Re: Solve::ivar: 0 is not a valid variable.

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• Subject: [mg129086] Re: Solve::ivar: 0 is not a valid variable.
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Fri, 14 Dec 2012 02:59:24 -0500 (EST)
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• References: <20121213090851.2041868B5@smc.vnet.net>

```Use exact values until the last moment to enable simplifications when possible.

gamma = 3/2;
m = 0;
C0 = 1;
CA = 2;
CF = Sqrt[C0^2 + CA^2];
CT = (C0 CA)/CF;
Ce = 1/2;
CAe = 5;
CFe = Sqrt[Ce^2 + CAe^2];
CTe = (Ce CAe)/CFe;
delta = (2 C0^2 + gamma CA^2)/(2 Ce^2 + gamma CAe^2);
kza = 1;

When using numeric techniques your function definitions should be
restricted to numeric arguments.

k0[x_?NumericQ] :=
Sqrt[-(((x^2 - C0^2) (x^2 - CA^2))/((C0^2 + CA^2) (x^2 - CT^2)))]

ke[x_?NumericQ] :=
Sqrt[-(((x^2 - Ce^2) (x^2 - CAe^2))/((Ce^2 + CAe^2) (x^2 - CTe^2)))]

You used the wrong syntax for NSolve

NSolve[(delta*(x^2 - CAe^2)*
k0[x]*BesselI[m + 1,
k0[x]*kza])/BesselI[m,
k0[x]*kza] +
((x^2 - CA^2)*ke[x]*
BesselK[m + 1, ke[x]*
kza])/BesselK[m,
ke[x]*kza] == 0, x, Reals]

NSolve::nsmet:This system cannot be solved with the methods available
to NSolve. >>

NSolve[
(4*(-25 + x^2)*BesselI[1,
k0[x]]*k0[x])/
(19*BesselI[0, k0[x]]) +
((-4 + x^2)*BesselK[1,
ke[x]]*ke[x])/BesselK[0,
ke[x]] == 0, x, Reals]

When NSolve cannot be used, use FindRoot but you will need estimates
of the roots. Plotting the function can help with this:

Plot[(delta (x^2 - CAe^2) k0[x] BesselI[m + 1, k0[x] kza])/
BesselI[m,
k0[x] kza] + ((x^2 - CA^2) ke[x] BesselK[m + 1, ke[x] kza])/
BesselK[m, ke[x] kza], {x, -5.1, 5.1}, PlotRange -> {-20, 20}]

Plot[(delta (x^2 - CAe^2) k0[x] BesselI[m + 1, k0[x] kza])/
BesselI[m,
k0[x] kza] + ((x^2 - CA^2) ke[x] BesselK[m + 1, ke[x] kza])/
BesselK[m, ke[x] kza], {x, 0.89, 0.94}]

FindRoot[(delta (x^2 - CAe^2) k0[x] BesselI[m + 1, k0[x] kza])/
BesselI[m,
k0[x] kza] + ((x^2 - CA^2) ke[x] BesselK[m + 1, ke[x] kza])/
BesselK[m, ke[x] kza] == 0, {x, #}] & /@
Flatten[{-1.9, -0.93, -0.904, Range[-.897, -.894, .001],
Range[.894, .897, .001], .904, .93, 1.9}] // Chop

{{x -> -2.}, {x -> -0.928239}, {x -> -0.898603}, {x -> -0.898603}, {x \
-> -0.895811}, {x -> -0.895102}, {x ->
0.928239}, {x -> -0.928239}, {x -> 0.895102}, {x ->
0.895811}, {x -> 0.898603}, {x -> 0.898603}, {x -> 0.928239}, {x ->
2.}}

The Chop is necessary to remove small imaginary artifacts arising from
the numerical techniques.

These are not all of the roots since the function is highly
oscillatory in some ranges.

Bob Hanlon

On Thu, Dec 13, 2012 at 4:08 AM, Ding Yuan <gardener_2003 at hotmail.com> wrote:
> Hello,
>
> I am fresh in Mathematica. Can anyone explain what is happening in this code?
>
> Cheers
>
> Ding
>
>
> ------------
> gamma = 1.5; m = 0;
> C0 = 1.; CA = 2;
> CF = (C0^2 + CA^2)^0.5; CT = (C0 CA)/CF;
> Ce = 0.5; CAe = 5 ;
> CFe = (Ce^2 + CAe^2)^0.5; CTe = (Ce CAe)/CFe;
> delta = (2 C0^2 + gamma CA^2)/(2 Ce^2 + gamma CAe^2);
> kza = 1.;
> k0[x_] := (-(((x^2 - C0^2) (x^2 - CA^2) )/((C0^2 + CA^2) (x^2 -
>       CT^2))))^0.5
> ke[x_] := (-(((x^2 - Ce^2) (x^2 - CAe^2))/((Ce^2 + CAe^2) (x^2 -
>       CTe^2))))^0.5
> NSolve[(delta (x^2 - CAe^2) k0[x] BesselI[m + 1, k0[x] kza])/
>    BesselI[m,
>     k0[x] kza] + ((x^2 - CA^2) ke[x] BesselK[m + 1, ke[x] kza])/
>    BesselK[m, ke[x] kza] == 0, {x, 0, 10}, Reals]
>
> NSolve::ivar: 0 is not a valid variable. >>
> NSolve[(0.0941502 (-25 +
>       x^2) (-(((-4 + x^2) (-1. + x^2))/(-0.8 + x^2)))^0.5 BesselI[1,
>      0.447214 (-(((-4 + x^2) (-1. + x^2))/(-0.8 + x^2)))^0.5])/
>    BesselI[0,
>     0.447214 (-(((-4 + x^2) (-1. + x^2))/(-0.8 + x^2)))^0.5] + (
>    0.199007 (-4 +
>       x^2) (-(((-25 + x^2) (-0.25 + x^2))/(-0.247525 +
>        x^2)))^0.5 BesselK[1,
>      0.199007 (-(((-25 + x^2) (-0.25 + x^2))/(-0.247525 +
>          x^2)))^0.5])/
>    BesselK[0,
>     0.199007 (-(((-25 + x^2) (-0.25 + x^2))/(-0.247525 +
>         x^2)))^0.5] == 0, {x, 0, 10}, Reals]
>

```

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