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Heat Equation on a surface sphere using NDSolve?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127470] Heat Equation on a surface sphere using NDSolve?
*From*: georgesabitbol4 at gmail.com
*Date*: Fri, 27 Jul 2012 04:58:16 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*Delivered-to*: mathgroup-newout@smc.vnet.net
*Delivered-to*: mathgroup-newsend@smc.vnet.net
Hello Folks,
I am new to Mathematica, but it seems to be the most suitable tool for my issue.
I have to solve a equation which is similar to heat equation on a sphere surface. The exact expression in a spherical coordinates system is (in LaTeX):
\frac{\partial{h}}{\partial t} = \frac{1}{n} \left[ \frac{1}{r^2 \sin^2{\phi}} \frac{\partial}{\partial \theta} \left( K \frac{\partial h}{\partial \theta} \right) + \frac{1}{r^2 \sin \phi} \frac{\partial}{\partial \phi} \left( K \sin \phi \frac{\partial h}{\partial \phi}\right) + s(\theta,\phi,t)\right]
K is kind of a diffusion coeff.
s a source term (varying in time and space)
and n a constant (although it might depends on \phi and \theta).
\theta and \phi are defined here in a mathematical way.
r the radius (constant as we deal with a perfect sphere).
(no variation regarding the radius, we only look at the surface of the sphere).
I found that in Mathematica, and in a Cartesian frame, such equation (somewhat not exactly the same but similar) would be:
N = 1
K = 34
myfunc = NDSolve[{D[h[x, y, t], t] ==
1/N*(D[h[x, y, t], x, x] + D[h[x, y, t], y, y]), h[x, y, 0] == 0,
h[0, y, t] == K*t, h[9, y, t] == K*t, h[x, 0, t] == K*t,
h[x, 9, t] == K*t}, h, {x, 0, 9}, {y, 0, 9}, {t, 0, 6}];
I'm kind of stuck right now, because I do not know how can I switch to the spherical frame. I would appreciate any ideas/suggestions.
Thanks a lot and sorry for bothering you guys with this.
G.
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