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Up: Radial-spherical coordinates. Transient 1-D.
Previous: Solid 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 -
) small:
GRS11(r, t r ,
) |
= |
![$\displaystyle {\frac{1}{8\pi
r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$](img314.gif)  exp
-
![$\displaystyle \left.\vphantom{ \exp \left[ -\frac{[2n(b-a)+r-r^{\prime }]^{2}}{4\alpha (t-\tau )}%
\right] }\right.$](img383.gif) |
|
|
|
+exp
-
![$\displaystyle \left.\vphantom{ +\exp \left[ -\frac{[2n(b-a)+r+r^{\prime }-2a)^{2}}{4\alpha (t-\tau
)}\right] }\right\}$](img388.gif) |
|
b. Best convergence for (t -
) large:
GRS11(r, t r ,
) |
= |
 |
|
|
|
x exp
-
sin(m )sin(m ) |
|
RS12 Hollow sphere, a < r < b, with G = 0 (Dirichlet) at r = a and
G/
r = 0 (Neumann) at r = b.
a. Best convergence for
(t -
)/(b - a)2 < 0.022:
GRS12(r, t r ,
) |
= |
![$\displaystyle {\frac{1}{8\pi
r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$](img314.gif) exp
-
![$\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right]
}\right.$](img396.gif) |
|
|
|
-exp
-
+ exp
-
![$\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al...
...+\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right]
}\right\}$](img402.gif) |
|
|
|
- exp B2
+ B22
![$\displaystyle \left.\vphantom{ B_{2}\frac{(2b-r-r^{\prime
})}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$](img405.gif) |
|
|
|
x erfc![$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}%
+\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$](img371.gif)
+
![$\displaystyle \left.\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}%
+\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right]$](img374.gif) |
|
b. Best convergence for (t -
) large:
GRS12(r, t r ,
) |
= |
 exp
-
![$\displaystyle \left.\vphantom{ -\frac{\beta
_{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right]$](img408.gif) |
|
|
|
sin( )sin( ) |
|
where the eigenvalues are given by positive roots of
cot
= - 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
G/
r + hG = 0 (convection) at r = b.
a. Best convergence for
(t -
)/(b - a)2 < 0.022
(note
B2 = h2b/k - 1):
GRS13(r, t r ,
) |
= |
![$\displaystyle {\frac{1}{8\pi
r\,r^{\prime }[\pi \alpha (t-\tau )]^{1/2}}}$](img314.gif) exp
-
![$\displaystyle \left.\vphantom{ \exp \left[ -\frac{(r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right]
}\right.$](img396.gif) |
|
|
|
-exp
-
+ exp
-
![$\displaystyle \left.\vphantom{ -\exp \left[ -\frac{(r+r^{\prime }-2a)^{2}}{4\al...
...+\exp \left[ -\frac{(2b-r-r^{\prime })^{2}}{4\alpha (t-\tau )}\right]
}\right\}$](img402.gif) |
|
|
|
- exp B2
+ B22
![$\displaystyle \left.\vphantom{ B_{2}\frac{(2b-r-r^{\prime
})}{b^{{}}}+B_{2}^{2}\frac{\alpha (t-\tau )}{b^{2}}}\right]$](img405.gif) |
|
|
|
x erfc![$\displaystyle \left[\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}%
+\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right.$](img371.gif)
+
![$\displaystyle \left.\vphantom{ \frac{2b-r-r^{\prime }}{[4\alpha (t-\tau )]^{1/2}}%
+\frac{B_{2}\left[ \alpha (t-\tau )\right] ^{1/2}}{b^{{}}}}\right]$](img374.gif) |
|
b. Best convergence for (t -
) large:
GRS13(r, t r ,
) |
= |
 exp
-
![$\displaystyle \left.\vphantom{ -\frac{\beta
_{m}^{2}\alpha (t-\tau )}{(b-a)^{2}}}\right]$](img408.gif) |
|
|
|
sin( )sin( ) |
|
where the eigenvalues are given by positive roots of
cot
= - H2 |
|
and where
H2 = B2R2; B2 = h2b/k - 1; R2 = (b - a)/b.
Next: Laplace Equation. Steady Heat
Up: Radial-spherical coordinates. Transient 1-D.
Previous: Solid Sphere, transient 1-D.
Kevin D. Cole
2002-12-31