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Next: Hollow Sphere, transient 1-D. Up: Radial-spherical coordinates. Transient 1-D. Previous: Infinite body with a

Solid Sphere, transient 1-D.

RS01 Solid sphere, 0 < r < b, with G = 0 (Dirichlet) at r = b. a. Best convergence for $ \alpha$(t - $ \tau$)/b2 small:
GRS01(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$  
    x $\displaystyle \sum_{n=-\infty }^{\infty }$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{ (2nb+r-r^{\prime })^{2}}{4\... ...\exp \left[ -\frac{ (2nb+r+r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{ (2nb+r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{ (2nb+r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{ (2nb+r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{ (2nb+r+r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{ (2nb+r+r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{ (2nb+r+r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{ (2nb+r-r^{\prime })^{2}}{4\a... ...exp \left[ -\frac{ (2nb+r+r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for $ \alpha$(t - $ \tau$)/b2 large:

GRS01(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 br\,r^{\prime }}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -m^{2}\pi ^{2}\alpha (t-\tau )/b^{2}}\right.$ - m2$\displaystyle \pi^{2}_{}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/b2$\displaystyle \left.\vphantom{ -m^{2}\pi ^{2}\alpha (t-\tau )/b^{2}}\right]$sin(m$\displaystyle \pi$$\displaystyle {\frac{r}{b}}$)sin(m$\displaystyle \pi$$\displaystyle {\frac{r^{\prime }}{b}}$)    


RS02 Solid sphere, 0 < r < b, with $ \partial$G/$ \partial$r = 0 (Newmann) at r = b. a. Approximate relation for $ \alpha$(t - $ \tau$)/b2 $ \leq$ 0.022 (here B2 = - 1):

GRS02(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}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{% (r-r^{\prime })^{2}}{4\alp... ...\exp \left[ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{% (r-r^{\prime })^{2}}{4\alph... ...\exp \left[ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% }\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right\}$  
    - $\displaystyle {\frac{B_{2}}{4\pi r\,r^{\prime }b}}$exp$\displaystyle \left[\vphantom{ B_{2}\frac{2b-r-r^{\prime }% }{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right.$B2$\displaystyle {\frac{2b-r-r^{\prime }% }{b^{{}}}}$ + B22$\displaystyle {\frac{\alpha (t-\tau )}{b^{2}}}$ $\displaystyle \left.\vphantom{ B_{2}\frac{2b-r-r^{\prime }% }{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$$\displaystyle {\frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}$ $\displaystyle \left.\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right]$  

b. Best convergence for $ \alpha$(t - $ \tau$)/b2 large:
GRS02(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{3}{4\pi b^{3}}}$ + $\displaystyle {\frac{1}{2\pi br\,r^{\prime }}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/b2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right]$  
    x $\displaystyle {\frac{\left( \beta _{m}^{2}+1\right) }{\beta _{m}^{2}}}$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{b}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r^{\prime }}{b}}$)  

The eigenvalues are given by the positive roots of

$\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = 1    


RS03 Solid sphere, 0 < r < b, with k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = b.
a. Approximate relation for $ \alpha$(t - $ \tau$)/b2 $ \leq$ 0.022 (here B2 = hb/k - 1):

GRS03(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}{8\pi r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{% (r-r^{\prime })^{2}}{4\alp... ...\exp \left[ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ - exp$\displaystyle \left[\vphantom{ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{% (r-r^{\prime })^{2}}{4\alph... ...\exp \left[ -\frac{% (r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% }\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}% \right] }\right\}$  
    - $\displaystyle {\frac{B_{2}}{4\pi r\,r^{\prime }b}}$exp$\displaystyle \left[\vphantom{ B_{2}\frac{2b-r-r^{\prime }% }{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right.$B2$\displaystyle {\frac{2b-r-r^{\prime }% }{b^{{}}}}$ + B22$\displaystyle {\frac{\alpha (t-\tau )}{b^{2}}}$ $\displaystyle \left.\vphantom{ B_{2}\frac{2b-r-r^{\prime }% }{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$$\displaystyle {\frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}}$ + $\displaystyle {\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}$ $\displaystyle \left.\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right]$  

b. Best convergence for $ \alpha$(t - $ \tau$)/b2 large:
GRS03(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 br\,r^{\prime }}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right.$ - $\displaystyle \beta_{m}^{2}$$\displaystyle \alpha$(t - $\displaystyle \tau$)/b2$\displaystyle \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right]$  
    x $\displaystyle {\frac{\left( \beta _{m}^{2}+B_{2}^{2}\right) }{\beta _{m}^{2}+B_{2}^{2}+B_{2}}}$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r}{b}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{% r^{\prime }}{b}}$)  

where B2 = hb/k - 1 and where the eigenvalues are given by the positive roots of

$\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = - B2    

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Next: Hollow Sphere, transient 1-D. Up: Radial-spherical coordinates. Transient 1-D. Previous: Infinite body with a
Kevin D. Cole
2002-12-31