# Solving a system of nonlinear equations self-consistently

I am trying to solve a set of three non-linear equations in Mathematica. I need help with them. The Mathematica code (in plain text format) is attached below. If you copy & paste the code below into a Notebook the actual input will become clear. It is too complicated to be typed up, hence I am displaying it in this way.

I cannot seem to converge to a solution which is correct. I can get answers but I don't think that makes sense based on the problem that I am doing. Neither χ nor Δ should go to zero.

Defining the functions first (copy & paste in a Notebook)

tχfn[x_?NumericQ,y_?NumericQ,μ_?NumericQ,χ_?NumericQ,Δ_?NumericQ] :=
(Cos[x]+Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y]-μ)^2/Sqrt[4 χ^2 (Cos[x]+
Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y]-μ)^2+4 Δ^2 (Cos[x]+Cos[x/2+Sqrt[3]/2 y]+
Cos[x/2-Sqrt[3]/2 y])^2]

tΔfn[x_?NumericQ,y_?NumericQ,μ_?NumericQ,χ_?NumericQ,Δ_?NumericQ] :=
(Cos[x]+Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y])^2/Sqrt[4 χ^2 (Cos[x]+
Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y]-μ)^2+4 Δ^2 (Cos[x]+Cos[x/2+Sqrt[3]/2 y]+
Cos[x/2-Sqrt[3]/2 y])^2]

tμfn[x_?NumericQ,y_?NumericQ,μ_?NumericQ,χ_?NumericQ,Δ_?NumericQ] :=
(χ^2 (Cos[x]+Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y]-μ))/Sqrt[4 χ^2 (Cos[x]+
Cos[x/2+Sqrt[3]/2 y]+Cos[x/2-Sqrt[3]/2 y]-μ)^2+4 Δ^2 (Cos[x]+Cos[x/2+Sqrt[3]/2 y]+
Cos[x/2-Sqrt[3]/2 y])^2]


Setting up the non-linear solution (copy paste in a notebook)

FindRoot[
{
NIntegrate[tχfn[x, y, μ, χ, Δ], {x, (4*Pi)/3, (10*Pi)/3}, {y, 0, (4*Pi)/Sqrt[3]},
Method -> "MultiPeriodic"] == 1.5,
NIntegrate[tΔfn[x, y, μ, χ, Δ], {x, (4*Pi)/3, (10*Pi)/3}, {y, 0, (4*Pi)/Sqrt[3]},
Method -> "MultiPeriodic"] == 1.5,
NIntegrate[tμfn[x, y, μ, χ, Δ], {x, (4*Pi)/3, (10*Pi)/3}, {y, 0, (4*Pi)/Sqrt[3]},
Method -> "MultiPeriodic"] == 0.125
},
{{μ, 0.02}, {χ, 1.}, {Δ, 1.}},
Evaluated -> False
]


I have tried without the "MultiPeriodic" option, with "MonteCarlo" option, and even used the basic default option.

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## migrated from stackoverflow.comOct 18 '12 at 16:20

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You might be interested in this question... – J. M. Oct 18 '12 at 16:25
I need the exact numbers (i.e. the solution). The visual way is an interesting way to look at it. I agree. – user1755051 Oct 18 '12 at 16:31
"I need the exact numbers" - you won't get exact numbers from NIntegrate[] or FindRoot[]; all they do is give approximations. The method in that post I linked to computes numerical solutions as well, though it uses plotting functions internally. – J. M. Oct 18 '12 at 16:34
Yes. I see you did provide the solution. But I have integrals which I need to solve before obtaining the solution to the non-linear equation. I am not quite sure how to manage that in the approach that you mentioned. Thanks. – user1755051 Oct 18 '12 at 17:29
@J.M. Oh! That fertilization moment snapshot! – Dr. belisarius Oct 18 '12 at 20:52