Changes between Version 6 and Version 7 of u/erica/PoissonSolver

Timestamp:

08/19/13 12:36:43 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/PoissonSolver

@@ -5 +5 @@
 Elliptic equations can be thought of as being the steady state limit of the diffusion equation,
 
-[[latex($u_t = (\kappa u_x)_x + (\kappa u_y)_y + \psi$)]]
+[[latex($u_t = (\kappa u_x)_x + (\kappa u_y)_y + f$)]]
 
-when both the 1) boundary conditions and 2) the source (or forcing term) [[latex($\psi$)]] are time-independent. Under these conditions, the time-dependent terms vanish as the system relaxes to steady state (in the limit t [[latex($\rightarrow \infty$)]]), and we are led to the elliptic equation (here in 2D),
+when both the boundary conditions and the source (or "forcing") term  are time in-dependent. Under these conditions, the time dependent terms vanish as the system relaxes to steady state (in the limit t [[latex($\rightarrow \infty$)]]), and we are led to the elliptic equation (here in 2D),
 
 [[latex($(\kappa u_x)_x + (\kappa u_y)_y = f $)]]
 
-where u is the dependent variable we are solving for, and here f is the forcing term. 
+where u is the dependent variable we are solving for, and f is the forcing term. 
 
-This equation needs to 1) be satisfied by all points within a bounding region and 2) satisfy the boundary conditions on that region. This then can be interpreted as an instantaneous constraint on the system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed.
+The solution to this equation needs to simultaneously 1) satisfy this equation at all points within a bounding region and 2) satisfy the boundary conditions on that region. This then can be interpreted as an instantaneous system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed.
 
-Some special cases for elliptic equations are when [[latex($\kappa = 1$)]]. When f is non-zero, we have the Poisson equation, 
+Some special cases for elliptic equations occur when [[latex($\kappa = 1$)]]; when f is non-zero, we have the Poisson equation, 
 
 [[latex($\triangledown ^2 u = f$)]]