Re: system of nonlinear ODE

• To: mathgroup at smc.vnet.net
• Subject: [mg23790] Re: system of nonlinear ODE
• From: Brian Higgins <bghiggins at ucdavis.edu>
• Date: Sat, 10 Jun 2000 02:59:39 -0400 (EDT)
• References: <8gv85p\$5mt@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```It is not clear from your question whether the problem is
mathematically correct. As others have pointed out on this news
group your problem is certainly not consistent with Mathematica
notation. Nevertheless it appears that you are trying to solve the
similarity solution for natural convection, where f(x) is realted to the
stream function and z(x) is the temperature of the fluid, while x is
the similarity variable.The parameter Fr would then be the Prandtl
number.  In the code below I illustrate how one can solve the free
convection problem for the classical boundary layer flow near a
vertical wall that is heated. Further details and an example solution
can be found at

http://www.bgh.ucdavis.edu/brian/convection/freeconvection.html

Here is the code:

ODE[G1_, G2_,
Pr_] := {f'''[x] + 3  f[x] f''[x] - 2  f'[x]^2 + z[x] ==
0, z''[x] + 3Pr\f[x] z'[x] == 0,
z[0] == 1, z'[0] == G1, f'[0] == 0,
f[0] == 0, f''[0] == G2}

myODESol[G1_, G2_, Pr_] := NDSolve[ODE[G1, G2, Pr], {f, z}, {x, 0,
20}]
fend[G1_, G2_, Pr_] := ({f'[20], z[20]} /. myODESol[G1, G2, Pr])
BC[Pr_] :=
Flatten[FindRoot[First[fend[G1, G2, Pr]] == 0, {G1, -.7, -.3}, {G2, .5,
1},
MaxIterations -> 100, DampingFactor -> 1]]
myPlot[Pr_] :=
Plot[Evaluate[{f[x], f'[x], z[x]} /.
myODESol[G1 /. BC[Pr], G2 /. BC[Pr], Pr]], {x, 0, 20},
PlotStyle -> {RGBColor[0, 0, 1], RGBColor[0, 1, 0], RGBColor[1,
0, 0]}, PlotRange -> All];
myPlot[.1]

See the web page for the plot. Hope this helps.

Brian

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