Why can't the two variables' equations be solved?

Question

In[1]:= Solve[{\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*\ k + 1)\)*\ \[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2\ *k + 1)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(398\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]*\ 0.5]\)\) == 0.58, \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*\ k + 1)\)*\ \[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2\ *k + 1)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(1019\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]*\ 0.5]\)\) == 0.42, \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*\ k + 1)\)*\ \[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2\ *k + 1)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(7709\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]*\ 0.5]\)\) == 0.15}, {d, \[Beta]}]

During evaluation of In1:= Solve::inex: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help.

Out[1]= Solve[{0.0606305 MittagLefflerE[\[Beta], -441 398^\[Beta] d \
\[Pi]^2] - 
    0.0670126 MittagLefflerE[\[Beta], -361 398^\[Beta] d \[Pi]^2] + 
    0.0748964 MittagLefflerE[\[Beta], -289 398^\[Beta] d \[Pi]^2] - 
    0.0848826 MittagLefflerE[\[Beta], -225 398^\[Beta] d \[Pi]^2] + 
    0.0979415 MittagLefflerE[\[Beta], -169 398^\[Beta] d \[Pi]^2] - 
    0.115749 MittagLefflerE[\[Beta], -121 398^\[Beta] d \[Pi]^2] + 
    0.141471 MittagLefflerE[\[Beta], -81 398^\[Beta] d \[Pi]^2] - 
    0.181891 MittagLefflerE[\[Beta], -49 398^\[Beta] d \[Pi]^2] + 
    0.254648 MittagLefflerE[\[Beta], -25 398^\[Beta] d \[Pi]^2] - 
    0.424413 MittagLefflerE[\[Beta], -9 398^\[Beta] d \[Pi]^2] + 
    1.27324 MittagLefflerE[\[Beta], -398^\[Beta] d \[Pi]^2] == 0.58, 
  0.0606305 MittagLefflerE[\[Beta], -441 1019^\[Beta] d \[Pi]^2] - 
    0.0670126 MittagLefflerE[\[Beta], -361 1019^\[Beta] d \[Pi]^2] + 
    0.0748964 MittagLefflerE[\[Beta], -289 1019^\[Beta] d \[Pi]^2] - 
    0.0848826 MittagLefflerE[\[Beta], -225 1019^\[Beta] d \[Pi]^2] + 
    0.0979415 MittagLefflerE[\[Beta], -169 1019^\[Beta] d \[Pi]^2] - 
    0.115749 MittagLefflerE[\[Beta], -121 1019^\[Beta] d \[Pi]^2] + 
    0.141471 MittagLefflerE[\[Beta], -81 1019^\[Beta] d \[Pi]^2] - 
    0.181891 MittagLefflerE[\[Beta], -49 1019^\[Beta] d \[Pi]^2] + 
    0.254648 MittagLefflerE[\[Beta], -25 1019^\[Beta] d \[Pi]^2] - 
    0.424413 MittagLefflerE[\[Beta], -9 1019^\[Beta] d \[Pi]^2] + 
    1.27324 MittagLefflerE[\[Beta], -1019^\[Beta] d \[Pi]^2] == 0.42, 
  0.0979415 MittagLefflerE[\[Beta], -13^(2 + \[Beta])
         593^\[Beta] d \[Pi]^2] + 
    0.0606305 MittagLefflerE[\[Beta], -441 7709^\[Beta] d \[Pi]^2] - 
    0.0670126 MittagLefflerE[\[Beta], -361 7709^\[Beta] d \[Pi]^2] + 
    0.0748964 MittagLefflerE[\[Beta], -289 7709^\[Beta] d \[Pi]^2] - 
    0.0848826 MittagLefflerE[\[Beta], -225 7709^\[Beta] d \[Pi]^2] - 
    0.115749 MittagLefflerE[\[Beta], -121 7709^\[Beta] d \[Pi]^2] + 
    0.141471 MittagLefflerE[\[Beta], -81 7709^\[Beta] d \[Pi]^2] - 
    0.181891 MittagLefflerE[\[Beta], -49 7709^\[Beta] d \[Pi]^2] + 
    0.254648 MittagLefflerE[\[Beta], -25 7709^\[Beta] d \[Pi]^2] - 
    0.424413 MittagLefflerE[\[Beta], -9 7709^\[Beta] d \[Pi]^2] + 
    1.27324 MittagLefflerE[\[Beta], -7709^\[Beta] d \[Pi]^2] == 
   0.15}, {d, \[Beta]}]

as you can see in the picture, what does the inexact numbers mean in the warning?

Answer 1

Clear["Global`*"]

Use exact numbers in equations (e.g., 1/2 rather than the inexact 0.5)

eqns = {Sum[(4/((2*k + 1)*Pi))*
             MittagLefflerE[β, (-d)*(2*k + 1)^2*Pi^2*
                 398^β]*Sin[(2*k + 1)*Pi*1/2], 
      {k, 0, 10}] == 29/50, 
       Sum[(4/((2*k + 1)*Pi))*
             MittagLefflerE[β, (-d)*(2*k + 1)^2*Pi^2*
                 1019^β]*Sin[(2*k + 1)*Pi*1/2], 
      {k, 0, 10}] == 21/50, 
       Sum[(4/((2*k + 1)*Pi))*
             MittagLefflerE[β, (-d)*(2*k + 1)^2*Pi^2*
                 7709^β]*Sin[(2*k + 1)*Pi*1/2], 
      {k, 0, 10}] == 3/20} //
   Simplify;

Length@eqns

(* 3 *)

Variables[Level[eqns, {-1}]]

(* {d, β} *)

There are three equations with only two variables. The system is overdetermined. Approach it as a minimization problem.

The constraints are

Cases[eqns, MittagLefflerE[a_, _] :> a > 0, Infinity] // Union

(* {β > 0} *)

Minimizing,

{min, arg} = NMinimize[
  {Total[(#[[1]] - #[[-1]])^2 & /@ eqns], β > 0}, {d, β}, 
  AccuracyGoal -> 10, PrecisionGoal -> 9, WorkingPrecision -> 20]

(* General::ovfl: Overflow occurred in computation.

General::ovfl: Overflow occurred in computation.

General::ovfl: Overflow occurred in computation.

General::stop: Further output of General::ovfl will be suppressed during this calculation.

NMinimize::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations.

{0.25640235187367699755, {d -> 0.70884195297684036358, β -> 
   0.0029224816962366914900}} *)

(#[[1]] - #[[-1]] & /@ eqns) /. arg

(* {-0.42886059095864565614, -0.26922282559129573706, 0.000124807520} *)

Ideally these would be zero. This is as close as the system gets.

Answer 2

I believe that the reason you get inexact coefficients is due to your use of decimals. Replacing all the decimals with their equivalent fractions results in this equation:

Solve[{\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k\.01 = \.010\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*k\.01 + \.011)\)*\[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2*k\.01 + \.011)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(398\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]/2]\)\) == 58/100, \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k\.01 = \.010\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*k\.01 + \.011)\)*\[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2*k\.01 + \.011)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(1019\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]/2]\)\) == 42/100, \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(k\.01 = \.010\), \(10\)]\(
\*FractionBox[\(4\), \(\((2*k\.01 + \.011)\)*\[Pi]\)]*
     MittagLefflerE[\[Beta], \((\((\(-d\))\)*\ 
\*SuperscriptBox[\((2*k\.01 + \.011)\), \(2\)]\ *
\*SuperscriptBox[\(\[Pi]\), \(2\)]\ *
\*SuperscriptBox[\(7709\), \(\[Beta]\)])\)]\ *
     Sin[\((2\ *k + 1)\)*\ \[Pi]/2]\)\) == 15/100}, {d, \[Beta]}]

It returns a warning saying that

I suspect that this equation is beyond the scope of Solve, but the solving algorithm is not sure if it is or not, so it asks you for symbolic representations instead of decimals. I tried using Reduce as well, but it doesn't return, so I would assume you would have to do some manual solving first before feeding it to Solve.