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### 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 - ) small:
 GRS11(r, t  r, ) = exp - +exp -

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, ) = exp - -exp - + exp - - expB2 + B22 x erfc +

b. Best convergence for (t - ) large:
 GRS12(r, t  r, ) = exp - 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 kG/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, ) = exp - -exp - + exp - - expB2 + B22 x erfc +

b. Best convergence for (t - ) large:
 GRS13(r, t  r, ) = exp - 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