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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

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