Find consistent initial conditions for first-order implicit ODE system with algebraic constraints

`[`

finds
consistent initial conditions for the system of first-order implicit
ordinary differential equations with algebraic constraints returned
by the `y0`

,`yp0`

]
= decic(`eqs`

,`vars`

,`constraintEqs`

,`t0`

,`y0_est`

,`fixedVars`

,`yp0_est`

,`options`

)`reduceDAEToODE`

function.

The call `[eqs,constraintEqs] = reduceDAEToODE(DA_eqs,vars)`

reduces
the system of differential algebraic equations `DA_eqs`

to
the system of implicit ODEs `eqs`

. It also returns
constraint equations encountered during system reduction. For the
variables of this ODE system and their derivatives, `decic`

finds
consistent initial conditions `y0`

, `yp0`

at
the time `t0`

.

Substituting the numerical values `y0`

, `yp0`

into
the differential equations ```
subs(eqs, [t; vars(t); diff(vars(t))],
[t0; y0; yp0])
```

and the constraint equations ```
subs(constr,
[t; vars(t); diff(vars(t))], [t0; y0; yp0])
```

produces zero
vectors. Here, `vars`

must be a column vector.

`y0_est`

specifies numerical estimates for
the values of the variables `vars`

at the time `t0`

,
and `fixedVars`

indicates the values in `y0_est`

that
must not change during the numerical search. The optional argument `yp0_est`

lets
you specify numerical estimates for the values of the derivatives
of the variables `vars`

at the time `t0`

.

Reduce the DAE system to a system of implicit ODEs. Then, find consistent initial conditions for the variables of the resulting ODE system and their first derivatives.

Create the following differential algebraic system.

syms x(t) y(t) DA_eqs = [diff(x(t),t) == cos(t) + y(t),... x(t)^2 + y(t)^2 == 1]; vars = [x(t); y(t)];

Use `reduceDAEToODE`

to convert this system
to a system of implicit ODEs.

[eqs, constraintEqs] = reduceDAEToODE(DA_eqs, vars)

eqs = diff(x(t), t) - y(t) - cos(t) - 2*x(t)*diff(x(t), t) - 2*y(t)*diff(y(t), t) constraintEqs = 1 - y(t)^2 - x(t)^2

Create an option set that specifies numerical tolerances for the numerical search.

options = odeset('RelTol', 10.0^(-7), 'AbsTol', 10.0^(-7));

Fix values `t0 = 0`

for the time and numerical
estimates for consistent values of the variables and their derivatives.

t0 = 0; y0_est = [0.1, 0.9]; yp0_est = [0.0, 0.0];

You can treat the constraint as an algebraic equation for the
variable `x`

with the fixed parameter `y`

.
For this, set `fixedVars = [0 1]`

. Alternatively,
you can treat it as an algebraic equation for the variable `y`

with
the fixed parameter `x`

. For this, set ```
fixedVars
= [1 0]
```

.

First, set the initial value `x(t0) = y0_est(1) = 0.1`

.

fixedVars = [1 0]; [y0,yp0] = decic(eqs,vars,constraintEqs,t0,y0_est,fixedVars,yp0_est,options)

y0 = 0.1000 0.9950 yp0 = 1.9950 -0.2005

Now, change `fixedVars`

to `[0 1]`

.
This fixes `y(t0) = y0_est(2) = 0.9`

.

fixedVars = [0 1]; [y0,yp0] = decic(eqs,vars,constraintEqs,t0,y0_est,fixedVars,yp0_est,options)

y0 = -0.4359 0.9000 yp0 = 1.9000 0.9202

Verify that these initial values are consistent initial values satisfying the equations and the constraints.

subs(eqs, [t; vars; diff(vars,t)], [t0; y0; yp0])

ans = 0 0

subs(constraintEqs, [t; vars; diff(vars,t)], [t0; y0; yp0])

ans = 0

`daeFunction`

| `findDecoupledBlocks`

| `incidenceMatrix`

| `isLowIndexDAE`

| `massMatrixForm`

| `odeFunction`

| `reduceDAEIndex`

| `reduceDAEToODE`

| `reduceDifferentialOrder`

| `reduceRedundancies`