# How to solve this equation numerically? [closed]

Two Gaussian functions: f1,f2 of the form f1 = a1*Exp[-t^2/s1^2]. Total area, Atot of (f1+f2) is known. Height of each peak (a1 and a2) is known (experimental data). I don't want to experimentally determine FWHM, although it could be done. Integration gives:

Atot=Sqrt[2 π] (a1 s1 Erf[1/(Sqrt s1)] + a2 s2 Erf[1/(Sqrt s2)])


By plotting the function for Atot, I know this has four solutions for variables {s1,s2} for a given Atot, a1, and a2. However, constraint is that s1 and s2 both > 0, which yields one unique solution. (i.e. Atot = 1.1001, a1 = 0.382, a2 = 0.384)

How to solve in Mathematica?

• Two unknowns require two equations. You need another one involving s1 and s2. – J. M.'s technical difficulties Mar 17 '19 at 12:38
• In addition to what was noted by @J.M., it looks like the total area is not accounting for overlap. It is not obvious from the problem statement whether that is what you want. – Daniel Lichtblau Mar 17 '19 at 13:47

As stated in the comments, the solution is not unique!

This code gives a range of values where both $$s_1$$ and $$s_2$$ are positive (there are other ranges also).

Atot[a1_, a2_][s1_, s2_] :=
Sqrt[2 \[Pi]] (a1 s1 Erf[1/(Sqrt s1)] + a2 s2 Erf[1/(Sqrt s2)]);
Manipulate[
With[
{s1 = q}, FindRoot[Atot[.382, .384][s1, s2] - 1.01, {s2, 1}]
], {q, .255, .735}
]


For example ($$s_1 = .255$$ and $$s_2 = 7.70763$$) or ($$s_1=0.5$$ and $$s_2=0.660332$$), etc. 