ON THE NIELD-KOZNETSOV INTEGRAL FUNCTION AND ITS APPLICATION TO AIRY ’ S INHOMOGENEOUS BOUNDARY VALUE PROBLEM

In this work, we provide a solution to a two-point boundary value problem that involves an inhomogeneous Airy’s differential equation with a variable forcing function. The solution is expressed in terms of the recently introduced NieldKoznetsov integral function, Ni(x), and another conveniently defined integral function, Ki(x). The resulting expressions involving these integral functions are then evaluated using asymptotic and ascending series. Indexing terms/


INTRODUCTION
Consider the boundary value problem (BVP) composed of solving the inhomogeneous, second order ordinary differential equation (ODE) .
… (1) Subject to the following boundary conditions (BC) …( 2) where and are real constants and ] , [ b a x  .Equation ( 1) is the well-known inhomogeneous Airy's differential equation.Rooted in Airy's nineteenth century work in optics, Airy's ODE continues to receive interest due to the reduction of many differential equations in mathematical physics to it by an appropriate change of variables (cf.[4,5,8,11,12] and the references therein).
Solution to the homogeneous part of ODE ( 1) is given by where and are arbitrary constants, and and are two linearly independent functions, known as Airy's functions and defined by the following integrals (cf.[1,8,11,12]): …( 5) The wronskian of and is given by, [1]: .
… (7) When f(x) is a constant function of x, the inhomogeneous ODE (1) has a particular solution given by the Scorer functions, [10], as given in what follows: , a particular solution is given by , a particular solution is given by .

…(9)
The functions and are known as Scorer's functions.It can be seen from ( 8), ( 9) and ( 6) that …(10) For real values of , Airy's and Scorer's functions are real-valued functions, [3].Extensive analysis has been carried out by Gil et.al., [3], when the argument is complex.Computations of the Airy and Scorer functions continue to receive attention in the literature, and excellent results have been documented (cf.[1,2,6,7,11,12] and the references therein].Now, when the forcing function is a constant other than  1  , or when the forcing function is a variable function of x, we need a consistent notation and methodology to find and express the particular solution to ODE (1), and hence the general solution and the solutions to boundary value problems.This is the objective of this work in which we express particular J a n u a r y 05, 2 0 1 6

GENERAL SOLUTION OF BVP
General solution to ODE (1) is the sum of the complementary function, given by equation ( 4) as the solution to the homogeneous Airy's ODE, and the particular solution, , which, using variation of parameters, is assumed to be of the form …(11) where the functions and are given by the following forms, respectively, with the help of (7): Equation ( 11) thus takes the form and the general solution, g y , to ODE (1) is thus written as Equation ( 15) is valid for both constant forcing function and variable forcing function, as discussed in the following two cases.

THE CASE OF CONSTANT FORCING FUNCTION
When the forcing function in ODE (1) is constant, say First and second derivatives of . Now, using the BC ( 2) and ( 3) in (17), we can determine the following values of the arbitrary constants and render the BVP completely solved:

THE CASE OF VARIABLE FORCING FUNCTION
When the forcing function in ODE ( 1) is a variable function f(x), the general solution is given by equation ( 15).The particular solution given by equation ( 14) involves the integrals  . Using integration by parts, we express these integrals as follows: Substituting ( 22) and ( 23) in (14), and using Using (18), equation ( 24) can be written as: where we define the integral function as: From ( 15) and (24), we can express the general solution to (1) as The right-hand-side of ( 28) is recognized as The first two derivatives of   For large x, we can truncate each of the above series after the first term, and develop the following asymptotic approximations to 18), ( 19), ( 29) and (30): In what follows we will evaluate the solution to ODE (1) subject to BC (2) and (3) when the forcing function is given by   2) exp( ) exp( 2) exp(


. When the values of Table 1 are used to construct the solutions to the given BVP, and the solution is plotted on the selected intervals of the xaxis, Figures 1 through 4 are obtained.In these Figures we also compare the solutions with those numerical solutions obtained using Maple's dsolve numerical built-in function (shown in Table 2).As a first approximation in the asymptotic solutions to equation (1) in terms of the recently introduced Nield-Koznetsov function , revisited in this work.
can be conveniently written in the following form that implements the newly introduced Nield and Koznetsov function, been discussed in detail by Hamdan and Kamel[4]: obtained by multiplying (23) by ) (x A i , and (22) by ) (x B i , then subtracting, to obtain J a n u a r y 05, 2 0 1 6 using the BC (2) and (3) in (27), we can determine the following values of the arbitrary constants and render the BVP completely solved:

i
Computing solutions (17) and (27), and evaluation of the arbitrary constants associated with the BVP (1) subject to conditions (2) and (3), necessitates evaluating ) interval [a,b].Since these functions are expressed in terms of Airy's functions, we will rely on approximations of Airy's functions to approximate) of methods are discussed in the literature to approximate and , (cf.[1,2, 12]).In the current work, we illustrate the calculations using the asymptotic series approximations.The following asymptotic series approximations have been developed for Airy's functions, their derivatives and integrals, as given in [12], wherein where

2 c
are obtained from equations (32) and (33).Values of ) points are once again approximated using the above asymptotic series of these functions.We carry out these approximations for a number of BC, shown in Table1, where we have chosen 0   and 1 