Changes between Version 27 and Version 28 of u/erica/PoissonSolver

Timestamp:

08/19/13 14:32:50 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/PoissonSolver

@@ -80 +80 @@
 = Laplace Equation (source term = 0) Physical Meaning =
 
-Note that the discretization equation with f=0 describes each point as being the average of its neighbors. This is in fact one of the key properties of the harmonic functions, which are the class of solution to the Laplace Equation. Since this is such an ubiquitious equation in physics, and a subset of the Poisson equation, I felt it was important to review some of the properties behind it. 
+Note that the discretization equation with f=0 describes each point as being the average of its neighbors. This is in fact one of the key properties of the harmonic functions, which are the class of solution to Laplace's Equation. Since this is such an ubiquitous equation in physics, and a subset of the Poisson equation, I felt it was important to review some of its defining properties. 
 
-- Boundary conditions
-- Minimizing distance b/w  boundaries (in 1d this is a line, in 2d this is soap bubble)
-- Physical interprestaion - no charge within the domain, but elsewhere. 
+The most interesting property of the harmonic function is that it minimizes the distance (in 1D), or the surface area (in 2D), (or the analogous topology in higher dimensions). So in 1D, the solution to Laplace's equation is a line, and in 2D it is a soap bubble (or a waxy film, pulled taught over box with curvy top edges). Thus, solutions to Laplace's equation DOES NOT contain any local mins or maxs.
+
+The equation describes regions that contain no charge (be it electrical, or gravitational), but describes how the potential due to charge elsewhere effects the region of interest. 
 
 = Poisson Equation Physical Meaning = 
 
-While the interpreation of the solution propertoies are not as clear cut and intuitive as Laplace, the solution can be thoguht of as composing green's functions (cite references). 
+While the interpretation of Laplace's equation and the harmonic functions are nice, the same does not seem to apply for the more general Poisson's equation. Instead, the solutions to Poisson's equation are called Green's functions, and they are stitched together out of the solutions to the homogenous part of the PDE (namely, Laplace's equation. Here are some references:
+
+= Boundary Conditions =
+
+In 1D, either the value of u is to be given at both boundaries (Dirchlet boundary condition), or EITHER a normal derivative (von Neumann) plus the value of u (but not 2 normal derivatives, for this would not satisfy the boundary conditions).
+
+In 2D (or higher), one can have either Dirchlet or von Neumann conditions all around the region.  
 
 
 = The code = 
 
-Explain what my thing does, post it for download
+Here is an html copy of my code - 
 
 = Tests and Results =