Investigation of the factor separation features for basic mathematical functions

FS for some fundamental functions In spite of wide use of the FS method for complex functions, in order to investigate the FS method analytically, we start with only a few examples of mathematical functions chosen from a wide variety of functions. In other words, the FS method is checked out on simple mathematical functions that build up a base for meteorological or, in general, physical fields. A bivariate polynomial A bivariate polynomial is described by the function Let the factors x_i0 be multiplied by a varying coefficient c_i: It is probably preferable to write the function f(x₁, x₂) as f(c₁x₁, c₁x₂), but here we focus on the variable c_i (just simplified writing). It can easily be seen that the values of f and in the different simulations are related through: This function is a form of Taylor series expansion by itself. Hence, this is a trivial case. The four different contributions are naturally expected since the factors' contributions are proportional to the corresponding variables, i.e., ax₁, bx₂, while the synergy is the term that includes both factors, i.e., kx₁x₂. An exponential function An exponential function f(x₁, x₂) = e ax₁+bx₂+d was chosen. In the same way, the factors x_i are multiplied by a varying coefficient c_i and consequently we have The four simulations in this case yield And the four contributions are To examine the consistency of the synergism term, the direct method using the Taylor series is applied.