Polynomial Functions Composed of Terms with Non-Integer Powers

Polynomial functions containing terms with non-integer powers are studied to disclose possible approaches for obtaining their roots as well as employing them for curve-fitting purposes. Several special cases representing equations from different categories are investigated for their roots. Curve-fitting applications to physically meaningful data by the use of fractional functions are worked out in detail. Relevance of this rarely worked subject to solutions of fractional differential equations is pointed out and existing potential in re-lated future work is emphasized.


Introduction
, arguably the most prolific mathematician of all times who contributed greatly to every branch of applied and pure mathematics, was probably the first to work on functions containing terms with fractional powers. Euler's most praised book Introductio in Analysin Infinitorum [1], which is regarded to establish the subject of mathematical analysis, opens with definitions and exercises concerning various functions including fractional exponents of variables [1]. Laguerre (1834-1886) gave a survey of the roots of polynomials in usual forms but also considered what may briefly be termed as polynomials of fractional powers [2]. Nevertheless, studies concerning functions with fractional exponents have been quite limited over the years and the relevant literature is scarce. On the other hand, works on differential and integral equations of fractional order are on the rise and some analytical solutions to these types of equa-tions are linked to functions containing terms with fractional powers.
Among numerous works on fractional derivatives and integrals, we mention only a few here for introductory purposes. Diethelm et al. [3] presented a selection of algorithms for numerical solution of definite governing equations with derivatives or integrals of fractional order. Daftardar-Gejji and Jafari [4] employed Adomian decomposition method as rectified by Wazwaz [5] for the solution of a multi-order fractional differential equation. Li [6] used cubic B-spline wavelet collocation for the solution of fractional differential equations and obtained excellent agreement with known analytical solutions containing terms of fractional powers. Yüzbai [7] solved fractional Riccati type equations by employing Bernstein polynomials as converted to contain fractional powers. Kawala [8] proposed a numerical approach based on Legendre polynomials for the solution of a class of fractional differential equations. An extensive review of fractional calculus in modelling biological phenomena can be found in Ionescu et al. [9]. In these and quite many similar works, it is possible to see a definite connection between factional derivatives and fractional functions. The present study may be viewed as a preliminary work investigating the functions with terms of non-integer powers. First, roots of selected functions are considered; particularly, terms with fractional and transcendental powers are examined and distinct differences between these two categories are pointed out. Then, practical use of such functions is explored for curve-fitting purposes. Satisfactory results are obtained for the fractional functions constructed by employing physically meaningful data. Future applications of these functions are likely to expand into wider domains.

Polynomial Functions of Non-Integer Powers
General form of a polynomial of non-integer powers can be written as where , j j a α ∈  or  or irrational or transcendental. For covering all cases, the coefficients, j a , and especially powers, j α , may briefly be called as non-integer.
Due to virtually unlimited possibilities a general treatment of (1) cannot be attempted; therefore, relatively simple particular cases would serve well to embark on the subject. For instance, one may inquire the total number of roots of the following equations.
Making the cases apparently more complicated the determination of all the roots of the below equations may be sought.
Still more, we may question the differences in number of roots between the transcendental powers π and e and those of their truncated counterparts such as 3 In this manner, we could easily go on proposing more and more complicated equations involving more terms and coefficients. However, as we would hardly make any progress by proceeding so, we are going to tackle with problems amenable to treatment at least to some extent while making only some comments on those that can be treated partially, and finally avoid all those which are beyond our capability. Concerning the practical use of these types of functions we present some curve-fitting applications and thereby reveal the challenges lie in extending them to general multi-term forms defined in (1).

Roots of Some Particular Functions of Non-Integer Powers
Three different equations containing variables with non-integer powers are considered for demonstrating how the roots can be obtained.

Example 1
Let us seek the roots of the following fractional polynomial equation.  x in which −1 in the natural logarithm is expressed as π π = ± ± + ±π ± π π π with 0,1, 2, k =  and then ± sign has been replaced by + sign here and hereafter without loss of generality. Note also that k is limited to 0,1, 2,3, 4 k = since the quintic roots are evaluated. The above method is of course identical to the usual way of evaluating the power or roots of a complex number. In this particular case the equation is simultaneously squared and its quintic roots are computed; that is, The corresponding numerical values are   Obviously all the roots must be considered when dealing with fractional equations. To clarify this point visually for the present example two graphs of ( ) 5 2 5 2 1 P x x = + are drawn together in Figure 1 by evaluating the square root in 5 2 x positive 0 k = and negative 1 k = . Note that for the latter case the function has a zero at 1 x = as computed. The second approach introduces a new variable 1 2 u x = so that (6) becomes which has the roots ( ) with 0,1, 2,3, 4 k = : x u = naturally renders the results given in (8). Note that according to this approach 2 1 u = − , which makes it clear that we must select −1 as the appropriate root of +1 in to satisfy the equation. Finally, reason for introducing this rather longer approach lies in its use in multi-term equations.

Example 2
Let us seek the roots of the following multi-term polynomial equation with fractional powers. As we have more than one unknown term the second approach of Example 1 is needed. Expressing the fractions under a common denominator gives As in the first example the roots 0 1 x = + and 1 38.7165505 give the false impression that they are not true solutions. The key point here hinges on taking the appropriate roots. Numerically, Likewise, for the other solutions , the appropriate roots must be used. This is a somewhat ambiguous point without a strict rule as to which mode, 0 k = or 1 k = , should be selected for the root. At present, we may regard it a peculiarity of fractional equations. In summary, Equation (11) has four solutions altogether and to show that these solutions satisfy the equation the fractional powers must be computed for the appropriate mode, which is 1 k = for the square root and 0 k = for the cubic root in this case. Also, for a given set of x values x u = .

Example 3
In this example we essentially tackle with the problem of obtaining and examining roots of transcendental equation which is going to prove to be a very interesting case. Let us first consider x + = , which may be viewed as an approximate form of (15). Expressing the exponent as fraction 31 10 1 0 x + = and introducing 1 10 u x = result in the polynomial 31 1 0 u + = with 31 roots. If we take one more decimal 3.14 1 0 x + =, which gives rise to 157 1 0 u + = with 1 50 u x = and requires the determination of 157 roots. We may then conclude that increasing the number of digits to define π would increase the number of roots. Continuing in this line of reasoning we may infer that the process would figuratively lead to an equation with infinite number of roots since π is a transcendental number. Correctness of this naive deduction can be demonstrated in a simple way with quite interesting results. We take 1 0 x π + = and proceed to solve it as done in the first example.
We have thus shown that the πth roots of a complex number can be computed just like any integer roots as customarily done. Moreover, it is observed that there are infinite number of roots for transcendental numbers as conjectured above. The zeroth mode Closeness of the corresponding roots in (18) and (19) are clearly visible in Figure 2 where the first three roots of 1 0 x π + = and all three roots of for 0,1, 2, ,156 k =  as solutions. Table 1 compares the numerical values of the first ten roots, 0,1, 2, ,9 k =  , with those of 1 0 x π + = . As expected the roots of 3.14 1 0 x + = are closer to 1 0 x π + = than 3.1 1 0 x + = .  x + = (blue). Table 1. First ten roots of 3.1 1 0 x + = (second column), 3.14 1 0 x + = (third column), and 1 0 x π + = (fourth column).  with terms of fractional powers and the special form 1 0 x α + = with transcendental and imaginary powers, solution of the multi-term functions with transcendental powers as given in (5) remains a challenging problem.

Curve-Fitting Applications of Functions with Fractional Powers
Curve-fitting to a given data is a possible application area for functions containing terms with fractional powers. A general form as in (1) might be considered; however, just as in the case of roots, a corresponding general approach does not seem possible. For this reason our treatment is again confined to few special cases with select data. A physically meaningful set of points, known as the righting moment lever or GZ values, which make up the most essential data representing a ship's stability, is used for demonstrations. To establish a GZ φ − curve quite complicated computations involving the underwater volume of a ship at definite heel angles φ , second moment of water plane area, etc. are necessary. Certain points of the GZ curve are particularly important for defining its characteristics. At the zero angle of heel normally GZ is zero and the first derivative of the GZ curve at zero angle is called the metacentric height GM. The maximum value of the righting lever is GZ m and the vanishing angle of heel v φ is the angle at which GZ becomes zero.
All the curve-fitting examples here use the GZ data points computed for an actual design given in Table 2

Example 1
We begin by selecting a simple fractional function of the form where 0 a , 1 a , 2 a , and 0 α are constants to be determined by imposing certain conditions according to the data available in Table 2. First, we assume 0 0 α > and set 2 0 At this stage it is obvious that α must be a real quantity greater than unity. Naturally, there is no guarantee that the conditions imposed to determine 0 a and 0 α are  For comparison purposes we also use a third-degree polynomial ( ) Conditions imposed on Equation (20) are now applied to (23) so that For the data in Table 2 Note that 0 3.75 α = is greater than unity as required for this particular application. Figure 3 depicts the curve-fittings given in (25) against the data Table 2. The polynomial function ( ) p GZ φ performs slightly better since the true maximum of the fractional function ( ) f GZ φ is shifted to the right more compared to the polynomial; nevertheless, the general characteristics of both curves are quite similar.

Example 2
In this application the data and functional forms are kept the same but instead of satisfying the maximum point where v φ φ φ = is defined for the simplicity of notation and 0 α is the only constant to be determined by employing the least-squares method with the data given in Table 2. For discreet data points the corresponding total squared error function is Obviously solving (28) for 0 α is not possible by conventional approaches.

Example 3
The last application increases the number of terms to satisfy more conditions hence produce a much better curve-fitting function.
( ) The constants are to be determined by applying six different conditions:    other parameters in such a way that the ultimate error is absorbed greatly to become imperceptible. Indeed, from Figure 6, we see not a definite minimum at 2.30 but a nearly constant region within the range 2.25 -2.35 where the total error remains very low. The point 2.30 is the precise result of computation under set tolerances; however, use of any value between 2.25 and 2.35 produces virtually the same curve, supporting the above argument. This interesting characteristic of the squared error curve is observed for the last curve-fitting shown in Figure 8 too and presumed to be peculiar to fractional functions. Polynomial representations for this case had to be abandoned due to extremely cumbersome algebra required for dealing with a fifth-order polynomial. Such a serious difficulty should be taken as an indicator of the obvious advantage of fractional functions for satisfying a number of conditions, six in this case. Figure 7 depicts Equation (35) against the data of Table 2. Agreement with  is plotted against the corresponding data in Figure 8. The overall result is quite agreeable especially if allowances are made for the varying characteristics of the data.

Concluding Remarks
Polynomial functions composed of terms of non-integer powers are considered for developing methods to obtain their roots. Several representative cases amenable to treatment are examined and some distinct properties of fractional and transcendental powers are revealed. Curve-fitting is recognized as a useful application area of fractional functions and physically meaningful data are employed for computations with satisfactory results. New applications of these functions to diverse fields are likely to emerge in the future.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.