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Next: Laplace Equation. Steady Heat Up: Radial-spherical coordinates. Transient 1-D. Previous: Solid Sphere, transient 1-D.

Hollow Sphere, transient 1-D.

RS11 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and G = 0 (Dirichlet) at r = b.
a. Best convergence for (t - $ \tau$) small:
GRS11(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}}}$$\displaystyle \sum_{n=-\infty }^{\infty }$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}% \right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}% }\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}% \right] }\right.$  
    $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right.$ +exp$\displaystyle \left[\vphantom{ -\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ +\exp \left[ -\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}\right] }\right\}$  

b. Best convergence for (t - $ \tau$) large:
GRS11(r, t $\displaystyle \left\vert\vphantom{ \,r^{\prime },\tau }\right.$ r$\scriptstyle \prime$,$\displaystyle \tau$ $\displaystyle \left.\vphantom{ \,r^{\prime },\tau }\right.$) = $\displaystyle {\frac{1}{2\pi (b-a)r\,r^{\prime }}}$  
    x $\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )% }{(b-a)^{2}}}\right.$ - $\displaystyle {\frac{m^{2}\pi ^{2}\alpha (t-\tau )% }{(b-a)^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{m^{2}\pi ^{2}\alpha (t-\tau )% }{(b-a)^{2}}}\right]$sin(m$\displaystyle \pi$$\displaystyle {\frac{r-a}{b-a}}$)sin(m$\displaystyle \pi$$\displaystyle {\frac{r^{\prime }-a% }{b-a}}$)  

RS12 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and $ \partial$G/$ \partial$r = 0 (Neumann) at r = b.
a. Best convergence for $ \alpha$(t - $ \tau$)/(b - a)2 < 0.022:
GRS12(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}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$  
    $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al... ... +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ -exp$\displaystyle \left[\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}% }\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al... ...+\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{B_{2}}{4\pi r\,r^{\prime }b}}$exp$\displaystyle \left[\vphantom{ B_{2}\frac{(2b-r-r^{\prime })}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right.$B2$\displaystyle {\frac{(2b-r-r^{\prime })}{b^{{}}}}$ + B22$\displaystyle {\frac{\alpha (t-\tau )}{b^{2}}}$ $\displaystyle \left.\vphantom{ B_{2}\frac{(2b-r-r^{\prime })}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$$\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 (t - $ \tau$) large:
GRS12(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 (b-a)r\,r^{\prime }}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right]$  
    $\displaystyle {\frac{\beta _{m}^{2}+H_{2}^{2}}{\beta _{m}^{2}+H_{2}^{2}+H_{2}}}$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r-a}{b-a}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r^{\prime }-a}{b-a}}$)  

where the eigenvalues are given by positive roots of

$\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = - H2    

and where H2 = B2R2;  B2 = - 1;  R2 = (b - a)/b.


RS13 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and k$ \partial$G/$ \partial$r + hG = 0 (convection) at r = b.
a. Best convergence for $ \alpha$(t - $ \tau$)/(b - a)2 < 0.022 (note B2 = h2b/k - 1):

GRS13(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}}}$$\displaystyle \left\{\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$exp$\displaystyle \left[\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$  
    $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al... ... +\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right.$ -exp$\displaystyle \left[\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}% }\right.$ - $\displaystyle {\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau )}% }\right]$ + exp$\displaystyle \left[\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right.$ - $\displaystyle {\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}$ $\displaystyle \left.\vphantom{ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}}\right]$ $\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al... ...+\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right] }\right\}$  
    - $\displaystyle {\frac{B_{2}}{4\pi r\,r^{\prime }b}}$exp$\displaystyle \left[\vphantom{ B_{2}\frac{(2b-r-r^{\prime })}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right.$B2$\displaystyle {\frac{(2b-r-r^{\prime })}{b^{{}}}}$ + B22$\displaystyle {\frac{\alpha (t-\tau )}{b^{2}}}$ $\displaystyle \left.\vphantom{ B_{2}\frac{(2b-r-r^{\prime })}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$  
    x erfc$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}% +\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$$\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 (t - $ \tau$) large:
GRS13(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 (b-a)r\,r^{\prime }}}$$\displaystyle \sum_{m=1}^{\infty }$exp$\displaystyle \left[\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right.$ - $\displaystyle {\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}$ $\displaystyle \left.\vphantom{ -\frac{\beta _{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right]$  
    $\displaystyle {\frac{\beta _{m}^{2}+H_{2}^{2}}{\beta _{m}^{2}+H_{2}^{2}+H_{2}}}$sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r-a}{b-a}}$)sin($\displaystyle \beta_{m}^{}$$\displaystyle {\frac{r^{\prime }-a}{b-a}}$)  

where the eigenvalues are given by positive roots of

$\displaystyle \beta_{m}^{}$cot$\displaystyle \beta_{m}^{}$ = - H2    

and where H2 = B2R2;  B2 = h2b/k - 1;  R2 = (b - a)/b.
next up previous
Next: Laplace Equation. Steady Heat Up: Radial-spherical coordinates. Transient 1-D. Previous: Solid Sphere, transient 1-D.
Kevin D. Cole
2002-12-31