Asymptotic Series Evaluation of Integrals Arising in the Particular
Solutions to Airy’s Inhomogeneous Equation with Special Forcing
Functions
M.H. HAMDAN
Department of Mathematics and Statistics,
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
CANADA
S. JAYYOUSI DAJANI
Department of Mathematics and Computer Science
Lake Forest College
Lake Forest, IL 60045
USA
D.C. ROACH
Department of Engineering,
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
CANADA
Abstract: - In this work, particular and general solutions to Airy’s inhomogeneous equation are obtained
when the forcing function is one of Airy’s functions of the first and second kind, and the standard Nield-
Kuznetsov function of the first kind. Particular solutions give rise to special integrals that involve
products of Airy’s and Nield-Kuznetsov functions. Evaluation of the resulting integrals is facilitated by
expressing their integrands in asymptotic series. Corresponding expressions for the Nield-Kuznetsov
function of the second kind are obtained.
Key-Words: - Airy’s inhomogeneous equation, special integrals, asymptotic series, Nield-Kuznetsov
functions
Received: June 27, 2021. Revised: March 29, 2022. Accepted: April 28, 2022. Published: May 31, 2022.
1 Introduction
The objective of this work is to consider solutions to
the inhomogeneous Airy’s ordinary differential
equation, ode, [1], when the right-hand-side forcing
functions is a special function. In particular, the
interest is in the right-hand-side being an Airy’s
function of the first and of the second kind, and
, respectively, [2,3], and the standard Nield-
Kuznetsov function of the first kind, , [4],
discussed below.
Airy’s ode and Airy’s functions are some of the
mathematical gems that arise in mathematical
physics due to the fact that many problems in this
field can be reduced to Airy’s ode, and a number of
special functions are rooted in Airy’s functions, [3,
5]. Furthermore, solutions to Airy’s ode give rise to
interesting integral functions and special integrals
that lead to advancements of modern day
mathematics. Inhomogeneity in Airy’s ode due to the
functions chosen in this work give rise to important
integrals that involve products of Airy’s functions
and other functions, [6,7]. Some of these products
represent solutions to interesting differential
equations, as discussed in this work.
In the following sections, an overview of Airy’s
ode is provided together with its forms of solution.
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This will be followed with solutions to Airy’s ode
with special forcing functions. A discussion and
evaluations of the arising integrals then follows.
2 Airy’s Equation and its Solutions
The celebrated Airy’s (ode) is rooted in the
nineteenth century and has various practical
applications and theoretical implications in
mathematical physics, [2]. Its homogeneous part
takes the form
(1)
where “prime” notation denotes ordinary
differentiation. General solution of (1) is given by,
[2,3]:
(2)
where are arbitrary constants,
are the linearly independent Airy’s
functions of the first and second kind, respectively,
and defined by the following integrals, [2,3]:
(3)
(4)
The non-zero Wronskian of and is
given by, [2]:
(5)
The twentieth century witnessed an interest in the
inhomogeneous Airy’s ode due to applications in
systems theory, solutions to Schrodinger and Stark
equations, and in fluid mechanics, among others (cf.
Scorer, [7],; Khanmamedov et.al., [8]; Alzahrani
et.al., [9]; Nield and Kuznetsov, [10]; Lee, [11];
Dunster, [12,13]; and the references therein).
The literature shows that particular solutions to
the inhomogeneous Airy’s equation of the form:
(6)
are given by
(7)
when
, and by
(8)
when
, where , and are
arbitrary constants. The functions
are known as the Scorer functions, [7,11], with
integral representation given by:
(9)
(10)
The literature also shows that writing solution to
ode (6) in terms of the Scorer functions for any
constant , requires non-trivial mathematical
manipulations, [3].
The twenty first century, however, witnessed the
introduction of a general methodology to find the
general solution to equation (6) for any constant .
This solution is given by Hamdan and Kamel, [4], as:
(11)
where are arbitrary constants, and the integral
function is called the Standard Nield-
Kuznetsov Function of the First Kind, and is given
by, [4]:
(12)
When
, solution (11) reduces to (8), and
, and when
, solution (11)
reduces to (9), and .
Clearly, relationship between and the
Scorer functions is given by
(13)
with integral representation obtained from (9), (10)
and (13) as
(14)
The main properties of the Standard Nield-
Kuznetsov function of the first kind, , and its
efficient computations have been discussed by
previously discussed, [4,14,15]. The case when the
right-hand-side of Airy’s inhomogeneous equation is
a function of , namely the ode
(15)
was elegantly discussed in the mid-twentieth century
work of Miller and Mursi, [16].
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They have shown that (15) might be solved when
(16)
with
(17)
and where are expressed as power
series. The solution may be expressed in the same
form or as a series of derivatives of .
The solution is also given in the case where
is itself expressed as a power series; in this case it is
of the form
(18)
where
(19)
and and are expressed as power series. The
series solution terminates if are
polynomials, or if is a polynomial.
It is clear that the method of Miller and Mursi,
[16], above, has some restrictions on the function
in addition to being time consuming in its
application.
A decade ago, a method was introduced by
Hamdan and Kamel, [4], to find the general solution
of (15) when is a differentiable function of .
They showed that the general solution to (15) can be
expressed as:
(20)
where in the Standard Nield-Kuznetsov
function of the first kind, defined in (14), and
has been referred to as the Standard Nield-Kuznetsov
function of the second kind, defined by the following
equivalent forms:
(21)
(22)
3 Forms of Particular Solutions
Solution (20) indicates that the particular solution to
(15) is written as:
(23)
This has proved to be convenient for computations
involving many forms of , [14,15].
Using (22) in (23), equation (23) in written the
following equivalent form:
(24)
Equations (23) and (24) reflect the dependence of
the particular solution on the forcing function
and on integrability of the product of and Airy’s
functions. Clearly, when is itself an Airy’s
function, then the integrals involve products of Airy’s
functions. In order to illustrate the arising integrals
and their evaluations, the following three examples
of are discussed and the particular solution is
obtained using both forms, (23) and (24), which
produce the same integrals.
3.1. Case 1: Using Form (24)
Example 1: If , equation (15) takes the
form
(25)
Particular integral (24) for ode (25) takes the form
(26)
and the general solution to (25) is written as
(27)
Example 2: If , equation (15) takes the
form
(28)
Particular integral (24) for ode (28) takes the form
(29)
and the general solution to (25) is written as
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(30)
Example 3: If , (15) takes the form
(31)
Particular integral (24) for ode (31) takes the form
(32)
and the general solution to (31) is written as
(33)
Equations (27), (30) and (33), involve the five
integrals:
,
,
and
.
Their method of evaluation will be discussed below.
It is worth noting here that the functions ,
and are three linearly
independent solutions of the homogeneous third-
order ode , with Wronskian
(cf.
Vallée and Soares, [3], Page 30).
3.2. Case 2: Using Form (23)
Equation (20) gives the general solution in terms of
and . Using the same three example,
above, we obtain the following general solutions.
Example 1: When in (15), general
solution to (15) takes the form
(34)
where is evaluated using (21) or (22),
respectively, as
(35)
or
(36)
Example 2: When in (15), general
solution to (15) takes the form
(37)
where is evaluated using (21) or (22),
respectively, as
(38)
or
(39)
Example 3: When in (15), general
solution to (15) takes the form
(40)
where is evaluated using (21) or (22),
respectively, as
(41)
or
(42)
5 Asymptotic Series Representation of
Arising Integrals
In order to evaluate the integrals and the Nield-
Kuznetsov functions arising in the solutions above,
the following asymptotic series expressions are used
when , [3-5]:
(43)
(44)
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wherein
. Hamdan and Kamel, [4],
obtained the following asymptotic series for the
Nield-Kuznetsov functions:
(45)
(46)
wherein
.
Using (43)-(45), the following values of the
integrals appearing in (36), (39), and (42) are
obtained, where some have been evaluated using
Wolfram Alpha:
(47)
(48)
(49)
(50)
(51)
where is the gamma function, is the
incomplete gamma function, and is the real
part of .
6 Expressions for the Nield-Kuznetsov
Function of the Second Kind
Using asymptotic series exressions (43)-(45), and
integrals (47)-(51), the following expressions are
obtained for the particular solution (32) for each of
three examples considered. Furthermore, using (36),
(39), and (42), expressions for are obtained.
Example 1: Using (36), the following expression is
obtained for :
(52)
Particular solution (26) and general solution (27)
take the following forms, respectively:
(53)
(54)
Example 2: Using (39), the following expression is
obtained for :
(55)
Particular solution (29) and general solution (30)
take the following forms, respectively:
(56)
(57)
Example 3: Using (42), the following expression is
obtained for :
(58)
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Particular solution (32) and general solution (33)
take the following forms, respectively:
(59)
(60)
7 Conclusion
In this work, a method of solving the inhomogeneous
Airy’s equation when the right-hand-side is a special
function (such as one of Airy’s functions or the
Nield-Kuznetsov function of the first kind), was
presented. Arising special integrals involve products
of these special functions. The integrands have been
expressed using asymptotic series, and integral
evaluations were carried out using Wolfram Alpha.
The particular and general solutions of Airy’s
inhomogeneous equation have been presented and a
derivation of expressions for the Nield-Kuznetsov
function of the second kind, corresponding to each
forcing function, have been obtained. Significance of
this work stems from the fact that Airy’s equation is
one of our mathematical gems and its solutions have
given rise to many special functions since its
inception. It also plays a role in the development and
optimization of computational algorithms designed
to provide efficient computations of Airy’s functions.
These same algorithms are of great value to the
numerical analysis literature. The arising integrals in
this work might find applications in mathematical
physics.
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[3] Vallée, O. and Soares, M., Airy functions and
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