Melt flow of biopolymer through the cavities of an extruder die: Mathematical modelling

Alexander N. Ostrikov; Abdymanap A. Ospanov; Vitaly N. Vasilenko; Nurzhan Zh. Muslimov; Aigul K. Timurbekova; Gulnara B. Jumabekova; Alexander N. Ostrikov; Abdymanap A. Ospanov; Vitaly N. Vasilenko; Nurzhan Zh. Muslimov; Aigul K. Timurbekova; Gulnara B. Jumabekova

doi:10.3934/mbe.2019142

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2875-2905. doi: 10.3934/mbe.2019142

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Research article

Melt flow of biopolymer through the cavities of an extruder die: Mathematical modelling

1.
Voronezh State University of Engineering Technologies, Voronezh, Russian Federation
2.
Kazakh National Agrarian University, Almaty, Republic of Kazakhstan
3.
Taraz innovative-humanities university, Taraz, Republic of Kazakhstan
4.
M.Kh. Dulaty Taraz State University, Taraz, Republic of Kazakhstan

Received: 13 December 2018 Accepted: 07 March 2019 Published: 09 April 2019

This is an analytical solution of the two-dimensional non-isothermal mathematical model describing the change in the velocity profile of a cylindrical extrusion die. This solution is based on the following assumptions. The two-dimensional melt flow is asymmetric. A melt viscosity anomaly may take place. Heat generated by viscous friction is a factor affecting the melt flow. The melt flow moving towards the metering section is in a steady state. Neither mass forces nor inertia forces are present. Velocity gradients along the channel are neglected. The mathematical model was built up from the incompressibility equation, motion equations, energy equation, and the rheological equation. This model depicted a non-isothermal flow of rheological fluid moving through the cylindrical extrusion die. A diagram was drawn. It depicts the melt velocities at a die entrance in different cross-sectional views. Computer testing was performed to verify the obtained solutions and compare them with the real extrusion process. Difference between calculated and experimental data was below 14%. Results allow concluding a matching of numerical results with experimental data, and so the possibility of using a built-up model in an extrusion die design for single-screw extruders.

Keywords:

Citation: Alexander N. Ostrikov, Abdymanap A. Ospanov, Vitaly N. Vasilenko, Nurzhan Zh. Muslimov, Aigul K. Timurbekova, Gulnara B. Jumabekova. Melt flow of biopolymer through the cavities of an extruder die: Mathematical modelling[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2875-2905. doi: 10.3934/mbe.2019142

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Abstract

1. Introduction and preliminaries

In applied sciences, many important functions are defined via improper integrals or series (or finite products). The general name of these important functions knows as special functions. In special function, one of the most important function (Bessel function) has gained importance and popularity due to its applications in the problem of cylindrical coordinate system, wave propagation, heat conduction in cylindrical object and static potential etc. In the recent years, some generalizations (unification) and number of integral transforms of Bessel functions have been given by many mathematicians and physicist as well as engineers. The Bessel-Maitland function $J_{\vartheta }^{\tau } \left(z\right)$ is a generalization of Bessel function, defined in ^[7] through a series representation as:

$\begin{equation} J_{\vartheta }^{\tau } \left(z\right) = \sum\limits _{n = 0}^{\infty }\frac{\left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +1\right)n{\rm !}} \end{equation}$

(1.1)

In fact, the application of Bessel-Maitland function are found in the diverse field of mathematical physics, engineering, biological, chemical in the book of Watson ^[26].

Further, generalization of the generalized Bessel-Maitland function defined by Pathak ^[13] is as follow:

$\begin{equation} J_{\vartheta , q}^{\tau , \varsigma } \left(z\right) = \sum\limits_{n = 0}^{\infty }\frac{\left(\varsigma \right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-z\right)^{n} }{n{\rm !}} \end{equation}$

(1.2)

where $\tau, \vartheta, \varsigma \in \mathbb{C}; \, \Re \left(\tau \right)\ge 0, \Re \left(\vartheta \right)\ge -1, \, \Re \left(\varsigma \right)\ge 0, \, q\in \left(0, 1\right)\, \cup \mathbb{N}.$

Motivated by the established potential for application of these Bessel-Maitland functions, we introduce here another interesting extension of the generalized Bessel-Maitland function as follow:

$\begin{equation} J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-z\right)^{n} }{n{\rm !}} \end{equation}$

(1.3)

where $\tau, \vartheta, \varsigma, \omega \in \mathbb{C}; \, p > 0, \, \Re \left(\tau \right) > 0, \Re \left(\vartheta \right)\ge -1, \, \Re \left(\varsigma \right) > 0, \Re \left(s\right) > 0, \, q\in \left(0, 1\right)\, \cup \mathbb{N},$ which will be known as extended generalized Bessel Maitland function (EGBMF).

Here, $\mathcal{B}_{\omega } \left(x, y; p\right)$ is an extension of extended beta function introduced by Parmar et al. ^[12] in the following way:

$\begin{equation} \mathcal{B}_{\omega } \left(x, y;p\right) = \sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{1}t^{x-{\tfrac{3}{2}} } \left(1-t\right)^{y-{\tfrac{3}{2}} } K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right)dt \end{equation}$

(1.4)

where $K_{\omega +{\tfrac{1}{2}} } (.)$ is the modified Bessel's function. The special case of (1.4) corresponding to $\omega = 0$ be easily seen to reduce to the extended beta function

$\begin{equation} B\left(x, y;p\right) = \int _{0}^{1}t^{x-1} \left(1-t\right)^{y-1} \exp \left(\frac{-p}{t(1-t)} \right)dt \end{equation}$

(1.5)

upon making use of (^[9], Eq (10.39.2)). If $p = 0$ in Eq (1.5), reduces in to the classical beta function. For a detailed account of various properties, generalizations and applications of Bessel-Maitland functions, the readers may refer to the recent work of the researchers ^{[3,15,21,22,23,24,25]} and the references cited therein.

2. Integral representation

Theorem 2.1. The extended generalized Bessel Maitland function will be able to depict:

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \frac{1}{B\left(\varsigma , \omega -\varsigma \right)} \sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{1}t^{\varsigma -{\tfrac{3}{2}} } \left(1-t\right)^{s-\varsigma -{\tfrac{3}{2}} }$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \times K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right) J_{\vartheta , q}^{\tau , s} \, \left(t^{q} z\right)\, dt \end{equation}$

(2.1)

Proof. Using Eq (1.4) in Eq (1.3), we obtain

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \sum\limits_{n = 0}^{\infty }\left\{\sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{1}t^{\varsigma +nq-{\tfrac{3}{2}} } \left(1-t\right)^{s-\varsigma -{\tfrac{3}{2}} } K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right)dt \right\}$

$\begin{equation} \times \frac{\left(s\right)_{nq} \left(-z\right)^{n} }{B\left(\varsigma, s-\varsigma \right)\Gamma \left(\tau n+\vartheta +1\right)\, n{\rm !}} \end{equation}$

(2.2)

Reciprocate the order of summation and integration, that is surd under the presumption given in the description of Theorem 2.1, we get

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \frac{1}{B\left(\varsigma , s-\varsigma \right)} \sum\limits_{n = 0}^{\infty }\sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{1}t^{\varsigma +nq-{\tfrac{3}{2}} } \left(1-t\right)^{s-\varsigma -{\tfrac{3}{2}} }$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \times K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right) \frac{\left(s\right)_{nq} \left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +1\right)n{\rm !}} dt \end{equation}$

(2.3)

Using Eq (1.2) in Eq (2.3), we obtain the desired result Eq (2.1).

Corollary 2.2. Let the condition of Theorem 2.1 be satisfied, the following integral representation holds:

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, = \frac{1}{B\left(\varsigma , s-\varsigma \right)} \sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{\infty }r^{\varsigma -{\tfrac{3}{2}} } \, \left(1+r\right)^{2-s} K_{\omega +{\tfrac{1}{2}} } \left(\frac{-p\left(1+r\right)^{2} }{r} \right)$

$\begin{equation} \times J_{\vartheta , q}^{\tau , s} \left(\left(\frac{r}{1+r} \right)^{q} z\right)dr\, . \end{equation}$

(2.4)

Proof. By taking $t = \frac{r}{1+r}$ in Theorem 2.1, After simplification, we obtain the desired result Eq (2.4).

Corollary 2.3. Assume the state of Theorem 2.1 is satisfied, the following integral representation holds:

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, = \frac{2}{B\left(\varsigma , s-\varsigma \right)} \sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{{\tfrac{\pi }{2}} }\left(\sin \theta \right)^{2(\varsigma -1)} \left(\cos \theta \right)^{2(\omega -\varsigma -1)}$

$\begin{equation} \, \times K_{\omega +{\tfrac{1}{2}} } \left(\frac{-p}{\sin ^{2} \theta \cos ^{2} \theta } \right)J_{\vartheta , q}^{\tau , s} \left(z\sin ^{2q} \theta \right)\, d\theta \, \, . \end{equation}$

(2.5)

Proof. If we set $t = \sin ^{2} \theta$ in Theorem 2.1, we acquire the above result.

3. Recurrence relation

Theorem 3.1. Let $\omega, \varsigma, \tau, \vartheta, \in \mathbb{C}; \, \Re \left(\tau \right) > 0, \Re \left(\vartheta \right)\ge -1, \, \Re \left(\varsigma \right) > 0, \Re \left(s\right) > 0, \, p > 0, \, q\in \left(0, 1\right)\, \cup \mathbb{N},$ then the recurrence relation holds true:

$\begin{equation} J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, \, = \, \, \left(\vartheta +1\right)J_{\vartheta +1, q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, +\, \tau \, z\frac{d}{dz} J_{\vartheta +1, q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, \end{equation}$

(3.1)

Proof. Employing Eq (1.3) in right hand side of Eq (3.1), we obtain

$\begin{eqnarray*} && \left(\vartheta +1\right)J_{\vartheta +1, q}^{\tau , \varsigma , s;\omega } \left(z;p\right)\, +\, \tau \, z\frac{d}{dz} J_{\vartheta +1, q}^{\tau , \varsigma , s;\omega } \left(z;p\right) \\& = & \left(\vartheta +1\right)\sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} \left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +2\right)n{\rm !}} \\&+& \tau \, z\frac{d}{dz} \sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} \left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +2\right)n{\rm !}} \\& = & \sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} \left(\tau n+\vartheta +1\right)\left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +2\right)n{\rm !}} \\& = & J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right). \end{eqnarray*}$

4. Derivative formulae

Theorem 4.1. For the extended generalized Bessel Maitland function, we have the following higher derivative formula:

$\begin{equation} \frac{d^{n} }{dz^{n} } J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \left(s\right)_{q} \left(s+q\right)_{q} ...\left(s+\left(n-1\right)q\right)_{q} J_{\vartheta +n\tau , q}^{\tau , \varsigma +nq, s+nq;\omega } \left(z;p\right). \end{equation}$

(4.1)

Proof. Taking the derivative with respect to $z$ in Eq (2.1), we get

$\begin{equation} \frac{d}{dz} J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \left(s\right)_{q} J_{\vartheta +\tau , q}^{\tau , \varsigma +q, s+q;\omega } \left(z;p\right) \end{equation}$

(4.2)

Again taking the derivative with respect to $z$ in Eq (6.5), we get

$\begin{equation} \frac{d^{2} }{dz^{2} } J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z;p\right) = \left(s\right)_{q} \left(s+q\right)_{q} J_{\vartheta +2\tau , q}^{\tau , \varsigma +2q, s+2q;\omega } \left(z;p\right) \end{equation}$

(4.3)

Ongoing the repetition of this technique $n$ times, we get the desired result Eq (4.1).

Theorem 4.2. For the extended generalized Bessel Maitland function, the following differentiation holds:

$\begin{equation} \frac{d^{n} }{dz^{n} } \left\{z^{\vartheta } J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\sigma \, z^{\tau } ;p\right)\right\} = z^{\vartheta -n} J_{\vartheta -n, q}^{\tau , \varsigma , s;\omega } \left(\sigma \, z^{\tau } ;p\right). \end{equation}$

(4.4)

Proof. Replace $z$ by $\sigma z^{\tau }$ in Eq (2.1) and take its product with $z^{\vartheta }$ , then taking $z$ -derivative $n$ times. We obtain our result.

5. Beta transform

Definition 5.1. The Beta transform ^[19] of a function $f(z)$ is defined as:

$\begin{equation} \mathfrak{B}\left\{f(z);\ a, b \right\} = \int _{0}^{1}z^{a-1} \left(1-z\right)^{b-1} \, f\left(z\right)dz \end{equation}$

(5.1)

$\left(a, b \in \mathbb{C}, \, \Re \left(a\right) \gt 0, \, \Re \, \left(b\right) \gt 0\right).$

Theorem 5.2. Let $\omega, \varsigma, \tau, \vartheta, \in \mathbb{C}; \, \Re \left(\tau \right) > 0, \Re \left(\vartheta \right)\ge -1, \, \Re \left(\varsigma \right) > 0, \Re \left(s\right) > 0, \, p > 0, \, q\in \left(0, 1\right)\, \cup \mathbb{N},$ Then the Beta transform of extended generalized Bessel Maitland function holds true:

$\begin{equation} \mathfrak{B}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda z^{\tau } ;p\right)\, ;\vartheta +1, 1\right\} = J_{\vartheta +1, q}^{\tau , \varsigma , s;\omega } \left(\lambda ;p\right). \end{equation}$

(5.2)

Proof. By definition of Beta transform (5.1) and (1.3), we get

$\mathfrak{B}\left(J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z^{\tau } ;p\right)\, ;\vartheta +1, 1\right)$

$\begin{equation} = \int _{0}^{1}z^{\vartheta } \left(1-z\right)\sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-\lambda z^{\tau } \right)^{n} }{n{\rm !}} dz, \end{equation}$

(5.3)

Upon interchanging the order of summation and integration in Eq (5.3), which can easily verified by uniform convergence under the constraint with Theorem 5.2, we get

$\mathfrak{B}\left(J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(z^{\tau } ;p\right)\, ;\vartheta +1, 1\right) = \sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-\lambda \right)^{n} }{n{\rm !}}$

$\times \int _{0}^{1}z^{\vartheta +\tau n} \left(1-z\right) dz,$

Using the familiar definition of beta function, and interpreting with Eq (1.3), we get the desired representation Eq (5.2).

6. Mellin transform

Definition 6.1. The Mellin transform ^[19] of the function $f\left(z\right)$ is defined as

$\begin{equation} \mathfrak{M}\left(f\left(z\right)\, ;\xi \right) = \int _{0}^{\infty }z^{\xi -1} f\left(z\right) dz = f^{*} \left(\xi \right), \, \, \, \, \, \, \, \left(\Re \left(\xi \right) \gt 0\right) \end{equation}$

(6.1)

then inverse Mellin transform

$\begin{equation} f\left(z\right) = \mathfrak{M}^{-1} [f^{*} \left(\xi \right)\, ;x] = \frac{1}{2\pi i} \int _{\lambda -i\infty }^{\lambda +i\infty }f^{*} \left(\xi \right)\, x^{-\xi } d\xi .\, \end{equation}$

(6.2)

In the next theorem, we give Mellin transform of the extended generalized Bessel Maitland function. Therefore, we require the definition of Wright generalized hypergeometric function ^[20] as:

$\begin{equation} {}_{p} \psi {}_{q} \left(z\right)\, = {}_{p} \psi {}_{q} \, \, \left. \left[\begin{array}{l} {\left(c_{1} , C_{1} \right)\, , \left(c_{2} , C_{2} \right)\, , ...\, , \left(c_{p} , C_{p} \right);\, } \\ {\left(d_{1} , D_{1} \right)\, , \left(d_{2} , D_{2} \right)\, , ...\, , \left(d_{q} , D_{q} \right);\, } \end{array}\right. \, z\right] = \sum\limits_{n = 0}^{\infty }\frac{\prod _{i = 1}^{p}\Gamma \left(c_{i} , C_{i} n\right)z^{n} }{\prod _{j = 1}^{q}\Gamma \left(d_{i} , D_{i} n\right)n{\rm !} } \end{equation}$

(6.3)

where the coefficients $C_{i} \left(i = 1, \, 2, \, ..., p\right)$ and $D_{j} \left(j = 1, \, 2, ..., q\right)$ are positive real numbers such that

$1+\sum\limits_{j = 1}^{q}D_{j} - \sum\limits_{i = 1}^{p}C_{i} \ge 0 .$

Theorem 6.2. The Mellin transform of the extended generalized Bessel Maitland function is given by

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{2^{\xi -1} \Gamma \left(\xi +s-\varsigma \right)}{\sqrt{\pi } \, \Gamma \left(\varsigma \right)\Gamma \left(s-\varsigma \right)} \Gamma \left(\frac{\xi -\omega }{2} \right)\Gamma \left(\frac{\xi +\omega +1}{2} \right)$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \times {}_{2} \psi _{2} \left. \left[\begin{array}{l} {\left(s, q\right), \, \left(\varsigma +\xi , q\right)\, ;} \\ {\left(\vartheta +1, \tau \right), \, \left(s+2\xi , q\right)\, ;} \end{array}\right. -z\right] \end{equation}$

(6.4)

where $\omega, \varsigma, \tau, \vartheta, \xi, \in \mathbb{C}; \, \Re \left(\tau \right) > 0, \Re \left(\vartheta \right)\ge -1, \, \Re \left(\varsigma \right) > 0, \Re \left(s\right) > 0, \Re \left(\xi \right) > 0, \, p > 0, \, q\in \left(0, 1\right)\, \cup \mathbb{N},$ and ${}_{2} \psi _{2}$ is the Wright generalized hypergeometric function.

Proof. Using the definition of Melllin transform (6.1) and (1.3), we obtain

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{1}{B\left(\varsigma , s-\varsigma \right)} \int _{0}^{\infty }p^{\xi -1} \left\{\sqrt{{\tfrac{2p}{\pi }} } \int _{0}^{1}t^{\varsigma -{\tfrac{3}{2}} } \left(1-t\right)^{s-\varsigma -{\tfrac{3}{2}} } \right. \,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \left. \times K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right)J_{\vartheta , q}^{\tau , s} \, \left(t^{q} z\right)\, dt\right\}dp, \end{equation}$

(6.5)

Interchanging the order of integration in Eq (6.5), which is admittable because of the conditions in the statement of the Theorem 3.4, we get

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{1}{B\left(\varsigma , s-\varsigma \right)} \sqrt{{\tfrac{2}{\pi }} } \int _{0}^{1}t^{\varsigma -{\tfrac{3}{2}} } \left(1-t\right)^{s-\varsigma -{\tfrac{3}{2}} } J_{\vartheta , q}^{\tau , s} \, \left(t^{q} z\right)\, dt,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \times \left\{\int _{0}^{\infty }p^{\xi -{\tfrac{1}{2}} } K_{\omega +{\tfrac{1}{2}} } \left(\frac{p}{t(1-t)} \right)\, dp\right\}dt \end{equation}$

(6.6)

Now taking $u = \frac{p}{t\left(1-t\right)}$ in Eq (6.6), we get

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{1}{B\left(\varsigma , s-\varsigma \right)} \sqrt{{\tfrac{2}{\pi }} } \int _{0}^{1}t^{\varsigma +\xi -1} \left(1-t\right)^{s-\varsigma +\xi -1} J_{\vartheta , q}^{\tau , s} \, \left(t^{q} z\right)\, dt$

$\begin{equation} \times \int _{0}^{\infty }u^{\xi -{\tfrac{1}{2}} } K_{\omega +{\tfrac{1}{2}} } \left(u\right)\, du, \end{equation}$

(6.7)

From Olver et al. ^[9]:

$\begin{equation} \int _{0}^{\infty }u^{\xi -{\tfrac{1}{2}} } K_{\omega +{\tfrac{1}{2}} } \left(u\right) du = 2^{\xi -{\tfrac{3}{2}} } \Gamma \left(\frac{\xi -\omega }{2} \right)\Gamma \left(\frac{\xi +\omega +1}{2} \right), \end{equation}$

(6.8)

Applying Eq (6.8) in Eq (6.7), we obtain

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{1}{B\left(\varsigma , s-\varsigma \right)} {\tfrac{2^{\xi -1} }{\sqrt{\pi } }} \int _{0}^{1}t^{\varsigma +\xi -1} \left(1-t\right)^{s-\varsigma +\xi -1} J_{\vartheta , q}^{\tau , s} \, \left(t^{q} z\right)dt$

$\times \Gamma \left(\frac{\xi -\omega }{2} \right)\Gamma \left(\frac{\xi +\omega +1}{2} \right),$

Using Eq (1.2), and interchanging the order of summation and integration which is valid for $\Re \left(\tau \right) > 0, \Re \left(\vartheta \right) > 0, \Re \left(s\right) > 0, \, \Re \left(s\right) > \Re \left(\varsigma \right) > 0, \, \, \Re \left(s+\xi -\varsigma \right) > 0,$ we obtain

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{2^{\xi -1} }{B\left(\varsigma , s-\varsigma \right)\, \sqrt{\pi } } \sum\limits_{n = 0}^{\infty }\frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)}\frac{\left(-z\right)^{n} }{n{\rm !}}\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \times \Gamma \left(\frac{\xi -\omega }{2} \right)\Gamma \left(\frac{\xi +\omega +1}{2} \right) \int _{0}^{1}t^{\varsigma +\xi +nq-1} \left(1-t\right)^{\xi +s-\varsigma -1} dt, \end{equation}$

(6.9)

Using the relation between Beta function and Gamma function, we obtain

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{2^{\xi -1} }{B\left(\varsigma , s-\varsigma \right)\, \sqrt{\pi } } \sum\limits_{n = 0}^{\infty }\frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \Gamma \left(\frac{\xi -\omega }{2} \right) \, \, \, \, \, \, \, \, \, \, \,$

$\times \Gamma \left(\frac{\xi +\omega +1}{2} \right) \frac{\Gamma \left(\varsigma +\xi +nq\right)\Gamma \left(s+\xi -\varsigma \right)}{\Gamma \left(s+2\xi +nq\right)} \frac{\left(-z\right)^{n} }{n{\rm !}},$

After simplification, we obtain

$\mathfrak{M}\left\{J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p);\xi \right\} = \frac{2^{\xi -1} \Gamma \left(\xi +s-\varsigma \right)\Gamma \left(\frac{\xi -\omega }{2} \right)\Gamma \left(\frac{\xi +\omega +1}{2} \right)}{\sqrt{\pi } \, \Gamma \left(\varsigma \right)\Gamma \left(s-\varsigma \right)}\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \times \sum\limits_{n = 0}^{\infty }\frac{\Gamma \left(s+nq\right)\Gamma \left(\varsigma +\xi +nq\right)\left(-z\right)^{n} }{\Gamma \left(\tau n+\vartheta +1\right)\, \Gamma \left(s+2\xi +nq\right)n{\rm !}} , \end{equation}$

(6.10)

In view of Eq (6.3), we arrived at our result Eq (6.4).

Corollary 6.3. Taking $\xi = 1$ in Theorem 6.2, we get

$J_{\vartheta , q}^{\tau , \varsigma , s;\omega } (z;p) = \frac{\Gamma \left(1+s-\varsigma \right)}{\sqrt{\pi } \, \Gamma \left(\varsigma \right)\Gamma \left(s-\varsigma \right)} \Gamma \left(\frac{1-\omega }{2} \right)\Gamma \left(\frac{\omega +2}{2} \right)\, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \times {}_{2} \psi _{2} \left. \left[\begin{array}{l} {\left(s, q\right), \, \left(\varsigma +1, q\right)\, ;} \\ {\left(\vartheta +1, \tau \right), \, \left(s+2, q\right)\, ;} \end{array}\right. -z\right]. \end{equation}$

(6.11)

7. Fractional calculus approach

In recent years, the fractional calculus has become a significant instrument for the modeling analysis and assumed a significant role in different fields, for example, material science, science, mechanics, power, science, economy and control theory. In addition, research on fractional differential equations (ordinary or partial) and other analogous topics is very active and extensive around the world. One may refer to the books ^[28,31], and the recent papers ^{[1,2,6,16,18,27,29,30,32,33,34,35]} on the subject. In this portion, we derive a slight interesting properties of EMBMF associated with the right hand sided of Riemann-Liouville (R-L) fractional integral operator $I_{a+}^{\zeta}$ and the right sided of R-L fractional derivative operator $D_{a+}^{\zeta}$ , which are defined for $\zeta\in \mathbb{C}, (\Re \left(\zeta\right) > 0),$ $x > 0$ (See, for details ^[5,17]):

$\begin{equation} \left(I_{a+}^{\zeta} f\right)(x)\, = \, \frac{1}{\Gamma \left(\zeta\right)} \int _{a}^{x}\frac{f(t)}{(x-t)^{1-\zeta} } dt, \end{equation}$

(7.1)

and

$\begin{equation} \left(D_{a+}^{\zeta} f\right)(x)\, = \, \left(\frac{d}{dx} \right)^{\ell } \left(I_{a+}^{\ell -\zeta} f\right)(x)\, \, \, \, \, \ell = \left[\Re \left(\zeta\right)+1\right]. \end{equation}$

(7.2)

where $\left[\Re \left(\zeta\right)\right]$ is the integral part of $\Re \left(\zeta\right)$ .

A generalization of R-L fractional derivative operator (7.2) by introducing a right hand sided R-L fractional derivative operator $D_{a+}^{\zeta, \, \sigma }$ of order $0 < \zeta < 1$ and $0\le \sigma \le 1$ with respect to $x$ by Hilfer ^[4] is as follows:

$\begin{equation} \left(D_{a+}^{\zeta, \sigma } f\right)(x)\, = \, \left(I_{a+}^{\sigma (1-\zeta)} \frac{d}{dx} \right)\left(I_{a+}^{(1-\sigma )(1-\zeta)} f\right)(x). \end{equation}$

(7.3)

The generalization Eq (7.3) yields the R-L fractional derivative operator $D_{a+}^{\zeta}$ when $\sigma = 0$ and moreover, in its special case when $\sigma = 1$ , the definition (7.3) would reduce to the familiar Caputo fractional derivative operator ^[5].

Theorem 7.1. Let $\zeta, \, \lambda, \tau, \, \vartheta, \varsigma, s\, \in \mathbb{C}$ be such that $\Re \left(\zeta\right) > 0,$ $p\ge 0$ and the conditions given in Eq (1.3) is satisfied, for $x > a,$ the following relation holds:

$\left(I_{a+}^{\zeta} \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = \, \left(x-a\right)^{\zeta+\vartheta } \, J_{\vartheta +\zeta, q}^{\tau , \varsigma , s;\omega } \left(\lambda (x-a)^{\tau } ;p\right). \end{equation}$

(7.4)

$\left(D_{a+}^{\zeta} \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = \, \left(x-a\right)^{\vartheta -\zeta} J_{\vartheta -\zeta, q}^{\tau , \varsigma , s;\omega } \, \left(\lambda (x-a)^{\tau } ;p\right). \end{equation}$

(7.5)

$\left(D_{a+}^{\zeta, \sigma } \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \,$

$\begin{equation} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = \, \left(x-a\right)^{\vartheta -\zeta} \, J_{\vartheta -\zeta, q}^{\tau , \varsigma , s;\omega } \left(\lambda (x-a)^{\tau } ;p\right). \end{equation}$

(7.6)

Proof. By virtue of the formulas Eq (7.1) and Eq (1.3), the term by term fractional integration and use of the relation ^[4]

$\begin{equation} \left(I_{a+}^{\zeta} \left(z-a\right)^{\vartheta -1} \right)(x)\, = \, \frac{\Gamma \left(\vartheta \right)}{\Gamma \left(\zeta+\vartheta \right)} \left(x-a\right)^{\zeta+\vartheta -1} \, \, \, \, \, \, \, \left(\vartheta , \zeta\in \mathbb{C}, \, \Re \left(\zeta\right) \gt 0, \Re \left(\vartheta \right) \gt 0\right) \end{equation}$

(7.7)

yield for $x > a.$

$\left(I_{a+}^{\zeta} \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\,$

$= \left(I_{a+}^{\zeta} \left\{\sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-\lambda \right)^{n} (z-a)^{\tau n+\vartheta } }{n{\rm !}} \, \right\}\right)(x),$

$\begin{equation} = \, \left(x-a\right)^{\zeta+\vartheta } \, J_{\vartheta +\zeta, q}^{\tau , \varsigma , s;\omega } \left(\lambda (x-a)^{\tau } ;p\right). \end{equation}$

(7.8)

Consequent, by Eq (7.5) and Eq (1.3), we find that

$\left(D_{a+}^{\zeta} \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\,$

$= \left(\frac{d}{dx} \right)^{\ell } \left(I_{a+}^{\ell -\zeta} \left\{\, \left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\,$

$\begin{equation} = \left(\frac{d}{dx} \right)^{\ell } \left(\left(x-a\right)^{\vartheta +\ell -\zeta-1} \, \, J_{\vartheta +\ell -\zeta, q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(x-a\right)^{\tau } ;p\right)\right)(x)\, , \end{equation}$

(7.9)

Applying Eq (4.4), we are led to the desired result Eq (7.5). Lastly, by Eq (7.3) and Eq (1.3), we becomes

$\left(D_{a+}^{\zeta, \sigma } \left\{\left(z-a\right)^{\vartheta } \, J_{\vartheta , q}^{\tau , \varsigma , s;\omega } \left(\lambda \left(z-a\right)^{\tau } ;p\right)\right\}\right)(x)\,$

$= \left(D_{a+}^{\zeta, \sigma } \left\{\sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-\lambda \right)^{n} (z-a)^{\tau n+\vartheta } }{n{\rm !}} \, \right\}\right)(x)$

$\begin{equation} = \sum\limits_{n = 0}^{\infty }\frac{\mathcal{B}_{\omega } \left(\varsigma +\, nq, s-\varsigma ;p\right)}{B\left(\varsigma , s-\varsigma \right)} \frac{\left(s\right)_{nq} }{\Gamma \left(\tau n+\vartheta +1\right)} \frac{\left(-\lambda \right)^{n} }{n{\rm !}} \left(D_{a+}^{\zeta, \sigma } \left\{\, (z-a)^{\tau n+\vartheta } \right\}\right)(x), \end{equation}$

(7.10)

Using the familiar relation of Srivatava and Tomovski ^[11]:

$\begin{equation} \left(D_{a+}^{\zeta, \sigma } \left\{\left(z-a\right)^{\vartheta -1} \right\}\right)(x)\, \, = \, \, \frac{\Gamma \left(\vartheta \right)}{\Gamma \left(\vartheta -\zeta\right)} \, \left(x-a\right)^{\vartheta -\zeta-1} \end{equation}$

(7.11)

$\left(x \gt a;\, 0 \lt \zeta \lt 1;\, 0\le \sigma \le 1, \, \Re \left(\vartheta \right) \gt 0\, \right)$

In Eq (7.10), we are led to the result Eq (7.6).

8. Conclusions

In the present paper, The properties, integral transform and fractional calculus of the newly defined extended generalized Bessel-Maitland type function are investigated here and find their connection with other functions scattered in the literature of special function. Various special cases of the derived results in the paper can be evaluate by taking suitable values of parameters involved. For example if we set $\omega = 0$ , $\beta = \beta-1$ and $z = -z$ in (1.3), we immediately obtain the result due to Mittal et al ^[18]. For various other special cases we refer ^[19,21] and we left results for the interested readers.

Acknowledgments

The authors are thankful to the referee's for their valuable remarks and comments for the improvement of the paper.

Conflict of interest

The authors declare no conflict of interest in this paper.

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