Solutions for Series of Exponential Equations in Terms of Lambert-W Function and Fundamental Constants

Series of exponential equations in the form of 1 n x y n x y y +         = were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution 2 ln10 10 10 137.129 ln10 10 W −       =   −     is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.


Introduction
Exponential equations are widely used in natural and social sciences.In this paper, we considered series of exponential equations and solved them graphically, numerically, and analytically in terms of Lambert-W function.One equation connected to the fine structure constant, was derived in terms of the fundamental constants and led to a new equation.The Lambert-W function for real variables is defined by the equation [4] and it has applications in Planks spectral distribution law [5] [6], QCD renormalization [7], solar cells [8], bio-chemical kinetics [9], optics [10], population growth Journal of Applied Mathematics and Physics and water movement in soil [11].
Considering the series of exponential equations defined by the following equation where x, y, n are real variables.
Taking log y on both sides of the Equation (1.1) Converting the Equation (1.2) to natural logarithm The trivial solution of the Equations (1.1) to (1.3) is In this paper, we are focusing on the non-trivial solutions.

Graphical Solutions
If y =10, the Equations (1.5) to (1.9) become  The non-trivial solutions imply the following equations:

Numerical Solutions
Higher precision non-trivial numerical solutions were obtained for the series of equations = using the iterative technique for n = 2, 1.5, 1, 0.5, 0, −0.5, −1, −2 and 1 ≤ y ≤ 15 (Table 1).The iterations do not converge on non-trivial solutions for y < e, and solutions in this range were obtained by trial and error.
The solutions in Table 1 for n = −2, −1, 0, 0.5, 1, 2 are plotted as x vs y with x axis in log scale (Figure 2).Sharp turning points in the plots are observed for y values in the range of 1 to 2.

Analytical Solution
Consider the Equation (1.3) If n = 0, the Equation (1.1) x y x y Using the solution in the Equation (3.1), the analytical solution in terms of Lambert-W function is .
Hence e x = , the result in Table 1.
If n =0 and y =2 in Equation (3.2), the solutions in Table 1 and   The W(x) has two real values for 1 e 0 x − ≤ < [1].
In  The lnx vs n lines for different y values are crossing near the point (0.5, 1.4).This indicates the solutions for n = 0.5 have little dependency on y for y ≥ e.This is also evident in the numerical results for n = 0.5 in Table 1 and in the plot of 0.5 1.5 x y y = in Figure 2.

Solutions x in Base y
The solutions x in Table 1 can be written in base y, (x y ) to indicate the general pattern.
For any valued of n, x y can be written as ( ) For n = 2, the solutions  written in base y, x y shown in Table 2.
For y > 11, the x y are written using the hex notation.
There is a sharp change in the value of the x y at y = 4.
For n = 2, plot x y vs y, for 5 ≤ y ≤ 11 is shown in Figure 4.

Connection to the Fine Structure Constant
In Equation (1.1), when n = 2 and y = 10, the equation becomes   The solution 137.129 is close to the inverse of the fine structure constant 137.036 [14]- [21] which is dimensionless.
The inverse of the fine structure constant 1 α − is given by the expression α − , dimensionless constant [22].
In a recent publication Eaves [23] suggested an equation relating G and α;   1.59947 10 10 By taking the power of (1/289.5) on both sides of the Equation (6.8) and writing the equation for The Equation (6.8) is approximately the same as the equation

Conclusions
An equation in the form of The numerical solutions can be written as ( ) The numerical solutions can also be written in base y as ( ) and analytically.The analytical solution was derived in terms of Lambert-W function.A general numerical solution for any y is found in terms of n or in base y.A solution fine structure constant.The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.

5 )
The curves and the straight line to obtain the graphical solutions of the Equations (2.1) to (2.5) are shown in Figure1.The intercepts of the curves and the straight line indicate the solutions.

Figure 1 .
Figure 1.Plots of the functions to obtain the graphical solutions for the Equations (2.1) to (2.5).
Journal of Applied Mathematics and Physics

Figure 2 . 1 and
Figure 2. Plots of x vs y for the series of equations, in the Equation (3.1), the analytical solution in terms of the Lambert-

10, 1 . 9 )
371289 10 n y x = = × Journal of Applied Mathematics and Physics Using the solution in Equation (3.1), for any y the solution x can be written as Plots of lnx vs n shown in Figure 3 are linear as expected from Equation (3.1).

Figure 4 .
Figure 4. Plot of x y vs y for n = 2.

=
By substituting the expression for α in Equation (6.3) we get is the 10 2 is 106.6 in Equation (6.8).But the Equation (6.8) based on the Equation (6.3) is only an approximate equation.The value 27 1.59947 10 − × in Equation (6.6) is approximately equal to the 1/1.5 , G α G α defined by Jentschura[20].
numerical solution x vs y indicate sharp turning points for y values in-between 1 to 2.The analytical solution was found in terms of Lambert-W function as inverse of the fine structure constant value, 137.036.solution close to the fine structure constant can be derived from the equation

Table 1 .
Non-trivial numerical solutions for the series of equations

Table 2 .
Solutions x in base y(x y ).