Direct and Transform Methods to Higher Derivatives of Ki(x)
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
Abstract: - Higher derivatives and associated polynomials of the standard Nield-Kuznetsov function of the second
kind are investigated in this work. Two approaches are introduced in this work. The first, is the direct method of
differentiation and generalization of the nth derivative. This approach is dependent on higher derivatives of the
Nield-Kuznetsov function of the first kind. The second is the transform method in which integral transforms
associated with the Nield-Kuznetsov function of the second kind are introduce first, and higher derivatives are
then obtained. The transform method is independent of the direct higher derivatives of the Nield-Kuznetsov
function of the first kind. Both approaches are important in practical and theoretical mathematical analysis, and
both give rise to associated Airy polynomials, discussed in this work.
Key-Words: - Higher derivatives, Ki(x), Airy’s polynomials.
Received: August 19, 2021. Revised: March 26, 2022. Accepted: April 28, 2022. Published: June 6, 2022.
1 Introduction
In a previous article, Hamdan et.al. [1] provided
analysis of the higher derivatives of the function
, and the associated Airy’s polynomials that
arise with, and define its nth derivative. The function
, better-known as the Nield-Kuznetsov
function of the first kind, arises in the general and
particular solutions to inhomogeneous equation when
homogeneity is due to a constant forcing function,
[2,3].
For the sake of clarity, Airy’s inhomogeneous
equation takes the form, [4,5]
(1)
where is a constant, and its general solution is of
the form
(2)
are arbitrary constants, and is defined as:
(3)
where and are Airy’s functions of the
first and second kinds, respectively, [5,6].
Studies of higher derivatives of Airy’s functions,
[7], and of the Nield-Kuznetsov functions, [1], are
imperative from both practical applications and
theoretical implications. While a knowledge of
higher derivatives and associated polynomials might
lead to further applications in mathematical physics,
quantum theory, and systems theory, [8], they also
further our understanding of infinite series
representations of the said functions and
polynomials.
Success of studies of Airy’s functions higher
derivatives and polynomials, [7], and of , [1],
motivate the current work in which higher derivatives
and associated polynomials of the Nield-Kuznetsov
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function of the second kind, , are investigated.
This function arises in the particular and general
solutions of Airy’s equation (1), with the right-hand-
side, , replaced by a continuously differentiable
function , [2]. This general solution takes the
form:
)
(4)
The function is defined in either of the
forms, [2]:
(5)
(6)
To this end, the derivative of is
obtained in two ways: a direct method in which (5) is
differentiated and a generalization is obtained for the
nth derivative, and a transform method in which
integral transforms are developed for then the
nth derivative is obtained.
2 Higher Derivatives of
Equation (5) shows that is defined in terms of
and Airy’s functions and .
Consequently, derivatives of must dpend on
derivatives of , and . Higher
derivatives of are discussed in what follows.
In Hamdan et.al., [1], the first two derivatives of
have been expressed as;
(7)
(8)
However, third and higher derivatives of
have been expressed in terms of the functions
and , and the Wronskian of and ,
.
The derivative of can then be expressed
as, [1]:
(9)
With the knowledge of the nth derivative, the
n+1st derivative can be obtained as:
(10)
Equation (10) takes the following form in terms of
and
(11)
Using (9) in (10) yields the n+1st derivative as:
(12)
Comparing (9) and (10), establishes the following:
(13)
(14)
(15)
It is worth noting that polynomials ,
are the same polynomials that arise in the
derivatives of and , respectively, as
obtained by Abramochkin and Razueva, [7].
Clearly, higher derivatives of Ni(x) can be
expressed in terms of Ni(x) and its first derivative
Ni’(x), and the Wronskian
,
whose coefficients are polynomials. Associated with
the derivative of are the polynomials
, and , wherein “n” denotes the
order of the derivative. For instance, coefficients of
and the Wronskian for sample
derivatives above are given in Table 1, below, for
, (Hamdan et.al. [1])
Table 1. Polynomial Coefficients of ,
and
n
2
x
0
1
3
1
x
0
5
4x
3
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10
15
3 Higher Derivatives of
Higher derivatives of , defined by (5) and (6),
can be obtained in two ways, one of which is
following the method used for obtaining higher
derivatives of , above, and involves derivatives
of , [9,10], while the second method is
independent of , but requires the introduction
of integral transforms. Both methods are discussed in
what follows.
Method 1: The Direct Method
Using (5), the first few derivatives of are
obtained as:
.
(16)
(17)
. (18)
Continuing in this manner, the derivative takes
the form:
(19)
where and are the polynomial
coefficients of the integral terms and of the
Wronskian that appear in the nth derivative, namely
(20)
(21)
is coefficient of
(22)
Following Alderson and Hamdan, [9], and
Jayyousi-Dajani and Hamdan, [10], relationships
between polynomials and are
given by:
(23)
(24)
(25)
The n+1st derivative of , obtained by
differentiating (19), takes the form:
(26)
Polynomials , and are
associated with the derivative of , where
refers to the order of the derivative and not the degree
of the polynomial. These polynomials are the
negatives of the polynomials associated with the
derivatives of Airy’s functions, and , and
the derivative of the standard Nield-Kuznetsov
function of the first kind, , [1]. Thus, for
, the following relationships hold:
(27)
(28)
(29)
Table 2, below, lists the polynomials ,
and , for .
Table 2. Coefficient Polynomials and Coefficient
Function
0
-1
0
0
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1
0
-1
0
2
-x
0
0
3
-1
-x
0
4
-2
5
6
7
8
9
10
Degrees of the coefficient polynomials may be
determined for arbitrary order of derivative, , and
are provided in the following Table 3 in terms of the
floor function.
Table 3. Degrees of Coefficient Polynomials
Polynomial
Degree
3
3
3
Now, using (5), expression (26) can be written in
the following form:
(30)
Replacing in (23)-(25), the
derivative of , obtained from (30), takes the
following form:
; n = 1,2,3,… (31)
The derivative of is thus given by (31),
and the above discussion furnishes the following
Theorem.
Theorem 1:
Let on . Then, the Nield-
Kuznetsov function of the second kind, defined by
is continuously differentiable with an
derivative given by
; n = 1,2,3,…
where , , and are given by
(3), (15)-(17), respectively.
Method 2: The Transform Method
Definition (6) of can be conveniently written
in terms of the following transforms.
Define
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(32)
(33)
then (6) can be written as
(34)
The first few derivatives of (32) and (32) take the
form:
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
Continuing this pattern, we see that the
derivatives of and take the forms:
(47)
(48)
The first few derivatives of (34) are as follows.
(49)
(50)
(51)
These derivatives generalize into the following
derivative of :
(52)
Using (47) and (48), we write:
(53)
(54)
Using the following general forms of derivatives
of , given in Hamdan et.al., [1], and
Abramochkin and Razueva, [7]:
(55)
(56)
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we write
(57)
(58)
(59)
) (60)
Using (57)-(60) in (52)-(54), we obtain the following
form of the derivative of :
(61)
wherein
(62)
(63)
With the knowledge of the derivative of
, we can obtain the derivative of as:
(64)
Using (13) and (14) in the form
(65)
(66)
Equation (64) takes the following form:
(67)
The polynomials and appearing
in (68) are of course known from the derivative
of . Now, replacing by in (67) gives the
final form of the derivative of , and
furnishes the following theorem.
Theorem 2:
Let on . Then, the Nield-
Kuznetsov function of the second kind, defined by
is continuously differentiable with an
derivative given by
; n=1,2,3,…
where and are given by (53) and (54),
respectively.
4 Values of the Derivatives at Zero
Although Theorems (1) and Theorem (2) provide
equivalent forms of the derivative of ,
computations using Theorem 1 are easier to perform.
Using Theorem 1, values at of the
derivative of are given by:
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; n = 1,2,3,… (68)
where , and
. (69)
5 Conclusion
In this work, general forms of the derivative of
the Standard Nield-Kuznetsov Function of the
Second Kind, have been obtained using two
approaches: the direct approach, which is dependent
on the Nield-Kuznetsov function of the first kind,
, and its higher derivatives, and the second is
based on the introduction of integral transforms for
. Both approaches are viable, yet the first
approach is more suitable for evaluation of the
derivatives. Airy’s polynomials arising for these
derivatives have been discussed and quantified, and
relationships between them have been investigated.
References:
[1] M.H. Hamdan, S. Jayyousi Dajani and M.S. Abu
Zaytoon, Higher Derivatives and Polynomials of
the Standard Nield-Kuznetsov Function of the
First Kind, Int. J. Circuits, Systems and Signal
Processing, Vol. 15, 2021, pp. 1737-1743.
[2] M.H. Hamdan and M.T. Kamel, On the Ni(x)
Integral Function and its Application to the Airy's
Non homogeneous Equation, Applied Math.
Comput., Vol. 21 No. 17, 2011, pp. 7349-7360.
[3] D.A. Nield and A.V. Kuznetsov, The Effect of a
Transition Layer between a Fluid and a Porous
Medium: Shear Flow in a Channel, Transport in
Porous Media, Vol. 78, 2009, pp. 477-487.
[4] G.B. Airy, On the Intensity of Light in the
Neighbourhood of a Caustic, Trans. Cambridge
Phil. Soc. Vol. 6, 1838, pp. 379-401.
[5] O. Vallée and M. Soares, Airy Functions and
Applications to Physics. World Scientific,
London, 2004.
[6] M. Abramowitz and I.A. Stegun, Handbook of
Mathematical Functions, Dover, New York
1984.
[7] E.G. Abramochkin and E.V. Razueva, Higher
Derivatives of Airy’s Functions and of Their
Products”, SIGMA, Vol. 14, 2018, pp. 1-26.
[8] A.Kh. Khanmamedov, M.G. Makhmudova1 and
N.F. Gafarova, Special Solutions of the Stark
equation, Advanced Mathematical Models &
Applications, Vol. 6(1), No.1, 2021, pp. 59-62.
[9] T.L. Alderson and M.H. Hamdan, Taylor and
Maclaurin Series Representations of the Nield-
Kuznetsov Function of the First Kind. WSEAS
Transactions on Equations. Vol. 2, 2022, pp. 38-
47.
[10] S. Jayousi Dajani and M.H. Hamdan, Higher
Derivatives of the Nield-Kuznetsov Integral
Function of the Second Kind. 2 International
Antalya Scientific Research and Innovative
Studies Congress. March 17-21, 2022, Antalya,
Turkey. pp. 518-525.
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
https://creativecommons.org/licenses/by/4.0/deed.en
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