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System of coupled PDE's initial value problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg38893] System of coupled PDE's initial value problem
  • From: "Alexander Yashin" <yashin at stanford.edu>
  • Date: Fri, 17 Jan 2003 05:39:26 -0500 (EST)
  • Reply-to: "Alexander Yashin" <yashin at stanford.edu>
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

I am trying to solve system of 3 coupled PDE's but getting error
NDSolve::"ndnum": "Encountered non-numerical value for a derivative at t==0"
apparentely the problem is in inital conditions (derivate of one function
that go into the equation
for another is not specified explicitely in the initial moment of time)
any que how to go around this would be greatly appreciated

best regards
alexander

following system of 3 PDE

eq11 = D[a0[t,x], {t, 2}] - D[a0[t, x], {x, 2}] == I*e*(Conjugate[f[t,
x]]*D[f[t, x], {t, 1}] - f[t, x]*Conjugate[D[f[t, x], {t, 1}]]) + e^2*(a0[t,
x] + el*x)*(Abs[f[t, x]])^2;
eq12 = D[ax[t, x], {t, 2}] - D[ax[t, x], {x, 2}] ==I*e*(Conjugate[f[t,
x]]*D[f[t, x], {x, 1}] - f[t, x]*Conjugate[D[f[t, x], {x, 1}]]) +e^2*(ax[t,
x] - el*t)*(Abs[f[t, x]])^2;
eq13 = D[f[t, x], {t, 2}] - D[f[t, x], {x, 2}] + (m^2 - e^2*((a0[t, x] +
el*x)^2))*f[t, x] -2*I*e*((a0[t, x])*D[f[t, x], {t, 1}]) == 0;

with the initial and boundary conditions

eq21 = a0[0, x] == 0; eq22 = Derivative[1, 0][a0][0, x] == 0; eq23 =
a0[t, -x0] == 0; eq24 = a0[t, x0] == 0;
eq31 = ax[0, x] == 0; eq32 = Derivative[1, 0][ax][0, x] == 0; eq33 =
ax[t, -x0] == 0; eq34 = ax[t, x0] == 0;
eq41 = f[0, x] == p; eq42 = Derivative[1, 0][f][0, x] == p1; eq43 =
f[t, -x0] == 0; eq44 = f[t, x0] == 0;

where

p = NIntegrate[Exp[I*k*x - k^2/2*a^2], {k, 0, Infinity}];
p1 = NIntegrate[-I*Sqrt[m^2 + k^2]*Exp[I*k*x - k^2/2*a^2], {k, 0,
Infinity}];

when I try to solve it numerically with

res = NDSolve[{eq11, eq12, eq13, eq21, eq22, eq23, eq24, eq31, eq32, eq33,
eq34, eq41, eq42, eq43, eq44},{a0[t, x], ax[t, x], f[t, x]}, {t, 0, 10},
{x, -x0, x0}, StartingStepSize -> 0.03, MaxSteps -> 4000];

it gives me
NDSolve::"ndnum": "Encountered non-numerical value for a derivative at t==0"
and quits

i nailed it down to presence of D[f[t, x], {x, 1}] derivative in eq12
equation which was not specified at t==0 in the initial conditions
explicitly
but should be known because f[0,x] is specified.




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