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Help with NDSolve

Posted 9 years ago

Hello, I'm having trouble getting NDSolve to solve a system of DEs. Here's my code below, can you give me an idea of what I need to correct please? THANKS SO MUCH!!!

*User inputs chart functions. ee = x (u, v); ff = y (u, v);

 gg = cos (v);*)

ee = Cos[u]*Sin[v]
ff = Sin[u]*Sin[v]
gg = Cos[v]

eeh[u_, v_] := ee[x][u, v];

ffh[u_, v_] := ff[x][u, v];

ggh[u_, v_] := gg[x][u, v]

(*User inputs min and max values for u and v*)

u0 = -Pi
u1 = Pi

v0 = -Pi
v1 = Pi

(*(User inputs  initial p0 and q0)*)

p0 = 1
q0 = 1

tmin = 0
tmax = 200

a = mat = {{D[ee, u], D[ee, v]}, {D[ff, u], D[ff, v]}, {D[gg, u], 
    D[gg, v]}}

(a) // MatrixForm
Print["Derivative Matrix"]

e = D[ee, u]*D[ee, u] + D[ff, u]*D[ff, u] + D[gg, u]*D[gg, u]
e = TrigReduce[e]

h = D[ee, u]*D[ee, v] + D[ff, u]*D[ff, v] + D[gg, u]*D[gg, v]
h = TrigReduce[h]

g = D[ee, v]*D[ee, v] + D[ff, v]*D[ff, v] + D[gg, v]*D[gg, v]
g = TrigReduce[g]

bb = mat = {{e, h}, {h, g}}
bb // MatrixForm
Print["Metric, g"]
bbb = Inverse[bb]
Print["Christoffel Equations: gamma_i,j,k"]

b = 0.5*(Part[bbb, 1, 1]*D[e, u] + 
    Part[bbb, 1, 2]*(2*D[h, u] - D[e, v]))

c = 0.5*(Part[bbb, 2, 1]*D[e, u] + 
    Part[bbb, 2, 2]*(2*D[h, u] - D[e, v]))


d = 0.5*(Part[bbb, 1, 1]*D[e, v] + Part[bbb, 1, 2]*D[g, u])


i = 0.5*(Part[bbb, 2, 1]*D[e, v] + Part[bbb, 2, 2]*D[g, u])

j = 0.5*(Part[bbb, 1, 1]*D[e, v] + Part[bbb, 1, 2]*D[g, u])

k = 0.5*(Part[bbb, 1, 1]*(2*D[h, v] - D[g, u]) + 
    Part[bbb, 1, 2]*D[g, v])

l = 0.5*(Part[bbb, 2, 1]*D[e, v] + Part[bbb, 2, 2]*D[g, u])

m = 0.5*(Part[bbb, 2, 1]*(2*D[h, v] - D[g, u]) + 
    Part[bbb, 2, 2]*D[g, v])

(* gdesolv[u0,v0, p0, q0, t0, tmin, tmax]:=*)
soln4 = NDSolve[ {u'[t] == p[t], v'[t] == q[t],
   p'[t] + b*p[t]^2 + 2*c*p[t]*q[t] + i*q[t]^2 == 0,
   q'[t] + j*p[t]^2 + 2*l*p[t]*q[t] + m*q[t]^2 == 0, u[t0] == u0, 
   v[t0] == v0, p[t0] == du0, q[t0] == dv0},
  {u[t], v[t], p[t], q[t]}, {t, tmin, tmax}]

ParametricPlot[Evaluate[{u[t], v[t]} /. soln4], {t, tmin, tmax}]

ParametricPlot3D[{ee, ff, gg}, {u, u0, u1}, {v, v0, v1}]
Posted 9 years ago

One of your problems seems to be that you are using v as a variable in all the code before the NDSolve and the results from part of that code are used as parameters within the NDSolve, but in the NDSolve you are using v[t] as a function of t. Mixing those two different uses of v is going to give you grief.

Another problem is that you don't appear in these code fragments to have assigned concrete numeric values to t0, du0 or dv0. All the "N functions", like NDsolve demand that all variables, except those being solved for, have been assigned numeric values first.

You are perhaps using NDSolve in a somewhat unconventional way. If you look at the examples in the documentation I don't think you will see it being used in some of the ways that you are trying. I cannot be certain, given the other problems, whether this is going to cause problems, but when NDSolve is being used in an unconventional way that seems to often enough result in confusing problems.

I'm not sure what to make of the rest of the code. I assume this is either a coding experiment in progress or a collection of fragments from something bigger.

If you use Simplify on the result of each of your intermediate calculations and look at those results then this might help you confirm that the values are correct or expose errors that need to be tracked down and corrected.

POSTED BY: Bill Simpson
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