TRANSCENDENTAL-HYPERBOLIC FUNCTIONS AND THEIR ADOMIAN POLYNOMIALS WITH NUMERICAL RESULTS FACILITATED BY NONLINEAR SHANKS TRANSFORM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we provide explicitly the Adomian polynomials (AP) for transcendental-hyperbolic functions in a linear functional and forced the convergence of inconsistent solution series when Adomian decomposition method (ADM) is deployed in related problems by nonlinear Shanks transform (NST). These were achievable by developing a theoretical background of AP for transcendental-hyperbolic functions based upon a thorough examination of the historical preceding of ADM. Application of the presented polynomials resulted to unreliable series solutions which was, however, upturned on using NST in the problems considered. This paper has unified the notion of modified AP for transcendental-hyperbolic nonlinear functions and its application to similar equations. It further presented a reliable technique that forced convergence in unpredictable and alternating series solutions that are obtain by ADM.


INTRODUCTION
Over the years ADM [1] and its improvement has become a very powerful technique for obtaining analytical and approximate analytical solutions to a generalised nonlinear equations.
Where L n is the nth order derivative of ϑ which is a combination of highest order derivative and other differential operator. Which is corresponding to L −n operator given as  (.)dt 1 dt 2 dt 3 ...dt n (2) N = N(ϑ ) is the nonlinear term which is transcendental-hyperbolic in this article and is to be decomposed into AP. And, ζ = ζ (t) is the source term. The ADM by [1], has been widely reported in [2] - [15], [17] and [18]. It is a systematic analytical and approximation method applied to a wide class of equations. The method provided solutions in convergent series form under physically appropriate conditions. Nonetheless, it successful application, especially on the nonlinear problems, requires the right AP to be used for ultimate desired results, see [1,8,9] and the literatures therein. However, the nonlinearity in a linear functional varies; polynomial nonlinearity has been vividly reported in [8], other are [17,11,6,12,9,10,2,13]. Trigonometric nolinearity has extensively been investigated in [9], see also [14] and the literature therein.
Exponential and logarithmic nonlinearity can be seen in [18], transcendental hyperbolic sine and cosine has been reported in [13].
Reported investigation of AP in the are of transcendental hyperbolic nonlinear terms are still minimal in literature. The fundamental goal in this paper is to decompose the transcendentalhyperbolic nonlinear term N(ϑ ) in a linear functional (1) into series of polynomials Σ ∞ n=0 A n using the modified AP as contain in [8]. Where the A n are the AP. These polynomials are in hyperbolic form and equivalently in their respective exponential form for each N(ϑ ). Due to the presence of noise term in each polynomial, convergence to a solution in the illustrative problems considered were far fetched by the traditional ADM. However, this was reversed on application of NST.

THE ADOMIAN POLYNOMIAL IN ADOMIAN DECOMPOSITION METHOD
These polynomials were introduced by [1] and it is given as λ is a parameter introduced for convenient. A n depends on ϑ 0 , ϑ 1 , ϑ 2 ...ϑ n−2 , ϑ n−1 , ϑ n , where n ∈ Z + . Implementation of (3) on (1) basically gives ... A n = A(ϑ 0 , ϑ 1 , ϑ 2 , ..., ϑ n−2 , ϑ n−1 , ϑ n ) From its inception till date, several modifications has been carried out on how these polynomials can be generated. These somewhat feasible form calculates the AP in a simple way using any known computer algebra software like Maple, Mathematica, etc. All these other forms of AP has been implemented and reported in [2] - [15], [17] and [18].

THEORY OF ADOMIAN DECOMPOSITION METHOD AND NONLINEAR SHANKS TRANSFORM ON TRANSCENDENTAL-HYPERBOLIC EQUATIONS
The nonlinear hyperbolic-trigonometric equation can be expressed as where N(ϑ ) is a nonlinear function and c is a constant. ADM considers the solution equation And, ϑ n is calculated recursively. The nonlinear term is decomposed as where A n is as defined in equation (3). See [15], [3], [4], [5], [7] and the literature therein.
Substituting equations (6) and (7) in equation (5), we have a recurrence relation Suppose {ϑ n } is a sequence of partial sum of the series in equation (6), then the Shanks nonlinear transform by [16], denoted T{ϑ n }, is given as which the first order iteration is given as These iterations, according to [16], often leads to the reasonable results. The more the iterations the better the results, this can be seen in the numerical illustrations in the following section.

MAIN RESULTS
In this section we give examples by adopting the technique stated in the previous section on the theory of ADM and NST. The numerical calculations were made using Maple Mathematical software to ensure double precision arithmetic in order to reduce the round-off errors to the barest minimum.
Numerical Example 1. Consider the equation The approximate analytical solution is x = 0.2487139369 . Applying the Adomian polynomial, the theory of ADM and NST, we have the result as presented in Table 1. Its easy to see from Table 1 that ϑ = ∑ ∞ n=0 ϑ n failed to yield reasonable converging result in column ϑ n . This was quickly upturn on application of NST in column ι n , κ n , µ n and finally through column ρ n . Numerical Example 2. Consider the equation x + sinh x = 1 2 + cosh x The approximate analytical solution is x = 0.9046738485 . Also, applying the Adomian polynomial, the theory of ADM and NST, we have the result as presented in Table 2. Its obvious from Table 2 that ϑ = ∑ ∞ n=0 ϑ n failed to yield reasonable converging result in column ϑ n . However, this was overturn on application of NST in column ι n , κ n , µ n and finally through column