Solution of the 2D Heat Equation Using the Method of Lines
Solution of the 2D Heat Equation Using the Method of Lines
Consider the unsteady-state heat conduction problem defined by
∂T
∂t
2
∂
∂
2
x
2
∂
∂
2
y
where is the temperature, is the thermal diffusivity, is the time, and and are the spatial coordinates.
T
α
t
x
y
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):
[0,1]×[0,1]
BC 1: , where and ,
T(0,y,t)=0
0<y<1
t⩾0
BC 2: , where and ,
T(1,y,t)=0
0<y<1
t⩾0
BC 3: , where and ,
T(x,1,t)=0
0<x<1
t⩾0
BC 4: , where and ,
T(x,0,t)=10
0<x<1
t⩾0
IC: , where and .
T(x,y,0)=0
0<x<1
0<y<1
The Demonstration gives a contour plot of the temperature for user-set values of and .
t
α