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Consider the temperature caused by a spatially-varying initial condition in an
infinite 1D body. The temperature satisfies the following equations:
This case has number X00T5. The solution for the temperature is
given by the initial condition term of the GF solution equation:
 |
(2) |
Because the initial condition is zero over most of the domain the integral
may be carried out over the non-zero portion:
 |
(3) |
Using the X00 Green's function evaluated at
,
![\begin{displaymath}
T(x,t) = T_1 \int_c^d [4 \pi \alpha t]^{-1/2}\exp \left[
-\frac{(x-x^{\prime })^{2}}{4 \alpha t}\right] dx^{\prime}
\end{displaymath}](img76.png) |
(4) |
Using the substitution
this
integral can be written as
where the error function is defined
 |
(7) |
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Up: EXAMPLES, TEMPERATURE FROM GF
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2004-01-31