# Issue with NDSolve

This could be a stupid issue, but I have been checking for ages. I cannot solve this apparently innocuous system of PDEs:

L = 10; \[Xi] = 1; DD = 1; Subscript[\[Phi], 0] = 1;
l = 1; w = 0.13;
v = 0.1; \[Eta] = 0.5; DD = 1; \[Lambda] = 0.1;

\[CapitalOmega] = l \[Pi]/ L ;

Equations

u[\[Alpha]aa_, a_, A_] = -(1/(2 \[Eta] + \[Lambda]^2*(a +A E^(-Subscript[\[Phi],
0] - \[Phi][x] Cos[\[CapitalOmega] x]))*
Sin[2 \[Theta][x] Sin[\[CapitalOmega] x]]^2))*((a +
A E ^(Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x])) \[Lambda] w (D[\[Theta][
x] Sin[\[CapitalOmega] x], x])^2 \[CapitalOmega]^2 Sin[
2  \[Theta][x] Sin[\[CapitalOmega] x]] Sin[\[Theta][
x] Sin[\[CapitalOmega] x]]^2 -
2*(a +A E^(-Subscript[\[Phi], 0] - \[Phi][x] Cos[\[CapitalOmega] x]))*
v*(Subscript[\[Phi], 0] + \[Phi][x] Cos[\[CapitalOmega] x])^2 \[Theta][
x]  \[CapitalOmega] Cos[\[CapitalOmega] x]  Sin[\[Theta][
x] Sin[\[CapitalOmega] x]] + (a +A E^(-Subscript[\[Phi],
0] - \[Phi][
x] Cos[\[CapitalOmega] x]))*\[Alpha]aa*(Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x])^2*
Sin[2 \[Theta][x] Sin[\[CapitalOmega] x]] +
D[\[Phi][x] Sin[\[CapitalOmega] x],
x] \[CapitalOmega]  Cos[\[Theta][
x] Sin[\[CapitalOmega] x]] (w - \[Lambda] w Cos[
2 \[Theta][x] Sin[\[CapitalOmega] x]]));

eqn1[\[Alpha]aa_, a_, A_] := (-D[\[Theta][x]*Sin[\[CapitalOmega] x], {x, 2}]*(1 -
w Cos[\[Theta][x]*Sin[\[CapitalOmega] x]]^2) (a +
A E^(Subscript[\[Phi], 0] + \[Phi][x] Cos[\[CapitalOmega] x])) +
1/2 (a + A E^(
Subscript[\[Phi],
0] + \[Phi][
x] Cos[\[CapitalOmega] x])) w^2 \[CapitalOmega]^2 (D[\[Theta][
x] Sin[\[CapitalOmega] x], x])^2 Sin[
2 \[Theta][x] Sin[\[CapitalOmega] x]] - (Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x]) (a +
A  E^(-Subscript[\[Phi],
0] - \[Phi][x] Cos[\[CapitalOmega] x]))  v   D[\[Theta][
x] Sin[\[CapitalOmega] x],
x] \[CapitalOmega]  Sin[\[Theta][x]*Sin[\[CapitalOmega] x]] -
u[\[Alpha]aa, a,
A] (1 - \[Lambda] Cos[2 \[Theta][x] Sin[\[CapitalOmega] x]]) - (a +
A E^(Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x])) w  \[CapitalOmega]  D[\[Phi][
x]  Sin[\[CapitalOmega] x],
x] Cos[\[Theta][x]*Sin[\[CapitalOmega] x]]);

eqn2[\[Alpha]aa_, a_, A_] := -D[(
v (a + A E^(-Subscript[\[Phi],
0] - \[Phi][x] Cos[\[CapitalOmega] x])) (Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x])^2 Sin[\[Theta][
x] Sin[\[CapitalOmega] x]] - (DD (1 - \[Xi] Sin[\[Theta][
x] Sin[\[CapitalOmega] x]]^2) -
w Cos[\[Theta][x] Sin[\[CapitalOmega] x]]^2) (a +
A E^(Subscript[\[Phi],
0] + \[Phi][x] Cos[\[CapitalOmega] x])) \[CapitalOmega] D[\[Phi][
x] Cos[\[CapitalOmega] x], x] + \[Lambda]  u[\[Alpha]aa, a,
A] Sin[\[Theta][x] Sin[\[CapitalOmega] x]] Sin[
2 \[Theta][x] Sin[\[CapitalOmega] x]]), x];

Solving

sol1 = {\[Theta], \[Phi]} /. First[NDSolve[{
eqn1[0.07, 0, 1] == 0,
eqn2[0.07, 1, 0] == 0,
\[Theta][0] == 0,
\[Theta][L] == 0,

\!$$\*SuperscriptBox["\[Phi]", TagBox[ RowBox[{"(", "1", ")"}], Derivative], MultilineFunction->None]$$[0] == 0,

\!$$\*SuperscriptBox["\[Phi]", TagBox[ RowBox[{"(", "1", ")"}], Derivative], MultilineFunction->None]$$[L] == 0}, {\[Theta], \[Phi]}, {x, 0, L}] ];

\[Infinity]::indet: Indeterminate expression 0. ComplexInfinity encountered. >>

\[Infinity]::indet: Indeterminate expression 0. ComplexInfinity encountered. >>

\[Infinity]::indet: Indeterminate expression 0. ComplexInfinity encountered. >>

General::stop: Further output of \[Infinity]::indet will be suppressed during this calculation. >>

NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.. >>

ReplaceAll::reps: {<<1>>==0,<<10>>+(0.1 <<2>> (<<1>>))/(1.+<<21>> <<1>>)==0,\[Theta][0]==0,\[Theta][10]==0,(\[Phi]^\[Prime])[0]==0,(\[Phi]^\[Prime])[10]==0} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

-
First of all, try not to use capital (DD), and subscripted symbols. Also, please remove non-code lines (like Equations) and the actual output from your code, and present output as separate block (using Ctrl+Q). Also, u should be defined with := instead of =`. –  István Zachar Dec 13 '13 at 8:37