Semi-infinite body heated at a type 1 boundary.

Consider the temperature in semi-infinite body with a specified temperature applied to the $x=0$ boundary. The temperature satisfies the following equations:

$$\frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}; \; \; 0 < x < \infty$$ (8)

$$T(x=0,t) = T_1$$

$$T(x,t=0) = T_0$$

The above problem has two non-homogeneous terms, however one may be eliminated to simplify the problem by normalizing the temperature. Let $T^+(x,t) = (T(x,t) - T_0)/(T_1 - T_0)$. If you replace $T^+$ 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:

$$\frac{T(x,t)-T_0}{T_1 - T_0} = \alpha \int_{\tau=0}^t \left... ...al G_{X10}}{\partial x^{\prime}} \right]_{x^{\prime}=0} d \tau$$ (9)

Note that here the sign of the derivative has been chosen by the outward normal according to $-\partial / \partial n^{\prime} = + \partial / \partial x^{\prime}$. The GF is given by

$$G_{X10}(x,t \mid x^{\prime},\tau ) =[4 \pi \alpha (t-\tau)]^... ...-\frac{(x+x^{\prime })^{2}}{4 \alpha (t-\tau)}\right] \right\}$$ (10)

The required derivative of the GF with respect to $x^{\prime}$ evaluated at $x^{\prime}=0$ is given by

$$\left[ \frac{\partial G_{X10}}{\partial x^{\prime}} \right]_... ...]^{3/2} } \exp \left[ \frac{-x^{2}}{4 \alpha (t-\tau)}\right]$$ (11)

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

$$\frac{T(x,t)-T_0}{T_1 - T_0} = \left[ 1- erf \left( \frac{x}{(4 \alpha t)^{1/2}} \right) \right ]$$ (12)

or as the complementary error function

$$\frac{T(x,t)-T_0}{T_1 - T_0} = erfc \left( \frac{x}{(4 \alpha t)^{1/2}} \right)$$ (13)

Alternately, this problem could have been solved as an initial condition problem if the temperature were normalized as $(T(x,t) - T_1)/(T_0 - T_1)$.