Fundamental Solution of Laplace Equation 2d vs 3d

In summary, the fundamental solution for Laplace's Equation behaves differently in 2 dimensions and 3 dimensions. In 2 dimensions, the solution becomes unbounded as r goes to infinity while in 3 dimensions, it goes to zero. This difference can be explained by looking at the "gravitational potential" in three dimensions and two. Additionally, the fundamental solution can also be understood in terms of electrostatics and incompressible fluid flow. It is interesting to note that changing the dimension for certain PDE's can significantly affect the solution's character, as seen in the wave equation.
  • #1
starzero
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When one compares the fundamental solution for Laplace's Equation one might note that in 2 dimensions this solution becomes unbounded as r goes to infinity while in 3 dimensions the solution goes to zero as r goes to infinity.

Now I understand both mathematical derivations so my question is not about that. What I would like to know is can someone give me a good explanation in terms of the possible physics being modeled by this equation that would explain this difference between the 2d and 3d case.

That is I would like a real world explanation as opposed to simply a mathematical explanation.
 
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  • #2
You mean, of course, Laplace's equation with boundary conditions given on a circle or sphere. Laplace's equation is related to "potential theory" and so you might look at the difference between the "gravitational potential" in three dimensions and two.
 
  • #3
Thanks...I will consider it from that point of view.

Further, I would like to understand the fundamental solution also in terms of
electrostatics and incompressible fluid flow.

It is interesting to note how changing the dimension for certain PDE's will significantly
affect the character of the solution. Another example concerns solutions to the wave equation for which there is a significant difference between even and odd dimensions.

If you or anyone else has more ideas I am all ears.
 

1. What is the fundamental solution of the Laplace equation in 2D and 3D?

The fundamental solution of the Laplace equation in 2D and 3D is a mathematical function that satisfies the Laplace equation, also known as the potential equation. In 2D, the fundamental solution is a logarithmic function, while in 3D it is a reciprocal function. This means that the solution in 2D depends on the natural logarithm of the distance from the source, while in 3D it depends on the inverse of the distance.

2. How is the fundamental solution of the Laplace equation used in science?

The fundamental solution of the Laplace equation is used in various scientific fields, including physics, engineering, and mathematics. It is particularly useful in solving problems related to potential theory, such as electrostatics, fluid dynamics, and heat transfer. It is also used in the study of partial differential equations and boundary value problems.

3. What is the difference between the 2D and 3D fundamental solutions?

The main difference between the 2D and 3D fundamental solutions lies in the dimensionality of the problem. In 3D, the solution is more complex and depends on the distance from the source in all three dimensions, while in 2D it only depends on the distance in two dimensions. Additionally, the 3D solution has a more rapid decay as the distance from the source increases compared to the 2D solution.

4. How is the fundamental solution of the Laplace equation derived?

The fundamental solution of the Laplace equation can be derived using various methods, such as the method of images, separation of variables, or Green's functions. These methods involve solving the Laplace equation in a specific domain, such as a circle or a sphere, and then finding a solution that satisfies the given boundary conditions. The resulting solution is the fundamental solution of the Laplace equation for that particular geometry.

5. Are there any limitations to using the fundamental solution of the Laplace equation?

While the fundamental solution of the Laplace equation is a powerful tool in solving certain problems, it does have its limitations. It is only applicable to problems that satisfy the Laplace equation, and it may not be suitable for problems with complex geometries or boundary conditions. Additionally, the solution may become unstable or inaccurate for certain situations, and alternative methods may be needed.

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