Connecting Einstein Functions to the Nield-Kuznetsov and Airy’s
Functions
D.C. ROACH
Department of Engineering
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
M.H. HAMDAN
Department of Mathematics and Statistics
University of New Brunswick
100 Tucker Park Road, Saint John, New Brunswick, E2L 4L5
CANADA
Abstract: - In this work, the problem of obtaining particular and general solutions to Airy’s inhomogeneous
equation when the forcing function is one of Einstein’s functions is examined. Expressions for the particular
solutions provide connections between the Nield-Kuznetsov and Einstein functions. Computations have been
carried out using Wolfram Alpha.
Key-Words: - Einstein functions, Nield-Kuznetsov functions, Airy’s inhomogeneous equation
Received: July 28, 2021. Revised: March 16, 2022. Accepted: April 19, 2022. Published: May 9, 2022.
1 Introduction
Interest in special functions stems in part from their
direct use in solving applied problems and problems
in mathematical physics; their connections to other
elementary and special functions; and their
contribution to the creation and expansion of our
mathematical horizon, (cf. [1-3] and the references
therein).
Of particular interest to the current work is a class
of functions referred to as Einstein functions, which
are combinations of exponential and logarithmic
functions. Einstein functions have been the subject
matter of various studies and tabulations due to their
applications in the study of distributions, the
determination of physical and chemical material
constants, and in the study of Einstein’s field
equations. For these and many other applications of
Einstein functions, one is referred to the works of
Abramowitz and Stegun, [1], Hilsenrath and Ziegler,
[4], Cezairliyan, [5], and the references therein.
Computations, series representations and some
properties of Einstein functions have been illustrated
in Wolfram’s Mathworld, [6].
Noteworthy in the study of Einstein functions is
their connection to polylogarithmic functions, [7,8],
which bridge a gap in our mathematical knowledge
between Airy’s inhomoheneous ordinary differential
equation (ODE) with homogeneities due to special
functions, such as the sigmoid logistic function. This
connection was recently studied and established by
Roach and Hamdan, [9], and Hamdan and Roach,
[10], whose work underscored the importance of
connections between Airy’s functions, special
functions and the Nield-Kuznetsov integral
functions. Their work has inevitably lead to the
current work where a connection is being sought
between the Einstein functions, the Nield-Kuznetsov
functions and the classic Airy’s functions, [11,12].
The objective of this work is to provide particular
and general solutions to Airy’s inhomogeneous
ordinary differential equation (1), below, when the
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inhomogeneity is due to Einstein functions. As
already has been established in the literature,
solutions to Airy’s inhomogeneous equation are
expressed in terms of Airy’s functions of the first and
second kind, and the standard Nield-Kuznetsov
functions of the first and second kind, [13]. In order
to put this in perspective, the interest is in the
following form of Airy’s inhomogeneous ODE:
(1)
wherein “prime” notation denotes ordinary
differentiation with respect to the independent
variable, and the forcing function is chosen in
this work to be one of Einstein’s functions, [1].
Solutions to the inhomogeneous ODE (1) are rare.
In their elegant analysis, Miller and Musri, [14],
introduced a specific-purpose method for solving (1);
alas, the method is hardly practical and imposes
restrictions on . A general-purpose approach
was introduced by Hamdan and Kamel,[13], and can
easily provide the general solution to (1) as long as
is a differentiable function. Hamdan and Kamel,
[13], expressed the general solution to (1) as:
) (2)
where in the Standard Nield-Kuznetsov
function of the first kind, defined by, [15]:
(3)
and is the Standard Nield-Kuznetsov function
of the second kind. The integral function, , is
defined by the following equivalent forms, [13]:
(4)
(5)
The particular solution to (1) can thus be written
in one of the equivalent forms:
(6)
(7)
This approach is robust and versatile, and both
forms, (6) and (7), have recently been used to obtain
particular solutions to (1) when
, [16], and when is the sigmoid
logistic function, [10].
It is worth noting that the work of Hamdan and
Roach, [10], helped to establish a mathematically
significant connection between Airy’s functions, the
sigmoid functions and the dilogarithm function, and
between the sigmoid function and the Nield-
Kuznetsov functions, and . These forms
of forcing functions gave rise to interesting integrals
involving products of these special functions, and
introduced new functions, such as the dilogarithm
function, as building blocks of solutions to Airy’s
ODE.
Those arising integrals enrich our mathematical
knowledge and expand applicability of Airy’s
inhomogeneous ODE to potential and arising
subfields of mathematical physics, [17]. They also
motivate the current work in which the interest is to
solve ODE (1) when is an Einstein’s function.
The objective is to establish a connection between
Einstein’s functions, Airy’s functions and the Nield-
Kuznetsov functions. The rise of polylogarithm
functions is inevitable and helps connect these
integral functions using Theorem 1 in this work.
2 Solution to Airy’s
Inhomogeneous ODE with Einstein
Forcing Functions
The general solution to the homogeneous part of
Airy’s ODE (1) is given by the following
complementary function:
(8)
where are arbitrary constants, and
are the linearly independent Airy’s
functions of the first and second kind, respectively,
and whose non-zero Wronskian is given by, [1]:
(9)
Now, consider Airy’s ODE (1) with
, where is an Einstein function, namely
(10)
General solution to (10) is of the form
(11)
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where is given by the following equivalent forms:
(12)
(13)
where
(14)
and
(15)
Using (14) and (15) in (13) yields
(16)
Using (9) in (16) yields
(17)
In what follows equation (17) is evaluated for the
following four Einstein functions:
(18)
(19)
(20)
(21)
Although graphs are provided for the functions
representing the particular solutions in intervals
around , it is emphasized here that the
particular solutions obtained for Airy’s
inhomogeneous ODE in this work are valid for
. In the notation below, refers to the natural
logarithm.
Case 1:
Integrating yields a convergent improper
integral that can be used to obtain , as follows.
(22)
(23)
Evaluating the limit, takes the following final
form:
(24)
and the following general solution is then obtained:
(25)
where is the zeta function and
.
Graph of the particular solution is shown in Fig.
1, below.
Fig. 1. Graph of when
Case 2:
Integrating yields a convergent improper
integral that can be used to obtain , as follows.
(26)
(27)
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Evaluating the limit, takes the following final
form:
(28)
and the following general solution is then obtained:
(29)
Graph of the particular solution is shown in Fig.
2, below.
Fig. 2. Graph of when
Case 3:
Integrating yields a convergent improper
integral that can be used to obtain , as follows.
(30)
(31)
Evaluating the limit, takes the following final
form:
(32)
and the following general solution is then obtained:
(33)
Graph of the particular solution is shown in Fig.
3, below.
Fig. 3. Graph of when
Case 4:
Integrating yields a convergent improper
integral that can be used to obtain , as follows.
(34)
(35)
Evaluating the limit, takes the following final
form:
(36)
and the following general solution is then obtained:
(37)
Graph of the particular solution is shown in Fig.
4, below.
Fig. 4. Graph of when
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3 Asymptotic Series Representations
When , Airy’s and the Nield-Kuznetsov
functions have the following asymptotic series
representations (cf. Hamdan and Kamel, [13],
and the references therein) that can be used in the
evaluation of the general solution of ODE (1):
(38)
(39)
(40)
(41)
wherein
and
.
While asymptotic series representations of
, , and are independent of the
forcing function of Airy’s ODE (1), that of
takes into account the forcing function. Equation
(41) illustrates how Einstein’s function, ,
fits into the asymptotic series representation of
.
4 Relationships between Einstein’s,
Airy’s and the Nield-Kuznetsov
Functions
Equations (12), (13) and (17) establish the following
theorem on relationships between Einstein functions,
, Airy’s functions, and , and the
Nield-Kuznetsov functions and :
Theorem 1: Airy’s functions of the first and
second kind, the Nield-Kuznetsov functions of the
forst and second kind, and Einstein’s function, are
related by:
(i)
(ii)
Relationships involving the polylogarithm
functions are developed using the obtained
particular solutions, and take the following
forms:
(42)
(43)
(44)
(45)
5 Conclusion
In this work, particular solutions and general
solutions to the inhomogeneous Airy’s ordinary
differential equation were obtained when the sources
of inhomogeneity are Einstein’s functions. The
obtained solutions were expressed in terms of Airy’s
functions, the Nield-Kuznetsov functions and the
polylogarithm functions. These solutions set the
stage for the challenging tasks of computing
solutions to initial and boundary value problems, and
establish a connection between Airy’s functions, the
Nield-Kuznetsov functions and Einstein’s functions.
References
[1] Abramowitz, M. and Stegun, I.A., Handbook of
Mathematical Functions, Dover, New York, 1984.
[2] Khanmamedov, A.Kh., Makhmudova1, M.G. and
Gafarova, N.F., Special Solutions of the Stark
Equation, Advanced Mathematical Models &
Applications, Vol. 6(1), 2021, pp. 59-62.
[3] Temme, N.M., Special Functions: An
Introduction to the Classical Functions of
Mathematical Physics, Wiley, New York, 1996.
[4] Hilsenrath, J. and Ziegler, G.G., Tables of
Einstein Functions, Nat. Bur. Stand. (U.S.),
Monograph 49, 258 pages, 1962.
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[5] Cezairliyan, A., Derivatives of the Grüneisen and
Einstein Functions, Journal of Research of the
National Bureau of Standards- B. Mathematical
Sciences, Vol. 74B #3, 1970, pp. 175-182.
[6] Wolfram Alpha, www.mathworld.wolfram.com.
[7] Maximon, L.C. The Dilogarithm Function for
Complex Argument, Proc. R. Soc. Lond. A, Vol. 459,
2003, pp. 2807–2819.
[8] Lewin, L., ed., Structural Properties of
Polylogarithms, Mathematical Surveys and
Monographs, Vol. 37, 1991, Providence, RI: Amer.
Math. Soc. ISBN 978-0-8218-1634-9.
[9] Roach, D.C. and Hamdan, M.H., On the Sigmoid
Function as a Variable Permeability Model for
Brinkman Equation, Trans. on Applied and
Theoretical Mechanics, WSEAS, Vol. 17, 2022, pp.
29-38.
[10] Hamdan, M.H. and Roach, D.C., The Sigmoid
Neural Network Activation Function and its
Connections to Airy’s and the Nield-Kuznetsov
Functions, Proof, WSEAS, Vol. 2, 2022, pp. 108-114.
[11] Airy, G.B., On the Intensity of Light in the
Neighbourhood of a Caustic, Trans. Cambridge Phil.
Soc., Vol. 6, 1838, pp. 379-401.
[12] Vallée, O. and Soares, M., Airy functions and
applications to Physics. World Scientific, London,
2004.
[13] Hamdan, M.H. and Kamel, M.T., On the Ni(x)
Integral Function and its Application to the Airy's
Non-homogeneous Equation, Applied Math. Comput.
Vol. 21(17), 2011, pp. 7349-7360.
[14] Miller, J. C. P. and Mursi, Z., Notes on the
solution of the equation , Quarterly J.
Mech. Appl. Math., Vol. 3, 1950, pp. 113-118.
[15] Nield, D.A. and Kuznetsov, A.V., The effect of
a transition layer between a fluid and a porous
medium: shear flow in a channel, Transp Porous
Med, Vol. 78, 2009, pp. 477-487.
[16] Jayyousi Dajani, S. and Hamdan, M.H., Airy’s
Inhomogeneous Equation with Special Forcing
Function, İSTANBUL International Modern
Scientific Research Congress –II, Istanbul,
TURKEY, Proceedings, ISBN: 978-625-7898-59-1,
IKSAD Publishing House, Dec. 23-25, 2021, pp.
1367-1375.
[17] Dunster, T.M., Nield-Kuzenetsov Functions and
Laplace Transforms of Parabolic Cylinder Functions,
SIAM J. Math. Anal., Vol. 53(5), 2021, pp. 915-
5947.
Contribution of individual authors
Both authors reviewed the literature, formulated the
problem, provided independent analysis, and jointly
wrote and revised the manuscript.
Sources of funding
No financial support was received for this work.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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