Basic properties of incomplete Macdonald function with applications

The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the basic properties of the incomplete Macdonald function, such as recurrence and differential relations, series and asymptotic expansions. This paper also shows that the incomplete Macdonald function, as a simple closed-form expression, is a particular solution to a parabolic partial differential equation, which arises naturally in a wide variety of transient and diffusion-related phenomena.


Introduction
As a particular solution to the modified Bessel equation [1], the Macdonald function [2] (also known as the modified Bessel function of the second kind, the Basset function, or the modified Hankel function) has been employed in wide-ranging fields to provide analytical solutions for many physical phenomena. The Macdonald function can be expressed in the form of integrals. A new class of incomplete special function, called the incomplete Macdonald function, is defined by having the variable endpoint of integration and arises in wide-ranging contexts such as viscous flow [3], heat conduction [4], groundwater hydrology [5], electromagnetism [6], galaxy [7], nuclear reactor [8], and doubly special relativity [9].
The several distinct incomplete versions of the Macdonald function are possible since various integral representations of the Macdonald function often seem unrelated under the variable endpoint of integration. In literature, the specific definition chosen among various forms of the incomplete Macdonald function is directed by the particular application. These various forms and interrelationships of the incomplete Macdonald function are summarized in the Table 1.
The difference in the names assigned to various forms of the incomplete Macdonald function is worth noting (as listed in Table 1). They are the Shu function of viscous flow, generalized incomplete gamma function of heat conduction, leaky aquifer function of groundwater hydrology, and incomplete modified Bessel function of electromagnetism. It is pretty apparent that there is a lack of communication among different research communities. The Shu function was first employed by Shu and Chwang [3] in the expression of the hydrodynamic force acting on a rigid circular cylinder translating in a time-dependent rotating flow field. The motivation of studying the generalized incomplete gamma function [4] was the role it played in the closedform solution to several problems in heat conduction. Groundwater hydrologists commonly refer to the leaky aquifer integral as the leaky aquifer function [5], which is useful for determining the hydraulic properties of leaky-confined aquifers. The incomplete modified Bessel function [6] was introduced to express the solution of electromagnetic problems in truncated cylindrical structures. It is worth mentioning at this end that these Shu function, generalized incomplete gamma function, leaky aquifer function, and incomplete modified Bessel function have a well-deserved reputation of being a computational challenge due to their integral representations. The increasing number of applications calls for a rigorous analysis of various forms of the incomplete Macdonald function. To avoid redundancy, the special function analyzed in this paper is mainly referred to as the Shu function, which is a representative of a large class of time-dependent problems.
In this paper, the key properties of the Shu function, such as recurrence and differential relations, series and asymptotic expansions, are derived. It is also shown that the Shu function is a particular solution to a parabolic partial differential equation (PDE), which occurs in various transient problems, for example, transient flow in porous media [10], electromagnetic waves in a cylindrical waveguide [11,12] and diffusion-related phenomena [8].

Definition
The integral form of the Macdonald function is given [13] by in which Re denotes the real part of a complex number, the symbols  , z , and t are, respectively, termed the order, argument, and endpoint of the Shu function and are generally taken to be complex quantities. Hereafter, the symbols n and 0  x are, respectively, annotated to denote integer order  and real argument z . The behavior of the Shu function   t x S n , of integer order n , real argument 0  x , and real endpoint 0  t is depicted in Figures 1 and 2. In view of that the Shu function has infinite when Substituting  2 z e w  in (2), we get [14]   By using the substitution: This can be expressed in the form of the generalized incomplete gamma function [4] ( Furthermore, the Shu function can be related to the leaky aquifer function [5] (Table 1) Using this relation, the algorithm [15] proposed for the computation of numerical value of

Recurrence Relations
Two recurrence relations of the Shu function are derived in this section.

The First Recurrence Relation
Integrating (2) by parts, we obtain Here, note that Hence, we obtain 1 1 On simplifying this expression, we obtain the first recurrence relation,

The Second Recurrence Relation
We differentiate (2) with respect to z , On simplifying, we obtain the second recurrence relation,

Differential Relations
Two differential relations of the Shu function are derived in this section.

The First Differential Relation
Adding the two recurrence relations (8) and (9), we obtain Premultiplying by  z , we get This differential relation may be extended as

Partial Differential Equation
Replacing  by 1   in (10), we obtain On differentiating (12) with respect to z , we get From (12), (14), and (15), it is a straightforward exercise to obtain the following PDE: The PDE can also be written as where  represents the modified Bessel operator: This elegant parabolic PDE (16) arises naturally in a wide variety of transient and diffusionrelated phenomena.

Series Expansions
The asymptotic expansion of the incomplete gamma function    The leading term approximation is given by The leading term approximation to the actual value of the Shu function   t x S n , of integer order n and real argument 0  x for small endpoint t is shown in Figure 3. As can be observed, the agreement is better for higher integer order n .
The leading term approximation is given by The leading term approximation to the actual value of the Shu function   t x S n , of integer order n and real argument 0  x for large endpoint t is shown in Figure 5. As can be observed, the agreement is excellent for all integer order n .
which agrees with the result of Cicchetti and Faraone [6], but their expression is extremely complicated. The leading term approximation to the actual value of the Shu function   t x S n , of integer order n and real endpoint 0  t for large argument x is shown in Figure 6. As can be observed, the agreement is better for lower integer order n .

Conclusions
A number of key properties of the incomplete Macdonald function have been derived, comprising recurrence and differential relations, series and asymptotic expansions. As can be observed graphically, the agreement is found between the actual value and the corresponding leading term approximation for all limiting cases. It is also shown that the incomplete Macdonald function is a particular solution to the parabolic PDE associated with a wide variety of transient natural phenomena. The Shu function and other distinct incomplete versions of the Macdonald function can be used to find simple closed-form expressions for various natural phenomena.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.