# Transient Heat Conduction Using Chebyshev Collocation

Transient Heat Conduction Using Chebyshev Collocation

Consider the one-dimensional heat equation given by

1

α

∂u

∂t

∂u

2

∂x

2

t≥0

0≤x≤1

This equation represents heat conduction in a rod. The boundary conditions are such that the temperature, , is equal to 0 at both ends of the rod:

u

u(x=0,t)=0

u(x=1,t)=0

t≥0

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

α

1cm/s

2

u(x,0)=1

0≤x≤1.

The temperature can be found using either NDSolve (solid colored curve) or the Chebyshev collocation technique (colored dots). Both methods give the same results, which are plotted in the same diagram at various values of time, ranging from 0.01 to 0.1 with a span of 0.01. You can set the number of interior points used by the Chebyshev collocation method.

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