Approximate expressions for solutions to two kinds of transcendental equations with applications

In a broad spectrum of physics and engineering applications, transcendental equations have to be solved in order to determine their roots. Exact and explicit algebraic expression of solutions to such equations is, in general, impossible. Analytical approximate solutions to two kinds of transcendental equations with wide applications are presented. These approximate root formulas are systematically established by using the Padé approximant and show high accuracy. As an application of the proposed approximations, a highly accurate expression of the effective mass of the spring for a spring-mass system is obtained. The method described in this paper is also applied to other transcendental equations in physics and engineering applications.


Introduction
The determination of roots of transcendental equations is a problem commonly encountered in a broad spectrum of physics and engineering applications. However, it is difficult to obtain analytical approximate root formulas for such equations. Though a wide variety of root finding algorithms are available to achieve the solutions to desired degree of accuracy, analytical approximate solutions, which provide explicit dependence of the roots on the physical parameters of problem compared with purely numerical solutions, are always desirable and preferable.
Consider the following two kinds of transcendental equations with various applications. The first kind of equation is This equation arises from the solution of a longitudinal vibration problem in a uniform bar with one fixed and one attached mass boundary condition [1]. A similar equation comes from the solution of buckling problem of a uniform column which has one free and one elastically hinged supported boundary condition and is subjected an axial compression load [2]. The infinite series solution to the one-dimensional transient chemical diffusion problem under certain boundary conditions [3] is also related to this equation. The second kind of equation is where   p p p n x n 2 2 2.This equation comes from the problem of a particle moving in a finite square well potential where the energy eigenvalues are its roots [4]. After the fabrication of quantum wells [5], the experimental observation of revivals and super-revivals [6] and the progress of the so-called 'ghost orbit spectroscopy' [7], the square wells also describe realistic physical systems or phenomena. The need for an explicit solution exceeds the level of solving simple and relevant problems of quantum mechanics.
Researchers are interested in positive roots of these equations because only they are related to the physical quantities. Note that the analytical solutions to equations (1) and (2) are absent until now. A method for formulating an expression for the roots of any analytical transcendental function was presented [8]. The method Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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is based on Cauchy's integral theorem and uses only basic concepts of complex integration. Numerical evaluation of solutions requires a complex Fourier transform. However, the computational efficiency of this procedure would not be expected to rival that of traditional approximate root finding techniques [8]. Recently, Luo et al [9] constructed the analytical approximate solution to equation (1) by rewriting its series expansion solution [10] in the form of a ratio of polynomials by a second-order Padé approximant. However, their results showed large errors. Based on the algebraic approximations of trigonometric functions, it is possible to transform a class of transcendental equations in approximate, tractable algebraic equations [4,11,12]. As the algebraization used in those papers is, to a certain extent, an ad hoc procedure, this approximation must be used with a certain caution in order to avoid the appearance of spurious roots or of roots with too large errors [12].
In this paper, highly accurate approximate expressions for solutions to equations (1) and (2) are systematically constructed by exploring the periodic properties of functions x tan and sinx, and using the Padé approximant [13][14][15] to them. These approximations are valid for small as well as large values of parameters. Furthermore, as an application of the proposed approximate expression, a highly accurate expression of the effective mass of the spring for the spring-mass system is also obtained.

Preliminaries
A Padé approximant is the 'best' approximation of a function by a rational function of given order-under this technique, the approximant's power series agrees with the power series of the function it is approximating. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge and has thus abundant applications in physics and engineering.
Given a function ( ) f x and two integers  p 0 and  q 1, the Padé approximant of order [ ] p q is the rational function [13] The Padé approximant is unique for given p and q, that is, the coefficients   a a a b b b , , , , , , , can be uniquely determined. It is known that in many cases a higher accuracy of approximation is achieved for small integers p and q; thus the degrees of both numerator and denominator are set to be small hereafter so that the analytical approximate roots of transcendental equations can be obtained. (1) 3.1. Derivation of the first approximate expressions for all roots Based on the periodic property of function x tan , a rational approximate expression for function x tan in interval p p -[ ] 2, 2 is first introduced, the simple approximate root formulas of equation (1) in terms of parameter a are then established.

Highly accurate approximate expressions of solutions to transcendental equation
Note that from the graphs of functions = ( ) Solving y from equation (7) and using equation (4) yields the first approximation to the Especially, for = n 0, equation (8) gives the first approximation to the first root of equation (1)  Substitution of equation (10) into equation (5) produces which can be written as a p a p a + + - Solving quadratic equation for y 2 in equation (12) and using equation (4) with = n 0 gives the second approximation to the first root of equation (1)  Equation (11) is a quartic one for > n 0, expressions of its roots are lengthy and complex. We will try to find approximate solutions as follows. Note that, p p -< < y 2 2 , so higher powers of y could eventually be neglected in equation (11). Keeping the constant, linear and quadratic terms, and neglecting the cubic and quartic ones in equation (11)  Taking y 20 as the initial guess value, applying the Newton method to equation (11), iterating one step and noticing that y 20 satisfies equation (14), give the second approximations to the positive roots of equation (11)  Finally, based on equation (4), the second approximate expression of the + ( ) n 1 th root of equation (1) is

Results
For given value of parameter a, the roots of equation (1) can numerically be calculated by using the Newton method. The corresponding analytical approximations to these roots can be obtained by utilizing equations (8), (13) and (17), respectively. Relative errors are then calculated against these numerical exact roots. Here, the relative error of the ith analytical approximation to the + ( ) n 1 th root to equation (1) is defined as ni ni a n N n N where x n N denotes the + ( ) n 1 th root obtained by using the Newton method.
For a = 1, the relative errors of the approximate roots in equations (8), (13) and (17) computed by the proposed method are shown in table 1. For comparison purposes, the relative errors of the approximate roots, equations (7) and (12b), in [9] are also listed in the same table. Note that Luo et al [9] used the series expansion solutions [10] to equation (1) to construct the corresponding Padé approximants of order [ ] 2 2 .It can be seen from table 1 that, except for the first approximation to the first root, the accuracy of the proposed approximate roots is much higher than that in [9].
In general, researchers are more interested in the case of  a 1 [3]. Relative errors of the two approximate expressions given in equations (8), (13) and (17) for the first three roots of equation (1) are shown in figures 1-3, respectively. These figures indicate that, compared with the first approximation in equation (8), the second approximations in equations (13) and (17) can provide more accurate results. For  a 1, the maximum relative errors of the first and second approximations to the first root are less than 0.662% and 0.000 176%, respectively; the maximum relative errors of the first and second approximations to the second and third roots are 0.001 09% and 0.000 0309%, and 0.000 0292% and 0.000 000 296%, respectively. For the nth root with  n 4, the accuracies of approximate roots in equations (8) and (17) are higher than those for the third root. It should be pointed that, for  a 1, the expression in equation (8) can provide high quality approximation to the nth root   (1) with a = 1 and the relative errors of the approximate roots proposed in this paper and in [9].

Relative errors (%)
n Exact roots equation (8) equation (13)   Based on the results above, the second approximate roots in equations (13) and (17) show excellent accuracy and they are valid for small as well as large value of parameter a. 3.4. Determination of the effective mass of the spring for a spring-mass system As an application of the proposed approximation, the effective mass of the spring for a spring-mass system will be established. When a longitudinal vibration in a uniform bar with one fixed and one attached mass boundary condition [1,16,17] is considered, equation (1) reads  where w, l, A, r and E are the natural frequency, length, cross-sectional area, density and modulus of elasticity of the bar, respectively, M is the attached mass, and a º m M s is the ratio of the mass r = m Al s of the bar to the attached mass.
The introduction of the correction to the spring oscillations due to including the mass m s of a spring has led to many researches, for examples, we refer readers to [1,16,17] and cited therein. When the vibration of the bar is reduced to that of a spring-mass with a spring stiffness = K EA l and an end mass equal to + M m , eff the effective mass m eff need to be determined. Based on equations (1), (9), (13) and (19), two approximations to the first frequency of the spring-mass system can be expressed as Here, Note that, in [16,17], the frequency of the spring-mass system was given by Based on the assumption that the velocity of a spring element located a distance from the fixed end to vary linearly with it and the use of Rayleigh method, the effective mass of the spring is found to be the one-third the mass of the spring [1,16,17]. Now, the proposed first approximate frequency w s for the effective mass of the spring in the spring-mass system for any given mass ratio a. (2) 4.1. Derivation for highly accurate approximate expressions for all roots Note that from the graphs of functions = ( ) z f x in equation (2) and b = z x , for each positive integer n, equation (2) (2) with b = 15 and relative errors of the approximate roots proposed in this paper and in [4].