Puiseux and Taylor Series of the Einstein Functions and Their Role in
the Solution of Inhomogeneous Airy’s Equation
M.H. HAMDAN
Department of Mathematics and Statistics
University of New Brunswick
100 Tucker Park Road, Saint John, N.B., E2L 4L5
CANADA
D.C. ROACH
Department of Engineering
University of New Brunswick
100 Tucker Park Road, Saint John, N.B., E2L 4L5
CANADA
Abstract: - The Einstein functions in generalized Puiseux and Taylor series are used as forcing functions in Airy’s
inhomogeneous equation, and particular and general solutions are obtained. Comparison are made with solutions
obtained using the Nield-Kuznetsov functions’ approach. For each of the Einstein’s functions, the standard Nield-
Kuznetsov function of the second kind is expressed in terms of Bessel functions. Computations and graphs in
this work were produced using Wolfram Alpha.
Key-Words: - Puiseux series, Einstein and Nield-Kuznetsov functions.
Received: August 9, 2021. Revised: May 3, 2022. Accepted: May 21, 2022. Published: June 15, 2022.
1 Introduction
In a recent article, Roach and Hamdan, [1],
investigated solutions to Airy’s inhomogeneous
equation when the inhomogeneity (the forcing
function in Airy’s equation) is due to Einstein’s
functions. Einstein functions are combinations of
logarithmic and exponential that arise in the study of
distributions, and the determination of physical and
chemical material constants arising in the study of
Einstein’s field equations. For these and many other
applications of Einstein functions, one is referred to
the elegant works of Abramowitz and Stegun, [2],
Hilsenrath and Ziegler, [3], Cezairliyan, [4], and the
references therein.
The main objective of the work of Roach and
Hamdan, [1], was to find a connection between
Airy’s functions, [5], of the first and second kind, and
the four Einstein functions, are
given by:
where is the natural logarithm.
In order to accomplish their objective, Roach and
Hamdan, [1], provided particular and general
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solutions to Airy’s inhomogeneous ordinary
differential equation (ODE) of the form
wherein “prime” notation denotes ordinary
differentiation with respect to the independent
variable, and is one of the four functions in (1).
The particular solution to (2) is given by, [1]:
and the general solution is given by
where are arbitrary constants, and
are the linearly independent Airy’s
functions of the first and second kind, respectively,
[6], with a non-zero Wronskian given by, [2,6]:
Roach and Hamdan, [1], obtained the following
particular solutions corresponding to (1),
respectively, by evaluating (3):
where is the zeta function and
, and and are
polylogarithmic functions, [7,8].
In this work, an alternative method is offered in
which the series form of Einstein functions is used in
the evaluation of particular solution (3). The use of
series and tis approach might offer an easier
alternative to computing particular and general
solutions when dealing with initial and boundary
value problems. In the process of this work, the
existing relationships between Einstein’s functions
and the standard Nield-Kuznetsov functions,
[1,9,10], will be utilized to express the standard
Nield-Kuznetsov function of the second kind in terms
of Bessel functions, [11].
2 Einstein and Bessel Functions
Roach and Hamdan, [1], obtained particular solution
(3) through the following alternate form, introduced
by Hamdan and Kamel, (2011):
where and are the standard Nield-
Kuznetsov functions of the first and second kind,
respectively, defined as, (Nield and Kuznetsov, 2009,
Hamdan and Kamel, [9]:
where in the work of Roach and
Hamdan, [1], and in the current work.
The following relationships between Einstein
functions, , Airy’s functions, and ,
and the standard Nield-Kuznetsov functions
and were established in a theorem introduced
by Roach and Hamdan, [1]:
Using (6) and (7), the following relationships
involving the polylogarithm functions are developed:
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With the knowledge of the expressions of
and , and their integrals in terms of Bessel’s
function of the first kind as, [11]:
wherein
, then using (16)-(19) in (8), the
function can be expressed in terms of Bessel’s
function, as, [11]:
Using (16)-(19) in (12)-(15), the function
in (12)-(15) can be expressed, respectively, in terms
of Bessel’s function, as
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Although (21)-(24) do not have direct
implications on solving Airy’s inhomogeneous
equation at present, they are of theoretical value and
are presented in this work for the sake of
completeness and to provide a connection of Bessel’s
function and Einstein’s functions.
3 Series Expressions of Einstein Functions
Some of the elementary properties of the Einstein
functions, their domains, ranges, graphs and series
representations are summarised in what follows, [12].
Case 1:
Domain of is the set of real numbers except
, and its range is the set of values
or . Its graph is given in Fig. 1
Fig. 1. Graph of
The following improper integral of converges:
and the function has a horizontal asymptote at ,
namely:
First derivative of and its indefinite integral are
given by:
can be approximated by the following
Maclaurin series:
Using (29) in (3), the following particular solution is
obtained for (2) when its forcing function is :
and the general solution can be obtained by
substituting (30) in (4).
Case 2:
Domain of is the set of positive real numbers
and its range is the set of negative real numbers. Its
graph is shown in Fig. 2.
Fig. 2. Graph of
The following improper integral of
converges:
and the function has a horizontal asymptote at ,
namely:
First derivative of and its indefinite integral are
given by:
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can be approximated by the following
Puiseux series:
Using (35) in (3), the following particular solution is
obtained for (2) when its forcing function is :
and the general solution can be obtained by
substituting (36) in (4).
Case 3:
Domain of is the set of positive real numbers
and its range is the set of positive real numbers. Its
graph is shown in Fig. 3.
Fig. 3. Graph of
The following improper integral of
converges:
and the function has a horizontal asymptote at ,
namely:
First derivative of and its indefinite integral are
given by:
can be approximated by the following Puiseux
series:
Using (41) in (3), the following particular solution
is obtained for (2) when its forcing function is :
and the general solution can be obtained by
substituting (42) in (4).
Case 4:
Domain of is the set of real numbers except
, and its range is the set of values
Its graph is given in Fig.
Fig. 4. Graph of
The following improper integral of converges:
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and the function has a horizontal asymptote at ,
namely:
First derivative of and its indefinite integral are
given by:
can be approximated by the following
Maclaurin series:
Using (47) in (3), the following particular solution is
obtained for (2) when its forcing function is :
and the general solution can be obtained by
substituting (48) in (4).
4 Results and Discussion
Polylogarithmic expressions for the particular
solutions of Airy’s inhomogeneous equation with
Einstein’s functions as its right-hand-side, as given
by equation (6), are graphed in Figs.5(a), 6(a), 7(a)
and 8(a). Corresponding particular solutions
obtained from Taylor and Puiseux series, as given by
equations (30), (36), (42) and (48), are plotted in
Figs.5(b), 6(b), 7(b) and 8(b).
For particular solutions obtained using Taylor
series expressions of the Einstein function, Fig. 5(a)
and 5(b) show similar trends in the graphs, although
their numerical values are different. Similar behavior
is observed in Fig. 8(a) and 8(b). Although no
solution to initial or boundary value problems has
been obtained in this work, hence solutions based on
the general solutions have not been computed, it is
expected that numerical values of the general
solutions in both approaches should be close.
For particular solutions in Fig. 6(a) and 6(b), and
in Fig. 8(a) and 8(b), differences occur in both the
graphical trends and in the numerical values. This
might be attributed to the use of Puiseux series in
these cases, or it might be possible that solutions to
initial and boundary value problems based on the
general solutions would adjust themselves
numerically. While this is inconclusive at present,
graphs of the particular solutions are meant to
provide information with regard to how close the
polylogarithm expressions are to series expressions.
Fig. 5(a)
Corresponding to in Equation (6)
Fig. 5(b) of Equation (30)
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Fig. 6(a)
Corresponding to in Equation (6)
Fig. 6(b) of Equation (36)
Fig. 7(a)
Corresponding to in Equation (6)
Fig. 7(b) of Equation (42)
Fig. 8(a)
Corresponding to in Equation (6)
Fig. 8(b) of Equation (48)
5 Conclusion
In this work, a method of solution of the
inhomogeneous Airy’s equation when the right-
hand-side is one of Einstein’s functions is
investigated. The method is based on expressing the
Einstein functions in series form followed by
obtaining particular solutions. The main conclusion
that can be drawn from this work is that Taylor series
expansions of Einstein’s functions produce particular
solutions with a trend that is similar to that of
particular solutions expressed in polylogarithmic
functions.
References:
[1] Roach, D.C. and Hamdan, M.H.,
M.H., Connecting Einstein’s functions to
the Nield-Kuznetsov functions. Transactions
on Equations. WSEAS, Vol. 2, 2022, pp. 48-
53.
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[2] Abramowitz, M. and Stegun, I.A., Handbook
of Mathematical Functions, Dover, New
York, 1984.
[3] Hilsenrath, J. and Ziegler, G.G., Tables of
Einstein Functions, Nat. Bur. Stand. (U.S.),
Monograph 49, 258 pages, 1962.
[4] 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.
[5] Airy, G.B., On the Intensity of Light in the
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[6] Vallée, O. and Soares, M., Airy functions and
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[7] Maximon, L.C. The Dilogarithm Function
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[8] Lewin, L., ed., Structural Properties of
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[9] 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.
[10] 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.
[11] Hamdan, M.H., Jayyousi Dajani, S., and
Abu Zaytoon, M.S., Higher derivatives
and polynomials of the standard Nield-
Kuznetsov function of the first kind. Int. J.
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[12] Wolfram MathWorld,
https://mathworld.wolfram.com/ 2022.
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
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International , CC BY 4.0)
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