Solve Laplace's Equation with Laplace Transform

[Another]

Can we solve Laplace's equation by Laplace transform ?

 

[jedishrfu]

How did you come to this thought? because Laplace is associated with it these topics?

Here's a writeup on the Laplace Transform:

https://en.wikipedia.org/wiki/Laplace_transform

where it says it was discovered during his work on probability theory.

And here's a writeup on the Laplace Equation:

https://en.wikipedia.org/wiki/Laplace's_equation

and its beauty:

https://www.wired.com/2016/06/laplaces-equation-everywhere/

This is a partial differential equation to which "Separation of Variables" is often applied. to extract a solution.

http://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx

I couldn't find any example online where the Laplace equation was solved by a Laplace Transform at some point in the solution but perhaps @fresh_42 or @Mark44 know of one.

 

[bigfooted]

the Laplace transform of the partial derivative is $L[\frac{\partial^2U}{\partial x^2}] = \frac{d^2u}{dx^2}$. This means that the Laplace transform is not useful in solving the Laplace equation, but it can be used to solve the heat equation, wave equation, and basically any 2D PDE for U(x,t) where one partial derivative is with respect to time and the other with respect to the spatial coordinate.

 

[hunt_mat]

If you're solving your equation on the half space and you know the value of the solution and it's derivative on the boundary, then yes.