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]))
Print["gamma_1,1,1"]
c = 0.5*(Part[bbb, 2, 1]*D[e, u] +
Part[bbb, 2, 2]*(2*D[h, u] - D[e, v]))
Print["gamma_1,1,2"]
d = 0.5*(Part[bbb, 1, 1]*D[e, v] + Part[bbb, 1, 2]*D[g, u])
Print["gamma_1,2,1"]
i = 0.5*(Part[bbb, 2, 1]*D[e, v] + Part[bbb, 2, 2]*D[g, u])
Print["gamma_1,2,2"]
j = 0.5*(Part[bbb, 1, 1]*D[e, v] + Part[bbb, 1, 2]*D[g, u])
Print["gamma_2,1,1"]
k = 0.5*(Part[bbb, 1, 1]*(2*D[h, v] - D[g, u]) +
Part[bbb, 1, 2]*D[g, v])
Print["gamma_2,2,1"]
l = 0.5*(Part[bbb, 2, 1]*D[e, v] + Part[bbb, 2, 2]*D[g, u])
Print["gamma_2,1,2"]
m = 0.5*(Part[bbb, 2, 1]*(2*D[h, v] - D[g, u]) +
Part[bbb, 2, 2]*D[g, v])
Print["gamma_2,2,2"]
(* 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}]