The heat-flow equation controls the diffusion of heat. This equation is a prototype for migration. The 15 migration equation is the same equation but the heat conductivity constant is imaginary. (The migration equation is really the Schroedinger equation, which controls the diffusion of probability of atomic particles).

(28) |

(29) |

(30) |

9cData Table | 1cx |
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i |
n |
i |
t |
i |
a |
l |
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6c | |||||||||

6c | |||||||||

6c | |||||||||

6c | |||||||||

s |
6c | s |
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6c | |||||||||

6c | |||||||||

i |
3c | 1cstar | 2c | i |
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6c | |||||||||

2|c| | 2|c| | ||||||||

d |
1c | 1c | d |
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2|c| | 2|c| | ||||||||

3|c| | 3|c| | ||||||||

e |
2c | 1 | 2c | e |
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3|c| | 3|c| | ||||||||

6c | |||||||||

6c | |||||||||

6c | |||||||||

6c | |||||||||

t |
6c |

(31) |

(32) |

Equation (32) can be interpreted geometrically as a computational
star in the (*x*,*t*)-plane, as depicted in Table .7.
By moving the star around in the data table you will note that it
can be positioned so that only one number at a time (the 1) lies
over an unknown element in the data table.
This enables the computation of subsequent rows beginning from the top.
By doing this you are solving the partial-differential equation by the
finite-difference method.
There are many possible arrangements of initial and side conditions,
such as zero-value side conditions.
Next is a computer program and its result.

# Explicit heat-flow equation real q(12), qp(12) nx = 12 do ia= 1, 2 { # stable and unstable cases alpha = ia*.3333; write(6,'(/"alpha =",f5.2)') alpha do ix= 1,6 { q(ix) = 0.} # Initial temperature step do ix= 7,12 { q(ix) = 1.} do it= 1, 6 { write(6,'(20f6.2)') (q(ix),ix=1,nx) do ix= 2, nx-1 qp(ix) = q(ix) + alpha*(q(ix-1)-2.*q(ix)+q(ix+1)) qp(1) = qp(2); qp(nx) = qp(nx-1) do ix= 1, nx q(ix) = qp(ix) } } call exit(0); end

alpha = 0.33 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.33 0.67 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.11 0.33 0.67 0.89 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.04 0.15 0.37 0.63 0.85 0.96 1.00 1.00 1.00 0.00 0.00 0.01 0.06 0.19 0.38 0.62 0.81 0.94 0.99 1.00 1.00 0.00 0.00 0.02 0.09 0.21 0.40 0.60 0.79 0.91 0.98 1.00 1.00

alpha = 0.67 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.67 0.33 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.44 0.00 1.00 0.56 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.30 -0.15 0.96 0.04 1.15 0.70 1.00 1.00 1.00 0.00 0.00 0.20 -0.20 0.89 -0.39 1.39 0.11 1.20 0.80 1.00 1.00 0.13 0.13 -0.20 0.79 -0.69 1.65 -0.65 1.69 0.21 1.20 0.87 0.87

10/31/1997