# Transient Two-Dimensional Heat Conduction Using Chebyshev Collocation

Transient Two-Dimensional Heat Conduction Using Chebyshev Collocation

Consider the two-dimensional heat equation given by

1

α

∂u

∂t

2

∂

∂

2

x

2

∂

∂

2

y

t≥0

0≤x,y≤1

which represents heat conduction in a two-dimensional domain. The boundary conditions are such that the temperature, , is equal to 0 on all the edges of the domain:

u

u(x=0,y,t)=0

u(x=1,y,t)=0

t≥0

u(x,y=0,t)=0

u(x,y=1,t)=0

t≥0

Without loss of generality, one can take the thermal diffusivity, , equal to . The initial condition is given by

α

1/s

2

cm

u(x,y,0)=1

0≤x,y≤1

The dimensionless temperature, , can be found using either NDSolve or the Chebyshev collocation technique. As shown in the table of data given at , the two methods give the same results. Also, three contours of the dimensionless temperature, , (i.e., 0.25, 0.5, and 0.75) at are shown using the red curves and dashed black curves for the solution obtained with NDSolve and Chebyshev collocation, respectively. Again, perfect agreement is observed. Finally, you can set the value of the dimensionless time, and the contour plot tab lets you display the contour plot of the solution obtained using the Chebyshev collocation method.

u

x=y=1/2

u

t=0.02