       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>

```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[m][t]*x[t]^2 +
m[t]*x[t]*Derivative[x][t] + (1/3)*Derivative[q][t]*
x[t]^3 +
q[t]*x[t]^2*Derivative[x][t] - (25.2/10^15)*q[t]*x[t] == 0,
0.444*Derivative[x][t]*0.5*r[t]*(y[t] - x[t])^2 + (1/3)*
Derivative[u][
t]*(y[t] - x[t])^3 + (Derivative[y][t] -
Derivative[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[v][t]*(y[t] - 140/10^16)^2 +
v[t]*(y[t] - 140/10^6)*Derivative[y][t] + (1/3)*
Derivative[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., q == 2.342*^8, r == 3.7843*^7,
u == -1.385*^13, v == 0., w == 1.9856*^7,
x == 1.7/10^6, y == 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]+1.63767*10^-18 q^\[Prime]+0.000676838
x^\[Prime]==0,7.68509*10^-8+6.70788*10^-18 u^\[Prime]+0.000062155
x^\[Prime]+0.46512
(-x^\[Prime]+y^\[Prime])==0,1.66614*10^-6+9.7682*10^-12
v^\[Prime]-8.30741*10^-13 w^\[Prime]+0. y^\[Prime]==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[m][t]*x[t]^2 +
>
>      m[t]*x[t]*Derivative[x][t] + (1/3)*Derivative[q][t]*
>
>       x[t]^3 + q[t]*x[t]^2*Derivative[x][t] -
>
>      (25.2/10^15)*q[t]*x[t] == 0,
>
>    0.444*Derivative[x][t]*0.5*r[t]*(y[t] - x[t])^2 +
>
>      (1/3)*Derivative[u][t]*(y[t] - x[t])^3 +
>
>      (Derivative[y][t] - Derivative[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[v][t]*(y[t] - 140/10^16)^2 +
>
>      v[t]*(y[t] - 140/10^6)*Derivative[y][t] +
>
>      (1/3)*Derivative[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., q == 2.342*^8, r == 3.7843*^7,
>
>    u == -1.385*^13, v == 0., w == 1.9856*^7,
>
>    x == 1.7/10^6, y == 4.42/10^6}, {m, q, r, u, v, w, x, y},
>
>   {t, 0, 3600}]
>
>

--
DrMajorBob at bigfoot.com

```

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