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I have simplified the equations and decrease the variables to 5, and changed the parameters' value as I think the equations in enter link description here is because of the improper parameters' value.

and new codes are as this:

equa={(AmI[1] - BmI[1]/R3^2) CmI[1] == k1 u0, 
 R3 AmI[1] DmI[1] == (BmI[1] DmI[1])/R3, 
 (AmI[1] - BmI[1]/R2^2) CmI[1] == (R1^(-((2 π)/β))
    R2^(-((π + β)/β)) (R1^((2 π)/β) - R2^((2 π)/β)) β Aki[1]* 
    (Sin[thetai] + Sin[thetai + β]))/(-π^2 + β^2), 
 (AmI[1] - BmI[1]/R2^2) DmI[1] == (R1^(-((2 π)/β))
    R2^(-((π + β)/β)) (R1^((2 π)/β) - R2^((2 π)/β)) β Aki[1]* 
    (Cos[thetai] + Cos[thetai + β]))/(π^2 - β^2), 
 Aki[1] == -((2 β (R2^2 AmI[1] + BmI[1]) ((Cos[thetai] + Cos[thetai + β]) DmI[1] - 
      CmI[1] (Sin[thetai] + Sin[thetai + β])))/(R2 (π - β) (π + β)))}

system = equa;

vars = {AmI[1], BmI[1], CmI[1], DmI[1], Aki[1]};
parameters = {u0 -> 4*π*10^(-7), R1 -> 4/100, R2 -> 7/100, 
  R3 -> 8/100, β -> π/4, k1 -> (11/10)^5, L -> 0.1, 
  N1 -> 50, K -> 50, thetai -> π/6};
givenPoint = {{AmI[1], 0.1}, {BmI[1], 0.1}, {CmI[1], 0.1}, 
  {DmI[1], 0.1}, {Aki[1], 0.1 + I}};

NMinimize[# . # &[equa /. Equal -> Subtract /. parameters], vars]

{1.53176*10^-12, {AmI[1] -> -1.75396*10^-6, BmI[1] -> 8.53681*10^-9, CmI[1] -> -0.410309, DmI[1] -> 0.317123, Aki[1] -> 2.23402*10^-13}}

And it seems that the object is approximate to 0, however it is not 0, so I cannot get the solution by Solve.

The most important question is how to analyze this nonlinear equations mathematically with 5 variables? For example using MatrixRank or other functions to make sure in which condition the equations will and will not have solution.

I do not know if it is effective to use MatrixRank for nonlinear equations.

By the way, I do not know which of vars should be real and which complex and maybe the initial values are improper.

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  • $\begingroup$ From your 2nd equation AmI==BmI/R3^2 With that your 1st equation is 0==k1*u0==(11/10)^5*4*π*10^(-7) and that is False $\endgroup$
    – Bill
    Jul 6, 2022 at 4:43
  • $\begingroup$ Do you know which of vars should be real and which complex? $\endgroup$ Jul 6, 2022 at 8:50
  • $\begingroup$ So MatrixRank is not useful for nonlinear equations, but your system is polynomial, and one can use GroebnerBasis to conclusively check if there are (complex) solutions. You can use a generic command such as Solve and Reduce but then it may be less clear what they do in the background. But GroebnerBasis has a clear definition. See here for examples, in particular under "Scope" the example "no common roots". See also weak Nullstellensatz. $\endgroup$
    – user293787
    Jul 6, 2022 at 10:23
  • $\begingroup$ @Bill The 2nd equation can alternatively be satisfied by DmI == 0. The whole system may still not have a solution, I have not checked. $\endgroup$
    – user293787
    Jul 6, 2022 at 10:27
  • $\begingroup$ Thanks @user293787 If DmI==0 and thus the second equation is True then Simplify shows the 4th equation is Aki==0 and the system is again False $\endgroup$
    – Bill
    Jul 6, 2022 at 14:15

2 Answers 2

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Try Reduce to analyze the nonlinear equations:

Reduce[equa /. parameters, vars]
(* False*)

The result confirms @Bill's helpful comment!

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1, Surely even if DmI[1]=0 and Aki[1]=0, there is still one equation that cannot satisfy.

2, I have tried to change the first 2 equations,

 (AmI[1] - BmI[1]/R3^2) CmI[1] == k1 u0,    
          R3 AmI[1] DmI[1] - (BmI[1] DmI[1])/R3==0, 

Even make k1 u0=0 and therefore the first 2 eqautions are equal to 0 respectively, there is no solution.

3, Changing the first 2 eqautions to

(AmI[1] - BmI[1]/R3^2) CmI[1] == C1, 
          R3 AmI[1] DmI[1] - (BmI[1] DmI[1])/R3==C2,

and C1is not equal to 0, C2is not equal to 0, there is still no solution.

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