1D infinite body with initial condition.

Consider the temperature caused by a spatially-varying initial condition in an infinite 1D body. The temperature satisfies the following equations:

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

$$T(x,t=0) = \left\{ \begin{array}{cc} T_1; & c < x < d \\ 0; & \mbox{otherwise} \end{array} \right.$$

This case has number X00T5. The solution for the temperature is given by the initial condition term of the GF solution equation:

$$T(x,t) = \int_{-\infty}^{\infty} F(x^{\prime} ) G_{X00}(x,t \mid x^{\prime},\tau=0) dx^{\prime}$$ (2)

Because the initial condition is zero over most of the domain the integral may be carried out over the non-zero portion:

$$T(x,t) = T_1 \int_c^d G_{X00}(x,t \mid x^{\prime},\tau=0) dx^{\prime}$$ (3)

Using the X00 Green's function evaluated at $\tau =0$,

$$T(x,t) = T_1 \int_c^d [4 \pi \alpha t]^{-1/2}\exp \left[ -\frac{(x-x^{\prime })^{2}}{4 \alpha t}\right] dx^{\prime}$$ (4)

Using the substitution $u = (x-x^{\prime })/(4\alpha t)^{1/2}$ this integral can be written as

$$T(x,t) = \frac{T_1}{\pi^{1/2} } \int_{(x-d)/[(4 \alpha t)^{1/2}]} ^{(x-c)/[(4 \alpha t)^{1/2}]} e^{-u^2} du$$ (5)

$$= \frac{T_1}{2} \left\{ erf \left[ \frac{(x-c)}{(4 \alpha t)^{1/2}} \right] - erf \left[ \frac{(x-d)}{(4 \alpha t)^{1/2}} \right] \right\}$$ (6)

where the error function is defined

$$erf(z) = \frac{2}{\pi^{1/2}} \int_0^z e^{-u^2} du$$ (7)