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

Three edges of an insulated metal plate are kept constant at 0 degrees #Celsius while a fourth edge is held at 100 degrees. 🌡

The #LaplaceEquation in 2 spatial dimensions gives the steady-state temperature distribution at all points within the plate under these conditions. Blue corresponds to cool temperatures, red corresponds to hot temperatures, and shades of green and yellow correspond to intermediate temperatures.
#PartialDifferentialEquations #HeatEquation #Elliptic #Math

The #HeatEquation in 2 spatial dimensions. Three edges of an (insulated!) rectangular plate are held at 0 degrees #Celsius, while the fourth edge is held at 100 degrees.🌡 The plate is initially 0 degrees throughout.
The #LaplaceEquation (depicted below the animated time-dependent solution) gives the steady-state temperature distribution within the plate, which the time-dependent solution approaches. 🍿🥤
#PartialDifferentialEquations