Consider the temperature in semi-infinite body with a specified temperature
applied to the boundary. The temperature satisfies the following equations:

(8) | |||

The above problem has two non-homogeneous terms, however one may be eliminated to simplify the problem by normalizing the temperature. Let . If you replace into the above problem, the initial condition is zero and the boundary temperature is unity. Then this example is described by number X10B1T0. The temperature is given by the type 1 boundary term of the GF solution equation, as follows:

(9) |

(10) |

The required derivative of the GF with respect to evaluated at is given by

(11) |

Replace the above GF derivative into the GF solution equation to find the temperature; the integral may be evaluated as an error function.

(12) |

(13) |