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Infinite body, cylindrical coordinate, transient 1-D.

R00 Infinite region, 0 < r < $ \infty$.

GR00(r, t | r$\scriptstyle \prime$,$\displaystyle \tau$) = $\displaystyle {\frac{1}{4\pi \alpha (t-\tau )}}$exp$\displaystyle \left[\vphantom{ -\frac{r^{2}+\left( r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{r^{2}+\left( r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{r^{2}+\left( r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}\right]$I0$\displaystyle \left[\vphantom{ \frac{rr^{\prime }}{2\alpha (t-\tau )}}\right.$$\displaystyle {\frac{rr^{\prime }}{2\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ \frac{rr^{\prime }}{2\alpha (t-\tau )}}\right]$    

Equivalent forms:
GR00(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{4\pi \alpha (t-\tau )}}$exp$\displaystyle \left[\vphantom{ -\frac{\left( r-r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{\left( r-r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{\left( r-r^{\prime }\right) ^{2}}{4\alpha (t-\tau )}}\right]$  
    x exp$\displaystyle \left[\vphantom{ -\frac{rr^{\prime }}{2\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{rr^{\prime }}{2\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{rr^{\prime }}{2\alpha (t-\tau )}}\right]$I0$\displaystyle \left[\vphantom{ \frac{rr^{\prime }}{2\alpha (t-\tau )}}\right.$$\displaystyle {\frac{rr^{\prime }}{2\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ \frac{rr^{\prime }}{2\alpha (t-\tau )}}\right]$  


GR00(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{2\pi }}$$\displaystyle \int_{0}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta ^{2}\alpha (t-\tau )}\right.$ - $\displaystyle \beta^{2}_{}$$\displaystyle \alpha$(t - $\displaystyle \tau$)$\displaystyle \left.\vphantom{ -\beta ^{2}\alpha (t-\tau )}\right]$$\displaystyle \beta$ J0($\displaystyle \beta$r) J0($\displaystyle \beta$r$\scriptstyle \prime$)d$\displaystyle \beta$  
    Note that $\displaystyle \beta$ in the above integral has units of meters-1.  

Special case when heat source is located at r'=0:

GR00(r, t $\displaystyle \left\vert\vphantom{ \,0,\tau }\right.$ 0,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,0,\tau }\right.$) = $\displaystyle {\frac{1}{4\pi \alpha (t-\tau )}}$exp$\displaystyle \left[\vphantom{ -\frac{r^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{r^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{r^{2}}{4\alpha (t-\tau )}}\right]$    


Kevin D. Cole
2002-12-31