In this study, the numerical solutions have been obtained by using the

asymptotic expansions.

They cover

asymptotic expansions and series, asymptotic methods for solving nonlinear equations, the perturbation of nonlinear oscillations, a nonlinear oscillator in a potential well, auto-resonances in nonlinear systems, asymptotics for loss of stability, and systems of coupled oscillators.

In particular, we will study existence, multiplicity and

asymptotic expansions of solutions when the competition parameter tends to infinity.

epsilon],[delta]](*) by constructing

asymptotic expansions of its probability densities, which are associated with the adjoint operator [L.

In a similar way, the

asymptotic expansions of [v'.

by allowing further terms in the

asymptotic expansions with exponents between [alpha] and [alpha] + 1, with additional assumptions on the coefficients.

Using

asymptotic expansions, Efron (1975) demonstrated that LLD can be less efficient than LDA when the assumptions of LDA are satisfied.

Olde Daalhuis,

Asymptotic expansions of q-gamma, q-exponential and q-bessel functions, Journal of Mathematical Analysis and Applications, 186 (1994), 896-913.

The main purpose of this paper is to obtain the first term in the

asymptotic expansions of [[phi].

Asymptotic expansions and analytic continuations of the H-function have been discussed by Braaksma ((1)).

In the context of computing

asymptotic expansions of sums and series, usually the most useful form of the Euler-Maclaurin formula is

4), we use the ideas from [6,10] of

asymptotic expansions of the Bessel functions.