2. How is a Green's function different in n-dim compared to 1-dim?

In n-dim, a Green's function is a function of several variables, while in 1-dim it is a function of only one variable. This means that in n-dim, the Green's function is a function of both space and time, while in 1-dim it is only a function of time.

3. Can Green's function be used for any type of differential equation?

Yes, Green's function can be used for any type of linear differential equation, including ordinary and partial differential equations. However, the method for constructing the Green's function may vary depending on the type of differential equation.

4. How is Green's function related to boundary value problems?

Green's function can be used to solve boundary value problems by taking advantage of its property of being the inverse of a differential operator. It allows for the solution of a differential equation to be written in terms of the boundary conditions, making it easier to find the solution to the problem.

5. Are there any real-world applications of Green's function in n-dim with one independent variable?

Yes, Green's function has various applications in physics and engineering, including solving problems in heat transfer, fluid dynamics, and electromagnetics. It is also commonly used in quantum mechanics to find the probability of a particle's position and momentum at any given time.

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