Normalerweise spielt Eichfreiheit bei Stokes-Fluss (Re=0) keine große Rolle, aber ich habe jetzt tatsächlich einen Fall gefunden, wo sie sehr nützlich ist… 😎 #Hydrodynamik #PDEs #elektrodynamik

The #WaveEquation in 2 spatial dimensions. A membrane stretched over a square frame is set to vibrating after some initial displacement. Warmer colors indicate greater displacement in the positive z-axis. #PDEs #mathtodon #math

#Poisson’s equation in 2 spatial dimensions. Heat is generated at each point within a rectangular plate at a uniform rate. The top and bottom edges are held constant at a certain temperature, while heat is lost out of the left and right edges; the boundary conditions are given in terms of both the temperature, and spatial derivatives of the temperature, along the edges of the plate. The solution gives the steady-state temperatures at each location within the plate. #PDEs #math

The #HeatEquation (left) and #LaplaceEquation (right) solved under the same boundary conditions over a square plate. Over time, the solution to the #HeatEquation approaches that of the #LaplaceEquation - in this case, giving the steady-state temperature distribution at each interior point of a square plate whose adjacent edges are held either at 100 degrees C or 0 degrees C (blue shows colder regions; red shows hotter regions; the plate is insulated so that heat does not escape).
#PDEs #math