Laplace equation with irregular boundaries

[physwiz222]

TL;DR

We can solve Laplace’a Equation on regular shapes like squares and circles but what if we have an irregular shape can it still be solved

Is there a way to solve Laplace’s Equation on irregular domains if the domain’s shape is given by a function for example a 2D parabolic plate. I keep seeing numerical methods but I want to know is there an ANALYTICAL method to solve it on an irregular domain. If there isn't are there approximate analytical methods to solve it. If it isn't solvable why not.

 

[vanhees71]

Usually you need coordinates, where the Laplacian "separates". Then you can find solutions of boundary-value problems by expansion in terms of the corresponding orthogonal function systems (generalized Fourier series).

The 2D case is special since there you can apply the very elegant formulation in terms of complex functions and conformal mappings.

A good book about all this is

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, 2 Vols., McGraw-Hill (1953)

 

[physwiz222]

Usually you need coordinates, where the Laplacian "separates". Then you can find solutions of boundary-value problems by expansion in terms of the corresponding orthogonal function systems (generalized Fourier series).

The 2D case is special since there you can apply the very elegant formulation in terms of complex functions and conformal mappings.

A good book about all this is

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, 2 Vols., McGraw-Hill (1953)

Interesting but one more question can separation of variables be used for a solution on an irregular boundary given by a function. And how common are irregular geometries encountered in real life problems

 

[physwiz222]

Usually you need coordinates, where the Laplacian "separates". Then you can find solutions of boundary-value problems by expansion in terms of the corresponding orthogonal function systems (generalized Fourier series).

The 2D case is special since there you can apply the very elegant formulation in terms of complex functions and conformal mappings.

A good book about all this is

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, 2 Vols., McGraw-Hill (1953)

How about the 3D case

 

[jasonRF]

Interesting but one more question can separation of variables be used for a solution on an irregular boundary given by a function. And how common are irregular geometries encountered in real life problems

For The Laplace equation you can only use separation of variables when the boundaries are on surfaces where one of the coordinates is a constant. In Cartesian coordinates, the boundaries must be x=constant or y=constant or z=constant. In cylindrical coordinates the boundaries must be r=constant, z=constant or theta=constant. Etc.

In real life it is common to have irregular boundaries, and we usually must resort to numerical techniques. However, solving the simple problems in textbooks and examining properties of the solutions often yields a lot of insight to help with understanding those real-life cases.