Optimization of liquefaction process based on global meta-analysis and machine learning approach: Effect of process conditions and raw material selection on remaining ratio and bioavailability of heavy metals in biochar

Li Ma; Likun Zhan; Qingdan Wu; Longcheng Li; Xiaochen Zheng; Zhihua Xiao; Jingchen Zou; Li Ma; Likun Zhan; Qingdan Wu; Longcheng Li; Xiaochen Zheng; Zhihua Xiao; Jingchen Zou

doi:10.3934/environsci.2024016

AIMS Environmental Science

2024, Volume 11, Issue 3: 342-359. doi: 10.3934/environsci.2024016

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Research article

Optimization of liquefaction process based on global meta-analysis and machine learning approach: Effect of process conditions and raw material selection on remaining ratio and bioavailability of heavy metals in biochar

1.
College of Environment and Ecology, Hunan Agricultural University, Changsha, Hunan, 410128, China
2.
Key Laboratory for Rural Ecosystem Health in Dongting Lake Area of Hunan Province, Changsha 410128, China
3.
Beijing Key Laboratory of Biodiversity and Organic Farming, College of Resources and Environmental Sciences, China Agricultural University, Beijing, China

Received: 22 March 2024 Revised: 09 May 2024 Accepted: 17 May 2024 Published: 28 May 2024

Although liquefaction technology has been extensively applied, plenty of biomass remains tainted with heavy metals (HMs). A meta-analysis of literature published from 2010 to 2023 was conducted to investigate the effects of liquefaction conditions and biomass characteristics on the remaining ratio and chemical speciation of HMs in biochar, aiming to achieve harmless treatment of biomass contaminated with HMs. The results showed that a liquefaction time of 1–3 h led to the largest HMs remaining ratio in biochar, with the mean ranging from 84.09% to 92.76%, compared with liquefaction times of less than 1 h and more than 3 h. Organic and acidic solvents liquefied biochar exhibited the greatest and lowest HMs remaining ratio. The effect of liquefaction temperature on HMs remaining ratio was not significant. The C, H, O, volatile matter, and fixed carbon contents of biomass were negatively correlated with the HMs remaining ratio, and N, S, and ash were positively correlated. In addition, liquefaction significantly transformed the HMs in biochar from bioavailable fractions (F1 and F2) to stable fractions (F3) (P < 0.05) when the temperature was increased to 280–330 ℃, with a liquefaction time of 1–3 h, and organic solvent as the liquefaction solvent. N and ash in biomass were positively correlated with the residue state (F4) of HMs in biochar and negatively correlated with F1 or F2, while H, O, fixed carbon, and volatile matter were negatively correlated with F4 but positively correlated with F3. Machine learning results showed that the contribution of biomass characteristics to HMs remaining ratio was higher than that of liquefaction factor. The most prominent contribution to the chemical speciation changes of HMs was the characteristics of HMs themselves, followed by ash content in biomass, liquefaction time, and C content. The findings of this meta-analysis contribute to factor selection, modification, and application of liquefied biomass to reducing risks.

Keywords:

Citation: Li Ma, Likun Zhan, Qingdan Wu, Longcheng Li, Xiaochen Zheng, Zhihua Xiao, Jingchen Zou. Optimization of liquefaction process based on global meta-analysis and machine learning approach: Effect of process conditions and raw material selection on remaining ratio and bioavailability of heavy metals in biochar[J]. AIMS Environmental Science, 2024, 11(3): 342-359. doi: 10.3934/environsci.2024016

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Abstract

1. Introduction

The Bessel function has immense applications in the field of engineering, physics, and applied mathematics. Baricz ^[1], Generalized Bessel functions of the first kind in $(2010)$ , which discussed the geometric properties, functional inequalities of generalized Bessel function and also the inequalities involving circular and hyperbolic functions to Bessel function and modified Bessel functions. Tumakov ^[2] investigated the numerical algorithms for fast computations of the Bessel functions of an integer order with the required accuracy. Choi and Agarwal^[3], Abramowitz and Stegun ^[4], Heymans and Podlubny ^[5], Watson ^[6] and Purohit et al. ^[7] studied the following Bessel function (Bf) defined by

$\begin{align} J_{\alpha}(y) = \sum^{\infty}_{n = 0}\frac{(-1)^{n}(y/2)^{\alpha+2n}}{n!\Gamma(\alpha+n+1)}. \end{align}$

(1.1)

Edward Maitland Wright ^[8] introduced the generalized form of Bessel function with the name of Bessel-Maitland function (B-M1)

$\begin{align} J^{\alpha}_{\beta}(y) = \sum^{\infty}_{n = 0}\frac{(-y)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{align}$

(1.2)

The properties of generalised Bessel function can be found in the work of Srivastava and Singh ^[9]. Suthar et al. ^[10,11] discussed the various properties of Bessel-Maitland function. Ali et al. ^[12] established some fractional operators with the generalized Bessel-Maitland function.

Waseem et al. ^[13] established the generalized Bessel-Maitland function (B-M11) and discuss the numerous integral formulas for $y\in\mathbb{C}/(-\infty, 0]$ ; $\alpha, \beta, \gamma \in \mathbb{C}$ , $\Re(\alpha)\geq0$ , $\Re(\beta)\geq-1$ , $\Re(\gamma)\geq0$ , $k\in(0, 1)\cup\mathbb{N}$ defined by

$\begin{align} J^{\alpha, \gamma}_{\beta, k}(y) = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-y)^{n}}{n!\Gamma(\alpha n+\beta+1)}. \end{align}$

(1.3)

Suthar et al. ^[14] studied the following generalized multi-index Bessel function (Gm-Bf) defined by

$\begin{align} J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(y) = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-y)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}. \end{align}$

(1.4)

Recently, fractional integrals are widely applied in different branches of mathematics, physics, engineering due to their wide applications (see e.g., ^{[5,15,16,17,18]}).

Riemann-Liouville fractional integral operators for $\Re(\rho) > 0$ are defined by

$\begin{align} &I^{\rho}_{a^{+}}h(u) = \frac{1}{\Gamma(\rho)}\int^{y}_{a} (y-u)^{\rho-1}h(u)du, a < y \end{align}$

(1.5)

$\begin{align} &I^{\rho}_{b}h(u) = \frac{1}{\Gamma(\rho)}\int^{b}_{y}(u-y)^{\rho-1}h(u)du, y < b. \end{align}$

(1.6)

Riemann-Liouville fractional differentials operators (RLDO) ^[12,19] for $\Re(\rho) > 0$ ; $n = [\Re(n)-1]$

$\begin{align} &D^{\rho}_{a^{+}}h(u) = (d/dy)^{n}I^{n-\rho}_{a^{+}}h(y) \end{align}$

(1.7)

$\begin{align} &D^{\rho}_{b}h(u) = (-d/dy)^{n}I^{n-\rho}_{b}h(y). \end{align}$

(1.8)

Srivastava and Singh ^[9] defined the following fractional integral operator for $\alpha_{1}, \beta_{1}, r\in \mathbb{C}$ , $\Re(\alpha_{1}) > 0$ , $\Re(\beta_{1})\geq-1$ by

$\begin{align} h(y) = ^{def}\int^{y}_{0}(y-t)^{\beta_{1}}J^{\alpha_{1}}_{\beta_{1}}(r(y-t)^{\alpha_{1}})h(u)du. \end{align}$

(1.9)

Srivastava and Tomovski ^[20] established the fractional integral operator (FIO) having Mittag-Leffler function as a kernel, discuss its boundedness and convergence of integral and also derive the product of FIO with Riemann-Liouville fractional integral operator defined for $r, \gamma \in \mathbb{C}$ , $\Re(\alpha_{1}) > \max\{0, \Re(k)-1\}$ ; $\min\{\Re(\beta_{1}), \Re(k)\} > 0$

$\begin{align} (\mathscr{E}^{r; \gamma, k}_{a^{+}; \alpha_{1}, \beta_{1}}h)(y) = \int^{y}_{a}(y-t)^{\beta_{1}-1}E^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}})h(t)dt. \end{align}$

(1.10)

Prabhakar fractional integral operators for $\gamma, \beta_{1} \in \mathbb{C}$ , $\Re(\alpha_{1}) > 0$ are defined in ^[21] by

$\begin{align} &\mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = ^{\circ}h(y) = \int^{y}_{a}(y-t)^{\beta_{1}-1}E^{\gamma}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}})h(t)dt, a < y, \end{align}$

(1.11)

$\begin{align} &\mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = \int^{b}_{y}(t-y)^{\beta_{1}-1}E^{\gamma}_{\alpha_{1}, \beta_{1}}(r(t-y)^{\alpha_{1}})h(t)dt, y < b. \end{align}$

(1.12)

Tilahun et al. ^[22] derived the generalized FIO for $\Re(\beta_{1}) > 0$ , $\Re(\alpha_{1}) > 0$ and $r, \gamma\in\mathbb{C}$ as

$\begin{align} \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y) = \int^{y}_{a}(y-t)^{\beta_{1}}J^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}}; p)h(t)dt, a < y \end{align}$

(1.13)

and

$\begin{align} \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y) = \int^{b}_{y}(y-t)^{\beta_{1}}J^{\gamma, k}_{\alpha_{1}, \beta_{1}}(r(y-t)^{\alpha_{1}};p)h(t)dt, \end{align}$

(1.14)

where $y < b$ .

Definition 1.1. (FIO)Fractional integral operator with generalized multi-index Bessel function (Gm-Bf) kernel for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ .

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h(y) = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})h(u)du, a < y \end{align}$

(1.15)

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h(y) = \prod^{m}_{j = 1}\int^{b}_{y}(u-y)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-y)^{\alpha_{j}})h(u)du, y < b. \end{align}$

(1.16)

Dirichlet formula (Fubini's theorem) Samko et al. in ^[23] and Kelelaw et al. in ^[22] is defined as

$\begin{align} \int^{d}_{b}dy\int^{y}_{b}h(y, z)dz = \int^{d}_{b}dz\int^{d}_{z}h(y, z)dy. \end{align}$

(1.17)

Kilbas ^[24] analyzed the generalized Wright function for $\xi_{i}, \zeta_{j}\in\mathbb{R}$ , $(i = 1, 2 \cdots r), (j = 1, 2 \cdots s)$ and $b_{i}, c_{j}\in \mathbb{C}$ as

$\begin{align} {_{r}\psi_{s}}(y)& = \sum^{\infty}_{n = 0}\frac{\prod^{r}_{i = 1}\Gamma(b_{i}+\xi_{i}n)}{\prod^{s}_{j = 1}\Gamma(c_{j}+\zeta_{j}n)}\frac{y^{n}}{n!}\\ & = {_{r}\psi_{s}}\left[ \begin{array}{c} (b_{i}, \xi_{i})_{1, r}\\ (c_{j}, \zeta_{j})_{1, s} \end{array}\middle|y \right]. \end{align}$

(1.18)

The integral representation of beta function ^[25,26] for $\Re(y) > 0$ , $\Re(z) > 0$ and also in gamma form appearance of beta function is defined as follows

$\begin{align} B(y, z) = \int^{1}_{0}u^{y-1}(1-u)^{z-1}du = \frac{\Gamma(y)\Gamma(z)}{\Gamma(y+z)}. \end{align}$

(1.19)

Pochhammer symbol and its properties can be found ^[25,26,27] as

$\begin{align} (\gamma)_n& = \left\{ \begin{array}{ll} \gamma(\gamma+1)(\gamma+2) \cdots (\gamma+n-1), & \hbox{for $ n\geq1 $} \\ 1, & \hbox{for $n = 0$, $\gamma\neq0$} \end{array} \right. \end{align}$

(1.20)

$\begin{align} & = \frac{\Gamma(\gamma+n)}{\Gamma(\gamma)} \,\,\ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; and \; \; \; \; \; \; \; \; \; \; \; \,\,\ (\gamma)_{kn} = \frac{\Gamma(\gamma+kn)}{\Gamma(\gamma)} \,\,\ \; \; \; \; \; (k > 0). \end{align}$

(1.21)

The space of Lebesgue measurable for complex and real valued functions defined by Kelelaw et al. ^[22] as follows

$\begin{align} L(a, y) = \bigg\{h: ||h||_{1}: = \int^{y}_{a}\big|h(u)\big|du < \infty\bigg\}. \end{align}$

(1.22)

1.1. Conditions of fractional integral operator

The following some conditions of fractional integral operators can be obtained by setting the integrals according to requirements:

$1).$ Setting $r = 0$ , $j = 1 = m$ and $\beta_{1} = \beta_{1}-1$ in (FIO) defined in Eqs (1.15) and (1.16), we get the Riemann-Liouville fractional integral operator defined in ^[28] as

$\begin{align} &\mathcal{I}^{0; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = I^{\beta_{1}}_{a^{+}}h(y) \end{align}$

(1.23)

$\begin{align} &\mathcal{I}^{0; \gamma, k}_{b; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = I^{\beta_{1}}_{b}h(y). \end{align}$

(1.24)

$2).$ Setting $j = m = 1$ , $\beta_{1} = \beta_{1}-1$ in Eq (1.15), we have a fractional integral defined in Eq (1.10) as

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = (\mathscr{E}^{r; \gamma, k}_{a^{+}; \alpha_{1}, \beta_{1}}h)(y). \end{align}$

(1.25)

$3).$ Setting $j = m = 1$ , $k = 1$ , $\beta_{1} = \beta_{1}-1$ , in Eqs (1.15) and (1.16), we get the FIO defined in Eqs (1.11) and (1.12) respectively

$\begin{align} &\mathcal{I}^{r; \gamma, 1}_{a^{+}; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = \mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y) = ^{\circ}h(y) \end{align}$

(1.26)

$\begin{align} &\mathcal{I}^{r; \gamma, 1}_{b; (\alpha_{1}, \beta_{1}-1)_{m}}h(y) = \mathfrak{E}^{*}(\alpha_{1}, \beta_{1}; \gamma; r)h(y). \end{align}$

(1.27)

$4).$ Setting $j = m = 1$ , $k = 0$ and limits from $[0, y]$ in Eq (1.15), we get a fractional integral defined in Eq (1.9) as

$\begin{align} &\mathcal{I}^{r; \gamma, 0}_{a^{+}; (\alpha_{1}, \beta_{1})_{m}}h(y) = \int^{y}_{0}(y-t)^{\beta_{1}}J^{\alpha_{1}}_{\beta_{1}}(r(y-t)^{\alpha_{1}})h(u)du = h(y). \end{align}$

(1.28)

$5).$ Setting $j = m = 1$ in Eq (1.15) then, we get the generalized fractional integral operator defined in Eq (1.13) as

$\begin{align} &\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{1}, \beta_{1})}h(y) = \bigg(\mathfrak{I}^{r; \alpha_{1}, \gamma}_{a^{+}, \beta_{1}, k}h\bigg)(y). \end{align}$

(1.29)

Lemma 1.1. Consider the Riemann-Liouville fractional integral operator with multi-index power function for $\alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $\Re(\rho) > 0$ as

$\begin{align} \prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}+\alpha_{j}n}](y)& = \prod^{m}_{j = 1} \Big[\frac{ \Gamma(\beta_{j}+\alpha_{j}n+1)}{\Gamma(\rho+\beta_{j}+\alpha_{j}n+1)}(y-a)^{\rho+\beta_{j}+\alpha_{j}n}\Big]. \end{align}$

(1.30)

Remark 1.1. Setting $j = m = 1$ in lemma 1.1 then we obtain the result that defined the Mathai Haubold ^[29] and Kelelaw et al. ^[22] as

$\begin{align} I^{\rho}_{a^{+}}[(u-a)^{\beta_{1}+\alpha_{1}n}](y)& = (y-a)^{\rho+\beta_{1}+\alpha_{1}n}\frac{ \Gamma(\beta_{1}+\alpha_{1}n+1)}{\Gamma(\rho+\beta_{1}+\alpha_{1}b+1)}. \end{align}$

(1.31)

2. Some properties of generalized multi-index Bessel function

The preliminary results for generalized multi-index Bessel function which used to proceed the new results is given in this section. We calculate the $nth$ -differential and also develop some results with the coordination of Riemann-Liouville fractional operator and (Gm-Bf).

Theorem 2.1. Consider the $nth$ -differential of generalized multi-index Bessel function with power function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ as

$\begin{align} (d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})] = (y-u)^{\beta_{j}-n}J^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}). \end{align}$

(2.1)

Proof. Let the $nth$ -differential of generalized multi-index Bessel function with power function as

$\begin{align} (d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})], \end{align}$

(2.2)

using the behavior of (1.4), we take as

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = (d/dy)^{n}\bigg[(y-u)^{\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-u)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{\beta_{j}+\alpha_{j}n}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\beta_{2}+ \cdots +\beta_{m})+(\alpha_{1}n+\alpha_{2}n+ \cdots +\alpha_{m}n)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\alpha_{1}n)+(\beta_{2}+\alpha_{2}n)+ \cdots +(\beta_{m}+\alpha_{m}n)}\bigg]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\\ &\times(d/dy)^{n}\bigg[(y-u)^{(\beta_{1}+\alpha_{1}n)}(y-u)^{(\beta_{2}+\alpha_{2}n)} \cdots (y-u)^{(\beta_{m}+\alpha_{m}n)}\bigg]. \end{align}$

(2.3)

Using the identity result for simplification of (2.3), we get

$\begin{align} &(d/dy)^{n}y^{\theta} = \frac{\Gamma(\theta+1)}{\Gamma(\theta-n+1)}y^{\theta-n}, \theta\geq n \end{align}$

(2.4)

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})] = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\\ &\times\frac{\Gamma(\beta_{1}+\alpha_{1}n+1)\Gamma(\beta_{2}+\alpha_{2}n+1) \cdots \Gamma(\beta_{m}+\alpha_{m}n+1)(y-u)^{\alpha_{j}n+\beta_{j}-n}}{\Gamma(\beta_{1}+\alpha_{1}n-n+1)\Gamma(\beta_{2}+\alpha_{2}n-n+1) \cdots \Gamma(\beta_{m}+\alpha_{m}n-n+1)}\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}\frac{\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}+1)}{\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}-n+1)}(y-u)^{\alpha_{j}n+\beta_{j}-n}\\ & = (y-u)^{\beta_{j}-n}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-u)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j} n+\beta_{j}-n+1)}\\ & = (y-u)^{\beta_{j}-n}J^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}). \end{align}$

(2.5)

Corollary 2.1. Suppose that $\alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ then theorem 2.1 can be expressed as

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = \frac{1}{\Gamma(\gamma)}(y-u)^{\beta_{j}-n} {_{2}\psi_{m+1}}\left\{ \begin{array}{c} (\gamma, k)(\beta_{j}+1, \alpha_{j})\\ (\beta_{j}-n+1, \alpha_{j})(\beta_{j}+1, \alpha_{j})|^{m}_{j = 1} \end{array}\middle|r(y-u)^{\alpha_{j}} \right\}. \end{align}$

(2.6)

Corollary 2.2. Suppose that $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $n\in\mathbb{N}$ and setting that $\beta_{j} = \beta_{j}-1$ , $(r(y-u)^{\alpha_{j}}) = (-r(y-u)^{\alpha_{j}})$ in theorem 2.1 we see that

$\begin{align} &(d/dy)^{n}[(y-u)^{\beta_{j}-1}J^{(\alpha_{j}, \beta_{j}-1)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})]\\ & = (y-u)^{\beta_{j}-1-n}E^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}}), \end{align}$

(2.7)

where $E^{(\alpha_{j}, \beta_{j}-n)_{m}}_{\gamma, k}(.)$ is generalized multi-index Mittag-Leffler function.

Theorem 2.2. Consider the Riemann-Liouville fractional integral operator defined in Eq (1.5) with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

(2.8)

Proof. Let (RLIO) with (Gm-Bf) is defined in Eq (1.4), we have

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})\bigg](y)\\ & = \prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(u-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}]\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}+\alpha_{j}n}]. \end{align}$

(2.9)

By using Lemma 1.1 in Eq (2.9) then we attain the equation as

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})](y)\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)} \prod^{m}_{j = 1}\frac{\Gamma(\alpha_{j}n+\beta_{j}+1)(y-a)^{\alpha_{j}n+\rho+\beta_{j}}} {\Gamma(\alpha_{j}n+\beta_{j}+\rho+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\rho+\beta_{j}}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+\rho+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(2.10)

Corollary 2.3. Consider the right-sided Riemann-Liouville fractional integral operator with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}I^{\rho}_{b}[(b-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(b-u)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(2.11)

Corollary 2.4. Consider the Riemann-Liouville fractional differential operator defined in Eq (1.7) with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}D^{\rho}_{a^{+}}[(u-a)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-a)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(y-a)^{\beta_{j}-\rho}J^{(\alpha_{j}, \beta_{j}-\rho)_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(2.12)

Corollary 2.5. Consider the right-sided Riemann-Liouville fractional differential operator with generalized multi-index Bessel function for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ and $y > a$ , $a\in\Re_{+}(0, \infty)$ , $\Re(\rho) > 0$ as

$\begin{align} &\prod^{m}_{j = 1}D^{\rho}_{b}[(b-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(b-u)^{\alpha_{j}})](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\beta_{j}-\rho}J^{(\alpha_{j}, \beta_{j}-\rho)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(2.13)

3. Properties of generalized fractional integral with non singular function as a kernel

In this section, we discuss some properties of the generalized fractional integrals with non singular function as a kernel.

Theorem 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for $\vartheta, \rho, \sigma, r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , when $h(u) = (u-a)^{\frac{\vartheta+\rho}{\sigma}-1}$ , then

$\begin{align} [\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}&(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}](y)\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma})_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(3.1)

Proof. (FIO) defined in Eq (1.15) with Eq (1.4), we obtain as

$\begin{align} &[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}](y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})(u-a)^{\frac{\vartheta+\rho}{\sigma}-1}du\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}\int^{y}_{a}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1} (y-u)^{\beta_{j}+\alpha_{j}n}du\\ & = \sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!}\frac{1}{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\prod^{m}_{j = 1}\int^{y}_{a}(u-a)^{\frac{\vartheta+\rho}{\sigma}-1} (y-u)^{\beta_{j}+\alpha_{j}n}du\\ \end{align}$

(3.2)

Substituting $u = y-z(u-a)$ and using the definition of beta function and the following relation in Eq (3.2), we get

$\begin{align*} \mathcal{I}^{\lambda}_{a^{+}}[(y-a)^{u-1}](y) = \frac{\Gamma(u)}{\Gamma(\lambda+u)}(y-a)^{\lambda+u-1}. \end{align*}$

$\begin{align} &[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}(u-a)^{\frac{\vartheta+\rho}{\sigma}}](y)\\ & = \prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r)^{n}}{n!}\frac{\Gamma(\frac{\vartheta+\rho}{\sigma})}{\prod^{m}_{j = 1}\Gamma(\frac{\vartheta+\rho}{\sigma}+\alpha_{j}n+\beta_{j}+1)} (y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}+\alpha_{j}n}\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}(-r(y-a)^{\alpha_{j}})^{n}}{n!\prod^{m}_{j = 1}\Gamma(\frac{\vartheta+\rho}{\sigma}+\alpha_{j}n+\beta_{j}+1)}\\ & = \prod^{m}_{j = 1}(y-a)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}}\Gamma(\frac{\vartheta+\rho}{\sigma})J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma})_{m}}_{\gamma, k}(r(y-a)^{\alpha_{j}}). \end{align}$

(3.3)

Corollary 3.1. Fractional integral operator having generalized multi-index Bessel function (Gm-Bf) kernel for $\vartheta, \rho, \sigma, r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , when $h(u) = (a-u)^{\frac{\vartheta+\rho}{\sigma}}$ , then

$\begin{align} [\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}&(b-u)^{\frac{\vartheta+\rho}{\sigma}}](y)\\ & = \prod^{m}_{j = 1}(b-y)^{\frac{\vartheta+\rho}{\sigma}+\beta_{j}+1}\Gamma(\frac{\vartheta+\rho}{\sigma}+1)J^{(\alpha_{j}, \beta_{j}+\frac{\vartheta+\rho}{\sigma}+1)_{m}}_{\gamma, k}(r(b-y)^{\alpha_{j}}). \end{align}$

(3.4)

Theorem 3.2. Consider the composition of Riemann-Liouville fractional integral operator with (FIO) for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y). \end{align}$

(3.5)

Proof. Let the left side of Eq (3.5), we seen as

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \frac{1}{\Gamma(\rho)}\int^{y}_{a}(y-u)^{\rho-1}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h](u)du\\ & = \frac{1}{\Gamma(\rho)}\int^{y}_{a}(y-u)^{\rho-1}\bigg[\prod^{m}_{j = 1}\int^{u}_{a}(u-t)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-t)^{\alpha_{j}})h(t)dt\bigg]du. \end{align}$

(3.6)

By interchanging the order of integrations and using the Eq (1.17) in Eq (3.6), we attain as

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\prod^{m}_{j = 1}\int^{y}_{t}(y-u)^{\rho-1}(u-t)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-t)^{\alpha_{j}})du\bigg] \times h(t)dt. \end{align}$

(3.7)

Setting $u-t = \eta$ , we have

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \int^{y}_{a}\bigg[\prod^{m}_{j = 1}\frac{1}{\Gamma(\rho)}\int^{y-t}_{0}(y-t-\eta)^{\rho-1}\eta^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\eta)^{\alpha_{j}})d\eta\bigg] \times h(t)dt\\ & = \int^{y}_{a}\prod^{m}_{j = 1}I^{\rho}_{0^{+}}\bigg[\eta^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\eta)^{\alpha_{j}})\bigg](y-t) \times h(t)dt. \end{align}$

(3.8)

applying theorem 2.2, we see

$\begin{align} &\bigg\{I^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[(y-t)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-t)^{\alpha_{j}})\bigg] h(t)dt\\ & = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y). \end{align}$

(3.9)

We start the right side to determine the second part of (3.5) with (FIO) defined in Eq (1.15) as

$\begin{align} &\bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})[I^{\rho}_{a^{+}}h](u)du\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-u)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(y-u)^{\alpha_{j}})\bigg[\frac{1}{\Gamma(\rho)}\int^{u}_{a}(u-t)^{\rho-1}h(t)dt\bigg]du. \end{align}$

(3.10)

Interchanging the order of integration and using Eq (1.17), we get

(3.11)

Setting $y-u = x$ and using theorem (2.2) then

$\begin{align} &\bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{a^{+}}h]\bigg\}(y)\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\int^{0}_{y-t}(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})(y-x-t)^{\rho-1}(-dx)\bigg]h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}\bigg[\frac{1}{\Gamma(\rho)}\int_{0}^{y-t}(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})(y-t-x)^{\rho-1}dx\bigg]h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}I^{\rho}_{0^{+}}\bigg[(x)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(x)^{\alpha_{j}})\bigg](y-t)\times h(t)dt\\ & = \prod^{m}_{j = 1}\int^{y}_{a}(y-t)^{\beta_{j}+\rho}J^{(\alpha_{j}, \beta_{j}+\rho)_{m}}_{\gamma, k}(r(y-t)^{\alpha_{j}})h(t)dt\\ & = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y). \end{align}$

(3.12)

Thus, we obtain the desired results by combining Eqs (3.9) and (3.12).

CCorollary 3.2. Composition of right-sided (FIO) with right-sided Riemann-Liouville fractional integral for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{I^{\rho}_{b}[\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j}+\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}}[I^{\rho}_{b}h]\bigg\}(y). \end{align}$

(3.13)

Corollary 3.3. Consider the composition of Riemann-Liouville fractional differential operator with (FIO) for $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ , $\Re(\rho) > 0$ then

$\begin{align} \bigg\{D^{\rho}_{a^{+}}[\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h]\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j}-\rho)_{m}}}\bigg\}(y) = \bigg\{{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}}[D^{\rho}_{a^{+}}h]\bigg\}(y). \end{align}$

(3.14)

Theorem 3.3. If $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ then

$\begin{align} &||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1}, \end{align}$

(3.15)

where

$\begin{align} A = \prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{|(y-a)^{\Re(\beta_{j})+1}||(\gamma)_{k n}||(-r (y-a)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}. \end{align}$

(3.16)

Proof. Let $V_{n}$ be the nth-terms of (3.16); we have

$\begin{align} \bigg|\frac{V_{n+1}}{V_{n}}\bigg|& = \prod^{m}_{j = 1}\bigg|\frac{(\gamma)_{k n+k}|}{(\gamma)_{k n}}\bigg|\frac{n!}{(n+1)!} \bigg|\frac{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}{\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\alpha_{j}+\beta_{j}+1)}\bigg|\\ &\times\frac{\Re(\alpha_{j})n+\Re(\beta_{j})+1}{\Re(\alpha_{j})n+ \Re(\alpha_{j})+\Re(\beta_{j})+1}\bigg|\frac{(-1)^{n+1}}{(-1)^{n}}\bigg| \bigg|r(y-a)^{\Re(\alpha_{j})}\bigg|\\ &\approx\prod^{m}_{j = 1}\frac{(kn)^{k}\big|-r(y-a)^{\Re(\alpha_{j})}\big|}{(n+1)\big|\prod^{m}_{j = 1}{|(\alpha_{j})|n}^{(\alpha_{j})}\big|}, \; \; \; as \; \; \; n \rightarrow \infty \end{align}$

(3.17)

hence, $\bigg|\frac{V_{n+1}}{V_{n}}\bigg| \rightarrow 0$ as $n \rightarrow \infty$ , and $k < \Re(\alpha_{j})$ which means that right hand side of (3.16) is convergent and finite under the given condition. The condition of boundedness of the integral operator $({\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h})(y)$ is discussed in the space of Lebesgue measure $L(a, y)$ of a continuous function, where $y > a$ . Consider the Lebesgue measurable space (1.22) and FIO (1.15), we have

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}& = \int^{y}_{a}\bigg|{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}\bigg|du\\ & = \int^{y}_{a}\bigg|\prod^{m}_{j = 1}\int^{u}_{a}(u-\tau)^{\beta_{j}}J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-\tau)^{\alpha_{j}})h(\tau)d\tau\bigg|du\\ &\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y}_{\tau}(u-\tau)^{\beta_{j}}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(u-\tau)^{\alpha_{j}})\big|du \bigg]|h(\tau)|d\tau. \end{align}$

(3.18)

Putting $u-\tau = \lambda, u = y \Rightarrow \lambda = y-\tau; u = \tau \Rightarrow \lambda = 0, du = d\lambda$ in Eq (3.18), we get

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}&\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y-\tau}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \bigg]|h(\tau)|d\tau\\ &\leq\int^{y}_{a}\bigg[\prod^{m}_{j = 1} \int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \bigg]|h(\tau)|d\tau, \end{align}$

where

$\begin{align} &\prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\big|J^{(\alpha_{j}, \beta_{j})_{m}}_{\gamma, k}(r(\lambda)^{\alpha_{j}})\big|d\lambda \\ \leq&\prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\beta_{j})}\bigg|\sum^{\infty}_{n = 0}\frac{(\gamma)_{k n}|(-r)^{n}(\lambda)^{\alpha_{j} b}|}{n!\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)}\bigg|d\lambda\\ \leq&\sum^{\infty}_{n = 0}\frac{|(\gamma)_{k n}||(-r)^{n}|}{n!|\prod^{m}_{j = 1}\Gamma(\alpha_{j}n+\beta_{j}+1)|} \prod^{m}_{j = 1}\int^{y-a}_{0}\lambda^{\Re(\alpha_{j})n+\Re(\beta_{j})}d\lambda\\ \leq&\prod^{m}_{j = 1}\sum^{\infty}_{n = 0}\frac{|(y-a)^{\Re(\beta_{j})+1}||(\gamma)_{k n}||(-r (y-a)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}\\ = &A. \end{align}$

(3.19)

Therefore,

$\begin{align} ||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}&\leq\int^{y}_{a}A|h(\tau)|d\tau\leq A||h||_{1}\\ &\Rightarrow||{\mathcal{I}^{r; \gamma, k}_{a^{+}; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1} \end{align}$

Corollary 3.4. If $r, \alpha_{j}, \beta_{j}, \gamma \in \mathbb{C}$ , $(j = 1, 2 \cdots m)$ , $\Re(\alpha_{j}) > 0$ , $\Re(\beta_{j}) > -1$ , $\sum^{m}_{j = 1}\Re(\alpha)_{j} > \max\{0; \Re(k)-1\}$ , $k > 0$ then

$\begin{align} &||{\mathcal{I}^{r; \gamma, k}_{b; (\alpha_{j}, \beta_{j})_{m}}h}||_{1}\leq A||h||_{1}, \end{align}$

(3.20)

where

$\begin{align} A = \prod^{m}_{j = 1}(b-y)^{\Re(\beta_{j})+1}\sum^{\infty}_{n = 0}\frac{|(\gamma)_{k n}||(-r(b-y)^{\Re(\alpha_{j})})^{n}|}{n!|\prod^{m}_{j = 1} \Gamma(\alpha_{j}n+\beta_{j}+1)|\Re(\alpha_{j})n+\Re(\beta_{j})+1}. \end{align}$

(3.21)

4. Conclusions

The results we discussed in this paper is creating a chain of fractional operators with kernels, having convergence and boundedness, continuity, symmetric properties and composition with Riemann-Liouville operators ^{[9,12,20,21,22,25,30,31]} in fractional calculus. We constructed the fractional operator with generalized multi-index Bessel function as a kernel, and discussed its properties, continuity, and check the behaviour with Riemann-Liouville fractional operators. We analyzed the generalized multi-index Bessel function $nth$ -derivative and integral in the field of fractional calculus.

Acknowledgments

Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

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