Solution of the 2D Heat Equation Using the Method of Lines

Consider the unsteady-state heat conduction problem defined by

,

where is the temperature, is the thermal diffusivity, is the time, and and are the spatial coordinates.

This Demonstration solves this partial differential equation–a two-dimensional heat equation–using the method of lines in the domain , subject to the following Dirichlet boundary conditions (BC) and initial condition (IC):

BC 1: , where and ,

BC 2: , where and ,

BC 3: , where and ,

BC 4: , where and ,

IC: , where and .

The Demonstration gives a contour plot of the temperature for user-set values of and .