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I have the following equation with two variables xi and nu.

funcGamma[m_, delta_, xi_] := 
  1/(m (delta + xi) + 1) + 1/(m - m (delta + xi) + 1);

Exp[-2 m k xi^2/(n + 1)] + 
 Exp[-2 funcGamma[
    m, delta, xi] ((n (nu - xi))^2 - 1)]==e

As an example, here are some values for the variables. This is just an example and the values for the variables might change from case to case. I have written the expected range as comment beside every variable.

n=10^5;(*n is a posiitve integer greater than 1*)
k=10^-2 n; (*k is a posiitve integer greater than 1 and less than n*)
delta=10^-1;(*delta is a real number between 0 and 1/2*)
e=10^-20;(*fixed value*)
m=n+k;

I tried FindInstance, Reduce, NSolve with no luck. Note that the equation is to be solved with the constraints given below.

FindInstance[
 Exp[-2 m k xi^2/(n + 1)] + 
    Exp[-2 funcGamma[
       m, delta, xi] ((n (nu - xi))^2 - 1)] == 
   e && delta > nu > xi > 0 && 
  m (delta + xi) \[Element] 
   PositiveIntegers && (n (nu - xi))^2 > 
   1, {nu, xi}, Reals]

How to solve this problem?

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1 Answer 1

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It seems there is no real solution in view of

NMinimize[{ (Exp[-2 m k xi^2/(n + 1)] + 
          Exp[-2 *(1/(m (delta + xi) + 1) + 
   1/(m - m (delta + xi) + 1))*((n (nu - xi))^2 - 1)] - 
    e)^2 , delta > nu > xi > 0 && m (delta + xi) == s &&
   s \[Element]  PositiveIntegers && (n (nu - xi))^2 > 
   1}, {nu, xi, s},WorkingPrecision->MachinePrecision]. 

{3.76882*10^-13, {nu -> 0.0893718, xi -> 0.0841485, s -> 18599}}

In the above the equation is solved as a minimization problem. The artificial variable s is introduced because NMinimize does not directly work with m (delta + xi) \[Element] PositiveIntegers. funcGamma[m, delta, xi] in the target function is replaced by 1/(m (delta + xi) + 1) + 1/(m - m (delta + xi) + 1) to speed up calculations.

Edit. Method->"DifferentialEvolution" is omitted.

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2
  • $\begingroup$ Of course, n=10^5;(*n is a posiitve integer greater than 1*) k=10^-2 n; (*k is a posiitve integer greater than 1 and less than n*) delta=10^-1;(*delta is a real number between 0 and 1/2*) e=10^-20;(*fixed value*) m=n+k; is executed before NMinimize. $\endgroup$
    – user64494
    Commented Nov 8, 2023 at 11:25
  • $\begingroup$ The command of Maple DirectSearch:-SolveEquations with bells cannot find even a feasible initial point. $\endgroup$
    – user64494
    Commented Nov 8, 2023 at 12:14

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