Hi everyone,
just signed up to the forum after failing to find a solution for quite a while.
Where I am stuck is the following. After a little of plasma physics I end up with two equations which are functions of two variables.
F[nccm, TcineV] == 0
G[nccm, TcineV] == 0
Now I am looking for a pair (nccm, TcineV) which satisfies both equations at the same time.
I hoped this would be straight forward, but it turns out not to be as you can see from the attached notebook;
NSolve does not give an answer (just repeats the question), NMinimize just sticks to a solution at the limit of the physical value. Finding a graphical solution sort of works, but is not very convenient, as I will have to evalute these equations for a large number of parameters.
Is there a more clever way to solve these equations/shuffle them in a more appropriate form for Mathematica?
Help would be very much appreciated!
Cheers,
Niels
(*Function F named EnEqZeroF as it originates from energy conservation: *)
EnEqZeroF = -2.46`*^18 -
1/(nccm3^4 TcineV^2)1.9920982277963014`*^61 E^(
49.2`/TcineV) (1.` + 0.01` TcineV)^4 (-404771.5405015526` +
TcineV^(7/2));
(*Function G named partBalZero as it originates from particle balance considerations: *)
partBalZero =
200000000000000000 -
1/(nccm3^3 TcineV^(3/2)) 2.6999484623235618`*^39 E^(
36.900000000000006`/
TcineV) (-4000000000000 + nccm3) (1.` +
0.01` TcineV)^3 \[Sqrt](((-4000000000000 + nccm3) (0.4` +
TcineV) -
4.`*^12 TcineV Log[
1.` + 2.5`*^-13 (-4000000000000 + nccm3)])/(-4000000000000 +
nccm3));
(* Does not work at all: *)
NSolve[{EnEqZeroF == 0, partBalZero == 0}, {TcineV, nccm3}, Reals]
(* Graphical solution seems to work: *)
ContourPlot[{EnEqZeroF == 0, partBalZero == 0}, {TcineV, 0,
15}, {nccm3, 1*^12, 1.5*^14},
AxesLabel -> {"\!\(\*SubscriptBox[\(T\), \(c\)]\) in eV",
"\!\(\*SubscriptBox[\(n\), \(c\)]\) in \
\!\(\*SuperscriptBox[\(cm\), \(-3\)]\)"},
PerformanceGoal -> "Quality"]
(* However, the the graphical solution, Tc~2.5 eV, nc ~ 1.2*10^14 cm^-3 is not found numerically:*)
NMinimize[{Abs[EnEqZeroF - partBalZero], 0 < TcineV < 40,
nccm3 > 4.0*^12}, {TcineV, nccm3}]
NMinimize[{Abs[EnEqZeroF - partBalZero], 0 < TcineV < 40,
nccm3 > 4.0*^12}, {TcineV, nccm3}, Method -> "NelderMead"]
NMinimize[{Abs[EnEqZeroF - partBalZero], 0 < TcineV < 40,
nccm3 > 4.0*^12}, {TcineV, nccm3}, Method -> "DifferentialEvolution"]
NMinimize[{Abs[EnEqZeroF - partBalZero], 0 < TcineV < 40,
nccm3 > 4.0*^12}, {TcineV, nccm3}, Method -> "SimulatedAnnealing"]
NMinimize[{Abs[EnEqZeroF - partBalZero], 0 < TcineV < 40,
nccm3 > 4.0*^12}, {TcineV, nccm3}, Method -> "RandomSearch"]
