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MathGroup Archive 2007

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Re: System of differential-algebraic equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78328] Re: [mg78271] System of differential-algebraic equations
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Thu, 28 Jun 2007 04:30:54 -0400 (EDT)
  • References: <20812896.1182941885849.JavaMail.root@m35>
  • Reply-to: drmajorbob at bigfoot.com

You don't necessarily need initial conditions for the derivatives, but you  
DO need initial conditions that are consistent. In this case you don't  
have that:

eqns = {m[t]*x[t] + q[t]*x[t]^2 ==
     2.75, (12.6/10^15)*m[t]^2 - (10.2/10^16)*m[t]*r[t] - (20.4/10^16)*
       m[t]*u[t]*y[t] + (20.4/10^16)*m[t]*u[t]*x[t] - (92.4084/10^15)*
       q[t] == 0,
    r[t]*(y[t] - x[t]) + u[t]*(y[t] - x[t])^2 ==
     0.444, (10.2/10^16)*r[t]^2 - (9.82/10^12)*r[t]*
       v[t] - (389.9256/10^16)*u[t] == 0,
    v[t] + 2*w[t]*(y[t] - 140/10^16) == 0,
    0.5*Derivative[1][m][t]*x[t]^2 +
      m[t]*x[t]*Derivative[1][x][t] + (1/3)*Derivative[1][q][t]*
       x[t]^3 +
      q[t]*x[t]^2*Derivative[1][x][t] - (25.2/10^15)*q[t]*x[t] == 0,
    0.444*Derivative[1][x][t]*0.5*r[t]*(y[t] - x[t])^2 + (1/3)*
       Derivative[1][u][
        t]*(y[t] - x[t])^3 + (Derivative[1][y][t] -
         Derivative[1][x][t])*(r[t]*(y[t] - x[t]) +
         u[t]*(y[t] - x[t])^2) - (20.4/10^16)*u[t]*(y[t] - x[t]) == 0,
    0.5*Derivative[1][v][t]*(y[t] - 140/10^16)^2 +
      v[t]*(y[t] - 140/10^6)*Derivative[1][y][t] + (1/3)*
       Derivative[1][w][t]*(y[t] - 140/10^6)^3 + (19.64/10^12)*
       w[t]*(140/10^6 - y[t]) + w[t]*(y[t] - 140/10^6)^2*y[t] == 0};

inits = {m[0] == 0., q[0] == 2.342*^8, r[0] == 3.7843*^7,
    u[0] == -1.385*^13, v[0] == 0., w[0] == 1.9856*^7,
    x[0] == 1.7/10^6, y[0] == 4.42/10^6};

initRules = inits /. Equal -> Rule;
eqns /. {t -> 0} /. initRules

{{False,False,False,False,False,-1.00331*10^-11+1.445*10^-12  
m^\[Prime][0]+1.63767*10^-18 q^\[Prime][0]+0.000676838  
x^\[Prime][0]==0,7.68509*10^-8+6.70788*10^-18 u^\[Prime][0]+0.000062155  
x^\[Prime][0]+0.46512  
(-x^\[Prime][0]+y^\[Prime][0])==0,1.66614*10^-6+9.7682*10^-12  
v^\[Prime][0]-8.30741*10^-13 w^\[Prime][0]+0. y^\[Prime][0]==0}}

As you can see, five equations are violated at the initial values, no  
matter WHAT the derivatives might be. The errors are not small, either:

five = Take[eqns /. t -> 0 /. Equal -> Subtract, 5];
five /. initRules

{-2.74932, -0.000021642, 0.02112, 2.00078, 175.527}

So... at minimum, you need initial conditions for which the equations are  
solvable at 0.

Bobby

On Wed, 27 Jun 2007 04:29:43 -0500, José Luis Gómez  
<jose.luis.gomez at itesm.mx> wrote:

> Dear Mathematica Group.
>
>
> A colleague has asked me help to solve a system of 8 algebraic and
> differential equations. The system is included below, at the end of this
> e-mil, in InputForm.
>
> Mathematica 6.0 NDSolve command replies with this message:
>
>
> NDSolve::icfail: Unable to find initial conditions which satisfy the
> residual function within specified tolerances.  Try giving initial
> conditions for both values and derivatives of the functions.
>
>
> Now, my colleague does not want to give initial conditions for the
> derivatives, because he does not have actual information about those  
> values.
> We fool around a bit in the documentation, play a little bit with
> AccuracyGoal, and PrecisionGoal, and with different methods specified by
> Method, but we were not able to obtain an answer.
>
>
> Does anyone have a suggestion for us? Can we avoid the use of initial  
> values
> for the derivatives?
>
>
> The system is included below. Thanks in advance for any advice.
>
>
> Jose Luis Gomez-Munoz
>
>
>
>
> NDSolve[{m[t]*x[t] + q[t]*x[t]^2 == 2.75,
>
>    (12.6/10^15)*m[t]^2 - (10.2/10^16)*m[t]*r[t] -
>
>      (20.4/10^16)*m[t]*u[t]*y[t] + (20.4/10^16)*m[t]*u[t]*x[t] -
>
>      (92.4084/10^15)*q[t] == 0,
>
>    r[t]*(y[t] - x[t]) + u[t]*(y[t] - x[t])^2 == 0.444,
>
>    (10.2/10^16)*r[t]^2 - (9.82/10^12)*r[t]*v[t] -
>
>      (389.9256/10^16)*u[t] == 0, v[t] + 2*w[t]*(y[t] - 140/10^16) ==
>
>     0, 0.5*Derivative[1][m][t]*x[t]^2 +
>
>      m[t]*x[t]*Derivative[1][x][t] + (1/3)*Derivative[1][q][t]*
>
>       x[t]^3 + q[t]*x[t]^2*Derivative[1][x][t] -
>
>      (25.2/10^15)*q[t]*x[t] == 0,
>
>    0.444*Derivative[1][x][t]*0.5*r[t]*(y[t] - x[t])^2 +
>
>      (1/3)*Derivative[1][u][t]*(y[t] - x[t])^3 +
>
>      (Derivative[1][y][t] - Derivative[1][x][t])*
>
>       (r[t]*(y[t] - x[t]) + u[t]*(y[t] - x[t])^2) -
>
>      (20.4/10^16)*u[t]*(y[t] - x[t]) == 0,
>
>    0.5*Derivative[1][v][t]*(y[t] - 140/10^16)^2 +
>
>      v[t]*(y[t] - 140/10^6)*Derivative[1][y][t] +
>
>      (1/3)*Derivative[1][w][t]*(y[t] - 140/10^6)^3 +
>
>      (19.64/10^12)*w[t]*(140/10^6 - y[t]) + w[t]*(y[t] - 140/10^6)^2*
>
>       y[t] == 0, m[0] == 0., q[0] == 2.342*^8, r[0] == 3.7843*^7,
>
>    u[0] == -1.385*^13, v[0] == 0., w[0] == 1.9856*^7,
>
>    x[0] == 1.7/10^6, y[0] == 4.42/10^6}, {m, q, r, u, v, w, x, y},
>
>   {t, 0, 3600}]
>
>



-- 
DrMajorBob at bigfoot.com


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