A novel study on the bifocusing method for imaging unknown objects in two-dimensional inverse scattering problem

Sangwoo Kang; Won-Kwang Park; Sangwoo Kang; Won-Kwang Park

doi:10.3934/math.20231386

AIMS Mathematics

2023, Volume 8, Issue 11: 27080-27112. doi: 10.3934/math.20231386

Previous Article Next Article

Research article Special Issues

A novel study on the bifocusing method for imaging unknown objects in two-dimensional inverse scattering problem

Sangwoo Kang ¹,
Won-Kwang Park ^{2
,
,}

1.
Advanced Defense Science & Technology Research Institute, Agency for Defense Development, Daejeon 34186, Korea
2.
Department of Information Security, Cryptology, and Mathematics, Kookmin University, Seoul 02707, Korea

Received: 14 July 2023 Revised: 19 September 2023 Accepted: 19 September 2023 Published: 25 September 2023
MSC : 78A46

In this paper, we consider the application of the bifocusing method (BFM) for a fast identification of two-dimensional circle-like small inhomogeneities from measured scattered field data. Based on the asymptotic expansion formula for the scattered field in the presence of small inhomogeneities, we introduce the imaging functions of the BFM for both dielectric permittivity and magnetic permeability contrast cases. To examine the applicability and the various properties of the BFM, we show that the imaging functions can be expressed by the Bessel function of orders zero and one, as well as the characteristics (size, permittivity, and permeability) of the inhomogeneities. To support the theoretical results, various numerical results with synthetic and experimental data are presented.

Keywords:

Citation: Sangwoo Kang, Won-Kwang Park. A novel study on the bifocusing method for imaging unknown objects in two-dimensional inverse scattering problem[J]. AIMS Mathematics, 2023, 8(11): 27080-27112. doi: 10.3934/math.20231386

Related Papers:

[1]	Dayang Dai, Dabuxilatu Wang . A generalized Liu-type estimator for logistic partial linear regression model with multicollinearity. AIMS Mathematics, 2023, 8(5): 11851-11874. doi: 10.3934/math.2023600
[2]	Muhammad Nauman Akram, Muhammad Amin, Ahmed Elhassanein, Muhammad Aman Ullah . A new modified ridge-type estimator for the beta regression model: simulation and application. AIMS Mathematics, 2022, 7(1): 1035-1057. doi: 10.3934/math.2022062
[3]	Sihem Semmar, Omar Fetitah, Mohammed Kadi Attouch, Salah Khardani, Ibrahim M. Almanjahie . A Bernstein polynomial approach of the robust regression. AIMS Mathematics, 2024, 9(11): 32409-32441. doi: 10.3934/math.20241554
[4]	Yanting Xiao, Wanying Dong . Robust estimation for varying-coefficient partially linear measurement error model with auxiliary instrumental variables. AIMS Mathematics, 2023, 8(8): 18373-18391. doi: 10.3934/math.2023934
[5]	Gaosheng Liu, Yang Bai . Statistical inference in functional semiparametric spatial autoregressive model. AIMS Mathematics, 2021, 6(10): 10890-10906. doi: 10.3934/math.2021633
[6]	Juxia Xiao, Ping Yu, Zhongzhan Zhang . Weighted composite asymmetric Huber estimation for partial functional linear models. AIMS Mathematics, 2022, 7(5): 7657-7684. doi: 10.3934/math.2022430
[7]	Xin Liang, Xingfa Zhang, Yuan Li, Chunliang Deng . Daily nonparametric ARCH(1) model estimation using intraday high frequency data. AIMS Mathematics, 2021, 6(4): 3455-3464. doi: 10.3934/math.2021206
[8]	Emrah Altun, Mustafa Ç. Korkmaz, M. El-Morshedy, M. S. Eliwa . The extended gamma distribution with regression model and applications. AIMS Mathematics, 2021, 6(3): 2418-2439. doi: 10.3934/math.2021147
[9]	Muhammad Amin, Saima Afzal, Muhammad Nauman Akram, Abdisalam Hassan Muse, Ahlam H. Tolba, Tahani A. Abushal . Outlier detection in gamma regression using Pearson residuals: Simulation and an application. AIMS Mathematics, 2022, 7(8): 15331-15347. doi: 10.3934/math.2022840
[10]	Zawar Hussain, Atif Akbar, Mohammed M. A. Almazah, A. Y. Al-Rezami, Fuad S. Al-Duais . Diagnostic power of some graphical methods in geometric regression model addressing cervical cancer data. AIMS Mathematics, 2024, 9(2): 4057-4075. doi: 10.3934/math.2024198

Abstract

1. Introduction

The field of mathematical analysis that deals with the study of arbitrary order integrals and derivatives is known as fractional calculus. Because of its numerous applications across a wide range of fields, this field has increased in importance and recognition over the past few years. According to researchers, this field is the most effective at identifying anomalous kinetics and has numerous uses in a variety of fields. Ordinary differential equations with fractional derivatives can be used to simulate a variety of issues, including statistical, mathematical, engineering, chemical, and biological issues. Several distinct forms of fractional integrals and derivative operators (see e.g., ^[1,2,3,4]), including Riemann-Liouville, Caputo, Riesz, Hilfer, Hadamard, Erdélyi-Kober, Saigo, Marichev-Saigo-Maeda and others, have been thoroughly investigated by researchers. From an application perspective, we suggest the readers to see the work related to the fractional differential equations presented by ^[5,6,7,8]. In ^[9], the authors studied symmetric and antisymmetric solitons in the defocused saturable nonlinearity and the PT-symmetric potential of the fractional nonlinear Schrödinger equation. In ^[10], the fractional exponential function approach is used to study a time-fractional Ablowitz-Ladik model, and bright and dark discrete soliton solutions, discrete exponential solutions, and discrete peculiar wave solutions are discovered. In ^[11], the authors presented the rich vector exact solutions for the coupled discrete conformable fractional nonlinear Schrödinger equations by taking into account the conformable fractional derivative.

On the other hand, special functions like Gamma, Beta, Mittag-Leffler, et al. play a vital part in the foundation of fractional calculus. Moreover, the Mittag-Leffler function is regarded as the fundamental function in fractional calculus. The Prabhakar fractional operator containing a three-parameter version of the aforementioned function in the kernel. The M-L function has been extensively studied to construct solutions of fractional PDEs, such as dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation, soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg-Landau equation and Abundant fractional soliton solutions of a space-time fractional perturbed Gerdjikov-Ivanov equation by a fractional mapping method, see ^[12,13,14]. Strong generalizations of the univariate and bivariate Mittag-Leffler functions, which are known to be important in fractional calculus, are the multivariate Mittag-Leffler functions.

The well-known one-parameter Mittag-Leffler (M-L) function is defined by ^[15,16] as follows

$\begin{equation} \varepsilon_{a}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+1)}\,\,\,\,\,\,\,\,\,(a\in\mathbb{C};\Re(a) > 0,z_1\in\mathbb{C}), \end{equation}$

(1.1)

where $\mathbb{C}$ represents the set of complex numbers and $\Re(a)$ denotes the real part of the complex number.

The generalization of (1.1) with two parameters is defined by ^[17,18] as

$\begin{equation} \varepsilon_{a,b}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{z_1^l}{\Gamma(al+b)}\,\,\,\,\,\,\,\,\,(a,b\in\mathbb{C};\Re(a) > 0,\Re(b) > 0), \end{equation}$

(1.2)

Later on, Agarwal ^[19], Humbert ^[20] and Humbert and Agarwal ^[21] studied the properties and applications of M-L functions. In ^[22], the generalization of (1.1) and (1.2) is defined by

$\begin{equation} \varepsilon_{a,b}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0). \end{equation}$

(1.3)

In ^[23], the following generalization of the M-L function is defined by

$\begin{equation} \varepsilon_{a,b}^{c,q}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c)_{lq}}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(a,b,c\in\mathbb{C};\Re(a) > 0,\Re(b) > 0, q > 0). \end{equation}$

(1.4)

In ^[24], Rahman et al. proposed the following generalized of M-L function by

$\begin{align} \varepsilon_{a,b,p}^{c,q,d}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{B_p(c +lq; d-c)(d)_{lq}}{B(c, d-c)\Gamma(a l+b)}\frac{z_1^l}{l!}, \end{align}$

(1.5)

where $a, b, c, d\in\mathbb{C}; \Re(c) > 0, \Re(a) > 0, \Re(b) > 0, q > 0$ and $B_p(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}e^{-t-\frac{p}{t}}dt$ is the extension of beta function (see ^[25]).

Gorenflo et al. ^[26] and Haubold et al. ^[27]) studied the various properties of generalized M-L function. In ^[28], a new generalization of M-L function (1.3) is presented by

$\begin{equation} \varepsilon_{a,b,p}^{c}(z_1) = \sum\limits_{l = 0}^{\infty}\frac{(c;p)_l}{\Gamma(a l+b)}\frac{z_1^l}{l!}\,\,\,\,\,\,\,\,\,(p\geq0, a,b,c\in\mathbb{C}; , \Re(a) > 0,\Re(b) > 0,), \end{equation}$

(1.6)

where $(\lambda; p)_l$ is the Pochhammer symbol defined by Srivastava et al. ^[29,30] as

$\begin{equation} (\lambda;p)_{\mu} = \begin{cases} \frac{\Gamma_{p}(\lambda+\mu)}{\Gamma(\lambda)}; &(p > 0, \lambda,\mu\in\mathbb{C} )\\ \\ (\lambda)_{\mu}; & (p = 0,\ {\lambda,\mu}\in \mathbb{C}\setminus\{{0}\}.\ \ \ \ \ \ \ \ \ \ \ \ \end{cases} \end{equation}$

(1.7)

The researchers examined the developments of these extension, (1.6) and (1.7) and studied their related features and applications. In ^[30], Srivastava et al. proposed the following generalized hypergeometric function

$\begin{equation} _sF_t[(\delta_1; p),\cdots, (\delta_s); (\zeta_1),\cdots, (\zeta_t); z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l\cdots (\delta_s)_l}{(\zeta_1)_l\cdots (\zeta_t)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$

(1.8)

where $\delta_{j}\in\mathbb{C}$ for j = 1, 2, $\cdots$ , s, $\zeta_{k}\in\mathbb{C}$ for $k = 1, 2, \cdots, t,$ and $\zeta_{k}\neq$ 0, -1, -2, $\cdots$ .

The integral representation of $(\mu; p)_\eta$ is explained by

$\begin{equation} (\mu;p)_{\eta} = \frac{1}{\Gamma(\mu)}\ \int_0^\infty\ s^{\mu+\eta-1}\ e^{-s-\frac{p}{s}} d s, \end{equation}$

(1.9)

where $\Re(\rho) > 0$ and $\Re(\mu+\eta) > 0$ . In particular, the related confluent hypergeometric function $_1F_1$ and the Gauss hypergeometric function $_2F_1$ are given by

$\begin{equation} _2F_1[(\delta_1; p),b;\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l (b)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, , \end{equation}$

(1.10)

and

$\begin{equation} _1F_1[(\delta_1; p);\lambda; z_1] = \Phi[(\delta_1; p);\lambda; z_1] = \sum\limits_{l = 0}^{\infty}\frac{(\delta_{1}; p)_l}{(\lambda)_l}\ \frac{z_1^l}{l!}\, . \end{equation}$

(1.11)

The expansion of the generalised hypergeometric function $_rF_s$ , which was studied by ^[30], has $r$ numerator and $s$ denominator parameters. Researchers recently developed several extensions of special functions, together with their corresponding characteristics and applications. Using extended beta functions as its foundation, Nisar et al. ^[31], Bohner et al. ^[32] and Rahman et al. ^[33] developed an enlargement of fractional derivative operators.

The multivariate M-L function is defined by ^[34] as follows:

$\begin{align} \mathcal{E}_{(a_j), b}^{(c_j)} (z_1, z_2, \dots, z_j)& = \mathcal{E}_{(a_1, a_2,\dots,a_j), b}^{(c_1, c_2 ,\dots, c_j)} (z_1, z_2, \dots z_j) \\ = & \sum\limits_{m_1, m_2, \dots, m_j = 0}^{\infty} \frac{(c_1)_{m_1}(c_2)_{m_2} \dots (c_j)_{m_j} (z_1)^{m_1} \dots (z_j)^{m_j}}{\Gamma(a_{1}m_1+a_2 m_2+\dots a_j m_j + b)m_1! \dots m_j!}, \end{align}$

(1.12)

where $z_i, a_i, b, c_i \in \mathbb{C}$ ; $i = 1, 2, \ldots, j$ , $\Re{(a_i)} > 0$ , $\Re{(b)} > 0$ and $\Re{(c_i)} > 0$ .

In ^{[35,36,37,38,39]}, the authors have studied various properties and applications of different type of generalized M-L functions. For real (complex) valued functions, the Lebesgue measurable space is defined by

$\begin{equation} \textsf{L}(r,s) = \left\{ \begin{array}{c} h: \|h\|_1 = \int_{r}^{s}|h(x)|dx < \infty \\ \end{array} \right\}. \end{equation}$

(1.13)

The left and right sides fractional integral operators of the Riemann-Liouville type are defined by ^[3,4] as follows:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{x}\frac{h(\varrho)}{(x-\varrho)^{1-\lambda}}d\varrho,\; \; \; (x > r), \end{equation}$

(1.14)

and

$\begin{equation*} (\mathfrak{I}_{s-}^{\lambda}h)(x) = \frac{1}{\Gamma(\lambda)}\int\limits_{x}^{s}\frac{h(\varrho)}{(\varrho-x)^{1-\lambda}}d\varrho,\; \; \; (x < s), \end{equation*}$

where $h\in \textsf{L}(r, s)$ , $\lambda\in\mathbb{C}$ and $\Re(\lambda) > 0$ .

The left and right sides Riemann-Liouville fractional derivatives for the function $h(x)\in \textsf{L}(r, s)$ , $\lambda\in\mathbb{C}$ , $\Re(\lambda) > 0$ and $n = [\Re(\lambda)]+1$ are defined in ^[3,4] by

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda}h)(x) = (\frac{d}{dx})^n(\mathfrak{I}_{r+}^{n-\lambda}h)(x) \end{equation}$

(1.15)

and

$\begin{equation*} (\mathfrak{D}_{s-}^{\lambda}h)(x) = (-\frac{d}{dx})^n(\mathfrak{I}_{s-}^{n-\lambda}h)(x), \end{equation*}$

respectively. The generalized differential operator $\mathfrak{D}_{r+}^{\lambda, v}$ of order $0 < \lambda < 1$ and type $0 < v < 1$ with respect to $x$ can be found in ^[2,4] as

$\begin{equation} (\mathfrak{D}_{r+}^{\lambda,v}h) = \left( \begin{array}{c} \mathfrak{I}_{r+}^{v(1-\lambda)}\frac{d}{dx}(\mathfrak{I}_{r+}^{(1-v)(1-\lambda)}h) \\ \end{array} \right)(x). \end{equation}$

(1.16)

In particular, if $v = 0,$ then (1.16) will lead to the operator $\mathfrak{D}_{r+}^{\lambda}$ defined in (1.15).

We also take into account the aforementioned well-known results.

Theorem 1.1. In ^[40], the following result for the fractional integral is presented by

$\begin{equation} \mathfrak{I}_{r+}^{\lambda}(\varrho-r)^{\eta-1} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}(x-r)^{\lambda+\eta-1}, \end{equation}$

(1.17)

where $\lambda$ , $\eta\in\mathbb{C}$ , $\Re(\lambda) > 0$ , $\Re(\eta) > 0$ ,

Theorem 1.2. ^[41] Suppose that the function $h(z)$ has a power series expansion $h(z) = \sum\limits_{k = 0}^{\infty}k_nz^k$ and it is analytic in the disc $|z| < R$ , then we have the following result

$\begin{equation*} \label{a11} \mathfrak{D}_{z}^{\lambda}\{z^{\eta-1}h(z)\} = \frac{\Gamma(\eta)}{\Gamma(\lambda+\eta)}\sum\limits_{n = 0}^{\infty}\frac{a_n(\eta)_n}{(\lambda+\eta)_n}z^n. \end{equation*}$

Lemma 1.1. (Srivastava and Tomovski ^[42]) Suppose that $x > r$ , $\lambda\in(0, 1)$ , $v\in[0, 1]$ , $\Re(\eta) > 0$ and $\Re(\lambda) > 0$ , then we have

$\begin{equation} \mathfrak{D}_{r+}^{\lambda}[(\varrho-r)^{\eta-1}](x) = \frac{\Gamma(\eta)}{\Gamma(\eta-\lambda)}(x-r)^{\eta-\lambda-1}. \end{equation}$

(1.18)

The generalized multivariate M-L function (1.12) is then defined in terms of the modified Pochhammer symbol (1.7) and its different features as well as the accompanying integral operators are examined. This is driven by the aforementioned modifications of special functions.

Motivated by the above results and literature, the paper has the following structure: First, we describe and investigate a novel generalization of the multivariate M-L function using a generalized Pochhammer symbol. Secondly, we offer a few differential and fractional integral formulas for the explored multivariate M-L function. By using the new form of the multivariate M-L function, a new generalization of the fractional integral operator is introduced, and some fundamental characteristics of the operator are discussed.

2. The generalized multivariate of M-L function

We are in a position to present the generalized multivariate M-L function by utilizing the extended Pochhammer symbol in (1.7) as follows:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+a_2l_2+\cdots+c_jl_j+b)}\frac{z_1^{l_1} z_2^{l_2}\cdots z_j^{l_j}}{l_1!\cdots l_j!}, \end{equation}$

(2.1)

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ . The special case for $a_1 = 1$ and $l_2 = \cdots = l_j = 0$ in (2.1) can be reduced to extended confluent hypergeometric function (1.11) as follows:

$\begin{equation} \varepsilon_{1,b;p}^{c_1}(z_1) = \frac{1}{\Gamma(b)} {_1F_1}[(c_1; p);b; z_1] = \frac{1}{\Gamma(b)} \Phi[(c_1; p);b; z_1]. \end{equation}$

(2.2)

In coming results, we demonstrate some fundamental characteristics and integral representations of the generalized multivariate M-L function.

Theorem 2.1. For the multivariate M-L function defined in (2.1), the following relation holds true:

$\begin{equation} \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = b \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,z_2,\cdots,z_j) \end{equation}$

(2.3)

$\begin{equation*} +[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}] \varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j), \end{equation*}$

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ .

Proof. From (2.1), we have

$\begin{align*} &b \varepsilon_{(a_j),b+1,p}^{(c_j)}(z_1,\cdots,z_j)+[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\varepsilon_{(a_j),b+1;p}^{(c_j)}(z_1,\cdots,z_j)\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1\frac{d}{dz_1}+\cdots+ a_jz_j\frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&[a_1z_1 \frac{d}{dz_1}+\cdots +a_jz_j \frac{d}{dz_j}]\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_jl_j}{l_1!\cdots l_j!}\\ = &b \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\+&\sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{\Gamma(a_1 l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j)\\ = & \sum\limits_{l_1,\cdots,l_j = 0}^{\infty}\frac{ (c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+1)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}(a_1l_1+\cdots+a_jl_j+b)\ \ (\mbox{using} \ \Gamma(z_1+1) = z_1 \Gamma(z_1))\\ = &\sum\limits_{l = 0}^{\infty}\frac{(c_1,p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j} }{ \Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\varepsilon_{(a_j),b,p}^{(c_j)}(z_1,z_2,\cdots,z_j), \end{align*}$

which is the desired result (2.3).

Theorem 2.2. For the generalized multivariate M-L function defined in (1.12), the following relations hold true:

$\begin{equation} \big(\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,z_2,\cdots,z_j) = (c_1)_m\cdots(c_j)_m \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,\cdots,z_j), \end{equation}$

(2.4)

and

$\begin{equation} \big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}))] = z_1^{b-m-1} \varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}), \end{equation}$

(2.5)

where $a_i, b, c_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, , p\geq0$ for $i = 1, 2, \cdots, j$ , and $\Re(b-m) > 0$ with $m\in\mathbb{N}.$

Proof. Differentiating (1.12) $m$ times with respect to $z_1, z_2, \cdots, z_j$ respectively, we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,z_j) = (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \sum\limits_{l_1 = l_2 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}(c_2)_{l_2}\cdots(c_j)_{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ = &\sum\limits_{l_1 = \cdots = l_j = m}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{l_1!\cdots l_j!\ z_1^{l_1-m}\cdots z_j^{l_j-m}}{(l_1-m)!\cdots(l_j-m)!\ l1!\cdots l_j!}\\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1+m}\cdots(c_j)_{l_j+m}}{\Gamma(a_1(l_1+m)+\cdots a_j(l_j+m)+b)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\ \ (\mbox{Replacing}\ l_i \ \mbox{by}\ l_i+m) \\ & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1)_m\cdots(c_j)_m\ (c_1+m;p)_{l_1}\cdots(c_j+m)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots+a_j)m)}\frac{z_1^{l_1}\cdots z_j^{l_j}}{l_1!\cdots l_j!}. \end{align*}$

Now using $(\lambda; \sigma)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu; \sigma)_p$ and $(\lambda)_{\mu+p} = (\lambda)_{\mu} (\lambda+\mu)_p$ , we get

$\begin{align*} & (\frac{d}{dz_1}\big)^m\cdots(\frac{d}{dz_j}\big)^m \varepsilon_{(a_j),b;p}^{(c_j)}(z_1,\cdots,c_j)\\ = &(c_1)_m \cdots(c_j)_m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1+m;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots a_jl_j+b+(a_1+\cdots +a_j)m)}\frac{z_1^{l}\cdots z_j^{l_j}}{l_1!\cdots l_j!}\\ & = (c_1)_m\cdots (c_j)_{m} \ \varepsilon_{(a_j),b+(a_j)m;p}^{(c_j)+m}(z_1,z_2,\cdots, z_j), \end{align*}$

which is the desired result (2.4). Similarly, to prove (2.5), we have

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\big(\frac{d}{dz_1}\big)^m z_1^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_jz_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &\big(\frac{d}{dz_1}\big)^m \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+ a_jl_j+b)} \frac{z_1^{b-1+a_1l_1+\cdots+a_jl_j}}{l_1!\cdots l_j!} \varpi_1^{l_1}\cdots\varpi_j^{l_j}\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}}{l_1!\cdots l_j!} \frac{(a_1l_1+\cdots+a_jl_j+b-1)!}{(a_1l_1+\cdots+a_jl_j+b-m-1)!}\ z_1^{a_1l_1+\cdots+a_jl_j+b-m-1}. \end{align*}$

Differentiating $m$ times and using the relation $l(l-1)! = l!$ , we get

$\begin{align*} &\big(\frac{d}{dz_1}\big)^m [z_1^{b-1} \varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots \varpi_j z_j^{a_j})]\\ = &\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b)} \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\, \varpi_1^{l_1}\cdots\varpi_j^{l_j} \, \frac{z_1^{a_1l_1+\cdots+a_jl_j+b-1-m}}{l_1!\cdots l_j!} \\ = & z_1^{b-m-1}\sum\limits_{l_1 = \cdots l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b-m)}\frac{(\varpi_1 z_1^{a_1})^{l_1}\cdots(\varpi_1 z_1^{a_j})^{l_j}}{l_1!\cdots l_j!}\\ = &z_1^{b-m-1}\varepsilon_{(a_j),b-m;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_j z_1^{a_j}). \end{align*}$

The proof is completed.

Corollary 2.1. The generalized multivariate M-L function has the following integral representations:

$\begin{equation*} \int_{0}^{z_1}t^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1 t^{a_1},\cdots,\varpi_j t^{a_j}) d t = z_1^{b}\varepsilon_{(a_j),b+1;p}^{(c_j)}(\varpi_1 z_1^{a_1},\cdots,\varpi_jz_1^{a_j}), \end{equation*}$

where $a_i, b, c_i, \varpi_i\in\mathbb{C}; \Re(a_i) > 0, \Re(b) > 0, p\geq0$ for $i = 1, 2, \cdots, j$ .

3. Fractional integral and differential formulas of generalized M-L function

In this section, we present some fractional integration and differentiation formulas of generalized M-L function given in (2.1).

Theorem 3.1. Suppose $x > r\, (r\in\mathbb{R_+} = [0, \infty))$ , $a_i$ , $b$ , $c_i$ , $\varpi\in\mathbb{C}$ , $\Re(a_i) > 0$ and $\Re(c_i) > 0$ , $\Re(b) > 0$ and $\Re(\lambda) > 0$ , then the following relations hold true:

$\begin{align} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_j),b;p}^{(c_j)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{\lambda+b-1} \varepsilon_{(a_i), b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align}$

(3.1)

$\begin{align} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{( c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \end{align}$

(3.2)

and

$\begin{align} &\mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &(x-r)^{b-\lambda-1} \varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align}$

(3.3)

Proof. Consider

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{(x-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_1(\varrho-r)^{a_j})} {(x-\varrho)^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\int_{r}^{x}(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}(x-\varrho)^{\lambda-1}d\varrho\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \left( \mathfrak{I}_{r+}^{\lambda}[(\varrho-r)^{b+a_1 l_1+\cdots+a_jl_j-1}] \right). \end{align*}$

By the use of (1.17), we have

$\begin{align*} &\mathfrak{I}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x)\\ = &\sum\limits_{n = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_n}\varpi^{l_1}\cdots\varpi^{l_j}}{\Gamma(a_1 l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} (x-r)^{b+\lambda+a_1l_1+\cdots+a_jl_j-1}. \frac{\Gamma(a_1l_1+\cdots+a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\sum\limits_{n = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)} \frac{[\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ = &(x-r)^{b+\lambda-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the proof of (3.1).

Next, we have

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right]\\ = &(\frac{d}{dx})^n \left\{ \mathfrak{I}_{r+}^{n-\lambda}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right\}, \end{align*}$

which on using (3.1) takes the following form:

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \\ = &(\frac{d}{dx})^n \left\{ (x-r)^{b-\lambda+n-1}\varepsilon_{(a_i),b-\lambda+n;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}) \right\}. \end{align*}$

Applying (2.5), we get

$\begin{align*} &\mathfrak{D}_{r+}^{\lambda}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j})\right](x) \\ = &\left\{ (x-r)^{\eta-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots, \varpi_j(x-r)^{a_j}) \right\}, \end{align*}$

which gives the proof of (3.2).

To obtain (3.3), we have

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})\right] \right)(x)\\ = &\left( \mathfrak{D}_{r+}^{\lambda,v}\left[\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}\right] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{a_1 l_1+\cdots+a_jl_j+b-1}] \right)(x). \end{align*}$

By applying (1.18), we get

$\begin{align*} &\left( \mathfrak{D}_{r+}^{\lambda,v}[(\varrho-r)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-r)^{a_1},\cdots,\varpi_j(\varrho-r)^{a_j})] \right)(x)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\frac{\varpi^{l_1}\cdots\varpi^{l_j}}{l_1!\cdots l_j!}\\ \times&\frac{\Gamma(a_1l_1+\cdots+ a_jl_j+b)}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}(x-r)^{a_1l_1+ \cdots+a_jl_j+b-\lambda-1}\\ = &(x-r)^{b-\lambda-1}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p,v)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j +b-\lambda)}\frac{\varpi^{l_1}(x-r)^{a_1}\cdots\varpi^{l_j}(x-r)^{a_j}}{l_1!\cdots l_j!}\\ = &(x-r)^{b-\lambda-1}\varepsilon_{(a_i),b-\lambda;p}^{(c_i)}(\varpi^{l_1}(x-r)^{a_1},\cdots,\varpi^{l_j}(x-r)^{a_j}), \end{align*}$

which completes the required proof.

Remark 3.1. Applying Theorem 3.1 for $p = 0$ , then we obtain the result presented in ^[34].

4. Generalization of fractional integral and its properties

In this section, we define a fractional integral involving newly defined multivariate M-L function and discuss its properties.

Definition 4.1. Let $b, a_i, c_i, \varpi_i\in\mathbb{C}$ , $\Re(c_i) > 0$ , $\Re(a_i) > 0$ and $\Re(b) > 0$ and $h\in L(r, s)$ . Then the generalized left and right sided fractional integrals are defined by

$\begin{align} &\left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho, (x > r) \end{align}$

(4.1)

and

$\begin{align} &\left(\mathfrak{R}_{s-;(a_i),b;p}^{(\varpi_i);(c_i)}h\right)(x) = \int_{x}^{s}(\varrho-x)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(\varrho-x)^{a_1},\cdots,\varpi_j(\varrho-x)^{a_j})h(\varrho)d\varrho, (x < s), \end{align}$

(4.2)

respectively.

Remark 4.1. If we consider $p = 0$ , then the operators defined in (4.1) and (4.2) will take the form defined earlier by ^[34]. Similarly, if we consider $p = 0$ and $j = 1$ , then the operators defined in (4.1) and (4.2) will take the form defined by ^[22]. If we take $j = 1$ , then the work done in this paper will lead to the work presented by ^[28]. Also, if we consider one of $\varpi_i = 0$ , for $i = 1, 2, \cdots, j$ , then the operators defined in (4.1) and (4.2) will take the form of the classical operators.

Next, we prove the following properties of integral operator defined in (4.1).

Theorem 4.1. Suppose that $b, a_i, \lambda, c_i, \varpi_i\in\mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(\lambda) > 0$ , $p\geq0$ and $\Re(c_i) > 0$ for $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = (x-r)^{\lambda+b-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda}^{(c_i);p}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$

Proof. By the use of definition (4.1), we have

$\begin{equation*} \left( \mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h \right)(x) = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho. \end{equation*}$

Therefore, we get

$\begin{align*} &\left( \mathfrak{R}_{r+;(a_i), b;p}^{(\varpi_i);(c_i)}[(\varrho-r)^{\lambda-1}] \right) (x) = \int_{r}^{x}(x-\varrho)^{b-1}(\varrho-r)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \left( \int_{r}^{x}(\varrho-r)^{\lambda-1}(x-\varrho)^{\lambda+a_1l_1+ \cdots+a_jl_j-1}d\varrho \right)\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))}\frac{\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {l_1!\cdots l_j!} \mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j+b}[(\varrho-r)^{\lambda-1}]\\ & = (x-r)^{b+\lambda-1} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}}{\Gamma(a_1l_1+\cdots+a_jl_j+b))} \frac{[\varpi_1(x-r)^{a_1l_1}\cdots\varpi_j(x-r)^{a_jl_j}]}{l_1!\cdots l_j!}\\ \times& \frac{\Gamma(\lambda)\Gamma(a_1l_1+\cdots+a_jl_j +b)}{\Gamma(a_1l_1+\cdots+a_jl_j+b+\lambda)}\\ = &(x-r)^{b+\lambda-1}\Gamma(\lambda)\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the desired proof.

Theorem 4.2. Suppose that $c_i, a_i, b, \varpi_i \in\mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0,$ $p\geq0$ for $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{equation*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq K\|\Phi\|_1. \end{equation*}$

Where

$\begin{align*} &K: = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}. \end{align*}$

Proof. By the use of (1.13) and (4.1) and by interchanging integration and summation order, we have

$\begin{align*} \|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i) }\Phi\|_1& = \int\limits_{r}^{s}|\int_{r}^{x}(x-\varrho)^{b-1} \varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) \Phi(\varrho)d\varrho|dx\\ \leq&\int_{r}^{s}\left[ \int_{\varrho}^{x} (x-\varrho)^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})|dx \right]|\Phi(\varrho)|d\varrho\\ = &\int_{r}^{s}\left[ \int_{0}^{x-\varrho} u^{\Re(b)-1}|\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1 u^{a_1},\cdots,\varpi_j u^{a_j})|du \right]|\Phi(\varrho)|d\varrho, \end{align*}$

by setting $u = x-\varrho.$ After simplification, we obtain

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\\ \leq&\int_{r}^{s} \Bigr[ \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j+b)}\frac{|\varpi_1^{a_1}\cdots\varpi_j^{l_j}|} {l_1!\cdots l_j!}\\ \times& \left( \frac{(u)^{\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j}}{(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)} \right)_{0}^{s-r} \Bigr] |\Phi(\varrho)|d\varrho. \end{align*}$

It follows that

$\begin{align*} &\|\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}\Phi\|_1\leq \Bigr\{ (s-r)^{\Re(b)} \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!} \Bigr\} \int\limits_{r}^{s}|\Phi(\varrho)|d\varrho\\ = &K||\Phi||_1, \end{align*}$

where

$\begin{align*} &K = (s-r)^{\mathfrak{Re}(b)}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{|(c_1;p)_{l_1}\cdots(c_j)_{l_j}|}{\Gamma(a_1l_1+\cdots+a_jl_j +b)(\Re(b)+\Re(a_1)l_1+\cdots+\Re(a_j)l_j)}\\ \times&\frac{|\varpi_1^{l_1}(s-r)^{a_1l_1}\cdots\varpi_j^{l_j}(s-r)^{a_jl_j}|}{l_1!\cdots l_j!}, \end{align*}$

which gives the desired result.

Corollary 4.1. If we take $a_i, b, c_i, \varpi_i \in \mathbb{C}$ , $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ with $i = 1, 2, \cdots, j$ , then the following result holds true:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) = (x-r)^{b}\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}). \end{align*}$

Proof. Consider

$\begin{align*} &\left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}1\right)(x) = \int_{r}^x(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-r)^{a_j})d\varrho\\ = &\int_{r}^x(x-\varrho)^{b-1}\left(\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j}(x-\varrho)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\right)d\varrho. \end{align*}$

It follows that

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}1\right)(x) & = \sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}\cdots\varpi_j^{l_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!} \int_{r}^x(x-\varrho)^{b+a_1l_1+\cdots+a_jl_j-1}d\varrho\\ = &(x-r)^{b}\sum\limits_{l_1 = \cdots = l_j = 0}^{\infty}\frac{(c_1;p)_{l_1}\cdots(c_j)_{l_j}\varpi_1^{l_1}(x-r)^{a_1l_1}\cdots\varpi_j^{l_j}(x-r)^{a_jl_j}} {\Gamma(a_1l_1+\cdots+a_jl_j+b)(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}\\ = &(x-r)^{b}\,\varepsilon_{(a_i),b+1;p}^{(c_i)}(\varpi_1(x-r)^{a_1},\cdots,\varpi_j(x-r)^{a_j}), \end{align*}$

which gives the desired proof.

Theorem 4.3. The generalized fractional operator can be represented in term of Riemann–Liouville fractional integrals for $c_i$ , $a_i$ , $b$ , $\varpi_i\in\mathbb{C}$ with $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ for $i = 1, 2, \cdots, j$ , $p\geq0$ and $x > r$ as follows:

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x) = \sum\limits_{l_1 = \cdots = l_j = 0}^\infty \frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1}\cdots\varpi_j^{a_j}} {\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1+\cdots+a_jl_j +b}h(x). \end{align*}$

Proof. By utilizing (2.1) in (4.1) and then interchanging the order of summation and integration, we have

$\begin{align*} \left(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h\right)(x)& = \int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)} (\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})h(\varrho)d\varrho\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{l_1} (x-\varrho)^{a_1l_1}\cdots\varpi_j^{l_j} (x-\varrho)^{a_jl_j}} {\Gamma(c_1)\Gamma(a_1l_1+\cdots+a_jl_j+b)l_1!\cdots l_j!}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!} \frac{1}{\Gamma(a_1l_1+\cdots+a_jl_j +b)}\\ \times&\int_{r}^{x}(x-\varrho)^{a_1l_1+\cdots+a_jl_j+b-1}h(\varrho)d\varrho\\ = &\sum\limits_{l_1 = \cdots = l_j = 0}^\infty\frac{\Gamma(c_1+l_1;p)(c_2)_{l_2}\cdots(c_j)_{l_j}\varpi_1^{a_1l_1}\cdots\varpi_j^{a_jl_j} }{\Gamma(c_1)l_1!\cdots l_j!}\mathfrak{I}_{r+}^{a_1l_1 +\cdots+a_jl_j+b}h(x), \end{align*}$

which gives the desired proof.

Theorem 4.4. For $\lambda$ , $c_i$ , $a_i$ , $b$ , $\varpi_i\in\mathbb{C}$ with $\Re(a_i) > 0$ , $\Re(b) > 0$ , $\Re(c_i) > 0$ , $\Re(\lambda) > 0$ , for $i = 1, 2, \cdots, j$ , $p\geq0$ and $x > r$ , then the following result holds true:

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = (\mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h)(x) = (\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x), \end{equation}$

(4.3)

where $h\in L(r, s)$ .

Proof. By employing (1.14) and (4.1), we have

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}\frac{[(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h)(\varrho)]}{(x-\varrho) ^{1-\lambda}}d\varrho\\ = &\frac{1}{\Gamma(\lambda)}\int_{r}^{x}(x-\varrho)^{\lambda-1} \left[ \int_{r}^{\varrho} (\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})h(u)du \right]d\varrho. \end{align*}$

It follows that

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}h])(x) \\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{u}^{x} (x-\varrho)^{\lambda-1}(\varrho-u)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) }(\varpi_1(\varrho-u)^{a_1},\cdots,\varpi_j(\varrho-u)^{a_j})d\varrho \right]h(u)du. \end{align*}$

By considering $\varrho-u = \theta$ , we get

$\begin{align*} \begin{aligned} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} (x-u-\theta)^{\lambda-1}\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i) } (\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right]h(u)du\\ = &\int_{r}^{x} \left[ \frac{1}{\Gamma(\lambda)}\int_{0}^{x-u} \frac{\theta^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})}{(x-u-\theta)^{1-\lambda}}d\theta \right]h(u)du.\end{aligned} \end{align*}$

Hence, from (1.14) and applying (3.1), we obtain

$\begin{align*} &(\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x)\\ = &\int_{r}^{x}\left[ \theta^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) \right]h(u)du\\ = &\int_{r}^{x} (x-u)^{\lambda+b-1}\varepsilon_{(a_i),b+\lambda}^{(c_i)}(\varpi_1(x-u)^{a_1},\cdots,\varpi_j(x-u)^{a_j})h(u)du. \end{align*}$

Thus, we have

$\begin{equation} (\mathfrak{I}_{r+}^{\lambda}[\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$

(4.4)

Next, consider the right hand side of (4.3) and employing (4.1) to derive the second part, we have

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}(x-\varrho)^{b-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j}) [\mathfrak{I}_{r+}^{\lambda}h] (\varrho)d\varrho\\ = &\int_{r}^{x}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})\left( \frac{1}{\Gamma(\lambda)}\int\limits_{r}^{\varrho}\frac{h(u)}{(\varrho-u)^{1-\lambda}}du \right)d\varrho. \end{align*}$

It follows that

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{u}^{x}(x-\varrho)^{b-1}(\varrho-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1(x-\varrho)^{a_1},\cdots,\varpi_j(x-\varrho)^{a_j})d\varrho \right] h(u)du. \end{align*}$

By setting $x-\varrho = \theta$ , we get

$\begin{align*} &(\mathfrak{R}_{r+;(a_i),b}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x)\\ = &\int_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{x-u}^{0}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j}) (-d\theta) \right] h(u)du\\ = &\int\limits_{r}^{x}\frac{1}{\Gamma(\lambda)} \left[ \int_{0}^{x-u}\theta^{b-1}(x-\theta-u)^{\lambda-1}\varepsilon_{(a_i),b;p}^{(c_i)}(\varpi_1\theta^{a_1},\cdots,\varpi_j\theta^{a_j})d\theta \right] h(u)du. \end{align*}$

Further, by using (1.14) and applying (3.1), we obtain

$\begin{equation} (\mathfrak{R}_{r+;(a_i),b;p}^{(\varpi_i);(c_i)}[\mathfrak{I}_{r+}^{\lambda}h])(x) = \left( \mathfrak{R}_{r+;(a_i),b+\lambda}^{(\varpi_i);(c_i)}h \right)(x). \end{equation}$

(4.5)

Thus, (4.4) and (4.5) gives the desired proof.

5. Conclusions

Nowadays, the theories are developed very rapidly. The scientists are introducing more advanced and generalized forms of the classical ones. In this present study, we introduced a generalized form of the multivariate M-L function (2.1) by employing the generalized Pochhammer symbol and its properties. By using this more extended form of M-L, we introduced a fractional integral operator and studied some of the basic properties of this operator. The special cases of the main results are if we take $p = 0$ , then the operators defined in (4.1) and (4.2) will reduce to the work done by ^[34]. Similarly, if we take $j = 1$ and $p = 0$ , then the operators defined in (4.1) and (4.2) will lead to the work done by ^[22]. If we take $j = 1$ , then the work done in this paper will lead to the work presented by ^[28]. Moreover, if we consider one of $\varpi_i = 0$ , for $i = 1, 2, \cdots, j$ , then the operators defined in (4.1) and (4.2) will reduce to the classical R-L operators. We believe that our proposed operator will be more applicable in the fields of fractional integral inequalities and operator theory.

Acknowledgments

The author T. Abdeljawad would like to thank Prince Sultan University for supporting through TAS research lab. Manar A. Alqudah: Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R14), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	S. Ahmad, T. Strauss, S. Kupis, T. Khan, Comparison of statistical inversion with iteratively regularized Gauss Newton method for image reconstruction in electrical impedance tomography, Appl. Math. Comput., 358 (2019), 436–448. https://doi.org/10.1016/j.amc.2019.03.063 doi: 10.1016/j.amc.2019.03.063
[2]	H. F. Alqadah, N. Valdivia, A frequency based constraint for a multi-frequency linear sampling method, Inverse Probl., 29 (2013), 095019. https://doi.org/10.1088/0266-5611/29/9/095019 doi: 10.1088/0266-5611/29/9/095019
[3]	H. Ammari, P. Garapon, F. Jouve, H. Kang, M. Lim, S. Yu, A new optimal control approach for the reconstruction of extended inclusions, SIAM J. Control Optim., 51 (2013), 1372–1394. https://doi.org/10.1137/100808952 doi: 10.1137/100808952
[4]	H. Ammari, J. Garnier, V. Jugnon, H. Kang, Stability and resolution analysis for a topological derivative based imaging functional, SIAM J. Control Optim., 50 (2012), 48–76. https://doi.org/10.1137/100812501 doi: 10.1137/100812501
[5]	H. Ammari, J. Garnier, H. Kang, W. K. Park, K. Sølna, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math., 71 (2011), 68–91. https://doi.org/10.1137/100800130 doi: 10.1137/100800130
[6]	H. Ammari, E. Iakovleva, S. Moskow, Recovery of small inhomogeneities from the scattering amplitude at a fixed frequency, SIAM J. Math. Anal., 34 (2003), 882–900. https://doi.org/10.1137/S0036141001392785 doi: 10.1137/S0036141001392785
[7]	H. Ammari, H. Kang, Reconstruction of small inhomogeneities from boundary measurements, Vol. 1846, Lecture Notes in Mathematics, Berlin: Springer-Verlag, 2004. https://doi.org/10.1007/b98245
[8]	H. Ammari, S. Moskow, M. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume, ESAIM: Control Optim. Calc. Var., 9 (2003), 49–66. https://doi.org/10.1051/cocv:2002071 doi: 10.1051/cocv:2002071
[9]	L. Audibert, H. Haddar, The generalized linear sampling method for limited aperture measurements, SIAM J. Imag. Sci., 10 (2017), 845–870. https://doi.org/10.1137/16M110112X doi: 10.1137/16M110112X
[10]	A. Baussard, D. Prémel, O. Venard, A Bayesian approach for solving inverse scattering from microwave laboratory-controlled data, Inverse Probl., 17 (2001), 1659. https://doi.org/10.1088/0266-5611/17/6/309 doi: 10.1088/0266-5611/17/6/309
[11]	K. Belkebir, M. Saillard, Special section: testing inversion algorithms against experimental data, Inverse Probl., 17 (2001), 1565. https://doi.org/10.1088/0266-5611/17/6/301 doi: 10.1088/0266-5611/17/6/301
[12]	K. Belkebir, A. G. Tijhuis, Modified $^2$ gradient method and modified Born method for solving a two-dimensional inverse scattering problem, Inverse Probl., 17 (2001), 1671. https://doi.org/10.1088/0266-5611/17/6/310 doi: 10.1088/0266-5611/17/6/310
[13]	E. Bergou, Y. Diouane, V. Kungurtsev, Convergence and complexity analysis of a Levenberg–Marquardt algorithm for inverse problems, J. Optim. Theory Appl., 185 (2020), 927–944. https://doi.org/10.1007/s10957-020-01666-1 doi: 10.1007/s10957-020-01666-1
[14]	R. F. Bloemenkamp, A. Abubakar, P. M. van den Berg, Inversion of experimental multi-frequency data using the contrast source inversion method, Inverse Probl., 17 (2001), 1611. https://doi.org/10.1088/0266-5611/17/6/305 doi: 10.1088/0266-5611/17/6/305
[15]	O. Bondarenko, A. Kirsch, X. Liu, The factorization method for inverse acoustic scattering in a layered medium, Inverse Probl., 29 (2013), 045010. https://doi.org/10.1088/0266-5611/29/4/045010 doi: 10.1088/0266-5611/29/4/045010
[16]	A. Carpio, T. G. Dimiduk, F. L. Louër, M. L. Rapún, When topological derivatives met regularized Gauss–Newton iterations in holographic 3D imagingg, J. Comput. Phys., 388 (2019), 224–251. https://doi.org/10.1016/j.jcp.2019.03.027 doi: 10.1016/j.jcp.2019.03.027
[17]	A. Carpio, M. Pena, M. L. Rapún, Processing the 2D and 3D Fresnel experimental databases via topological derivative methodss, Inverse Probl., 37 (2021), 105012. https://doi.org/10.1088/1361-6420/ac21c8 doi: 10.1088/1361-6420/ac21c8
[18]	X. Chen, Subspace-based optimization method for inverse scattering problems with an inhomogeneous background medium, Inverse Probl., 26 (2010), 074007. https://doi.org/10.1088/0266-5611/26/7/074007 doi: 10.1088/0266-5611/26/7/074007
[19]	M. Cheney, The linear sampling method and the MUSIC algorithm, Inverse Probl., 17 (2001), 591. https://doi.org/10.1088/0266-5611/17/4/301 doi: 10.1088/0266-5611/17/4/301
[20]	S. Coşğun, E. Bilgin, M. Çayören, Microwave imaging of breast cancer with factorization method: SPIONs as contrast agent, Med. Phys., 47 (2020), 3113–3122. https://doi.org/10.1002/mp.14156 doi: 10.1002/mp.14156
[21]	L. Crocco, T. Isernia, Inverse scattering with real data: detecting and imaging homogeneous dielectric objects, Inverse Probl., 17 (2001), 1573. https://doi.org/10.1088/0266-5611/17/6/302 doi: 10.1088/0266-5611/17/6/302
[22]	A. J. Deveney, Super-resolution processing of multi-static data using time-reversal and MUSIC, unpublished work, 2002.
[23]	O. Dorn, D. Lesselier, Level set methods for inverse scattering, Inverse Probl., 22 (2006), R67. https://doi.org/10.1088/0266-5611/22/4/R01 doi: 10.1088/0266-5611/22/4/R01
[24]	B. Duchêne, Inversion of experimental data using linearized and binary specialized nonlinear inversion schemes, Inverse Probl., 17 (2001), 1623. https://doi.org/10.1088/0266-5611/17/6/306 doi: 10.1088/0266-5611/17/6/306
[25]	L. Fatone, P. Maponi, F. Zirilli, An image fusion approach to the numerical inversion of multifrequency electromagnetic scattering data, Inverse Probl., 17 (2001), 1689. https://doi.org/10.1088/0266-5611/17/6/311 doi: 10.1088/0266-5611/17/6/311
[26]	M. Q. Feng, F. D. Flaviis, Y. J. Kim, Use of microwaves for damage detection of fiber reinforced polymer-wrapped concrete structures, J. Eng. Mech., 128 (2002), 172–183. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:2(172) doi: 10.1061/(ASCE)0733-9399(2002)128:2(172)
[27]	A. Franchois, C. Pichot, Microwave imaging-complex permittivity reconstruction with a Levenberg-Marquardt method, IEEE Trans. Antenn. Propag., 45 (1997), 203–215. https://doi.org/10.1109/8.560338 doi: 10.1109/8.560338
[28]	J. F. Funes, J. M. Perales, M. L. Rapún, J. M. Vega, Defect detection from multi-frequency limited data via topological sensitivity, J. Math. Imaging Vis., 55 (2016), 19–35. https://doi.org/10.1007/s10851-015-0611-y doi: 10.1007/s10851-015-0611-y
[29]	R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Probl., 27 (2011), 085005. https://doi.org/10.1088/0266-5611/27/8/085005 doi: 10.1088/0266-5611/27/8/085005
[30]	B. B. Guzina, F. Cakoni, C. Bellis, On the multi-frequency obstacle reconstruction via the linear sampling method, Inverse Probl., 26 (2010), 125005. https://doi.org/10.1088/0266-5611/26/12/125005 doi: 10.1088/0266-5611/26/12/125005
[31]	I. Harris, D. L. Nguyen, Orthogonality sampling method for the electromagnetic inverse scattering problem, SIAM J. Sci. Comput., 42 (2020), B722–B737. https://doi.org/10.1137/19M129783X doi: 10.1137/19M129783X
[32]	K. Huang, K. Sølna, H. Zhao, Generalized Foldy-Lax formulation, J. Comput. Phys., 229 (2010), 4544–4553. https://doi.org/10.1016/j.jcp.2010.02.021 doi: 10.1016/j.jcp.2010.02.021
[33]	D. Ireland, K. Bialkowski, A. Abbosh, Microwave imaging for brain stroke detection using Born iterative method IET Microw. Antenn. Propag., 7 (2013), 909–915. https://doi.org/10.1049/iet-map.2013.0054
[34]	K. Ito, B. Jin, J. Zou, A direct sampling method to an inverse medium scattering problem, Inverse Probl., 28 (2012), 025003. https://doi.org/10.1088/0266-5611/28/2/025003 doi: 10.1088/0266-5611/28/2/025003
[35]	L. Jofre, A. Broquetas, J. Romeu, S. Blanch, A. P. Toda, X. Fabregas et al., UWB tomographic radar imaging of penetrable and impenetrable objects, Proc. IEEE, 97 (2009), 451–464. https://doi.org/10.1109/JPROC.2008.2008854 doi: 10.1109/JPROC.2008.2008854
[36]	S. Kang, S. Chae, W. K. Park, A study on the orthogonality sampling method corresponding to the observation directions configuration, Results Phys., 33 (2022), 105108. https://doi.org/10.1016/j.rinp.2021.105108 doi: 10.1016/j.rinp.2021.105108
[37]	S. Kang, M. Lambert, Structure analysis of direct sampling method in 3D electromagnetic inverse problem: near- and far-field configuration, Inverse Probl., 37 (2021), 075002. https://doi.org/10.1088/1361-6420/abfe4e doi: 10.1088/1361-6420/abfe4e
[38]	S. Kang, M. Lambert, C. Y. Ahn, T. Ha, W. K. Park, Single- and multi-frequency direct sampling methods in limited-aperture inverse scattering problem, IEEE Access, 8 (2020), 121637–121649. https://doi.org/10.1109/ACCESS.2020.3006341
[39]	S. Kang, M. Lambert, W. K. Park, Direct sampling method for imaging small dielectric inhomogeneities: analysis and improvement, Inverse Probl., 34 (2018), 095005. https://doi.org/10.1088/1361-6420/aacf1d doi: 10.1088/1361-6420/aacf1d
[40]	S. Kang, M. Lambert, W. K. Park, Analysis and improvement of direct sampling method in the mono-static configuration, IEEE Geosci. Remote Sens. Lett., 16 (2019), 1721–1725. https://doi.org/10.1109/LGRS.2019.2906366
[41]	S. Kang, M. Lim, W. K. Park, Fast identification of short, linear perfectly conducting cracks in the bistatic measurement configuration, J. Comput. Phys., 468 (2022), 111479. https://doi.org/10.1016/j.jcp.2022.111479 doi: 10.1016/j.jcp.2022.111479
[42]	S. Kang, W. K. Park, Application of MUSIC algorithm for a fast identification of small perfectly conducting cracks in limited-aperture inverse scattering problem, Comput. Math. Appl., 117 (2022), 97–112. https://doi.org/10.1016/j.camwa.2022.04.015 doi: 10.1016/j.camwa.2022.04.015
[43]	S. Kang, W. K. Park, S. H. Son, A qualitative analysis of bifocusing method for a real-time anomaly detection in microwave imaging, Comput. Math. Appl., 137 (2023), 93–101. https://doi.org/10.1016/j.camwa.2023.02.017 doi: 10.1016/j.camwa.2023.02.017
[44]	Y. J. Kim, L. Jofre, F. D. Flaviis, M. Q. Feng, Microwave reflection tomographic array for damage detection of civil structures, IEEE Trans. Antenn. Propag., 51 (2003), 3022–3032. https://doi.org/10.1109/TAP.2003.818786 doi: 10.1109/TAP.2003.818786
[45]	A. Kirsch, The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Probl., 18 (2002), 1025. https://doi.org/10.1088/0266-5611/18/4/306 doi: 10.1088/0266-5611/18/4/306
[46]	A. Kirsch, S. Ritter, A linear sampling method for inverse scattering from an open arc, Inverse Probl., 16 (2000), 89. https://doi.org/10.1088/0266-5611/16/1/308 doi: 10.1088/0266-5611/16/1/308
[47]	R. Kress, Newton's method for inverse obstacle scattering meets the method of least squares, Inverse Probl., 19 (2003), S91. https://doi.org/10.1088/0266-5611/19/6/056 doi: 10.1088/0266-5611/19/6/056
[48]	O. Kwon, J. K. Seo, J. R. Yoon, A real-time algorithm for the location search of discontinuous conductivities with one measurement, Commun. Pure Appl. Math., 55 (2002), 1–29. https://doi.org/10.1002/cpa.3009 doi: 10.1002/cpa.3009
[49]	L. J. Landau, Bessel functions: monotonicity and bounds, J. London Math. Soc., 61 (2000), 197–215. https://doi.org/10.1112/S0024610799008352 doi: 10.1112/S0024610799008352
[50]	Z. Liu, A new scheme based on Born iterative method for solving inverse scattering problems with noise disturbance, IEEE Geosci. Remote Sens. Lett., 16 (2019), 1021–1025. https://doi.org/10.1109/LGRS.2019.2891660
[51]	F. L. Louër, M. L. Rapún, Topological sensitivity for solving inverse multiple scattering problems in 3D electromagnetism. Part I: one step method, SIAM J. Imaging Sci., 10 (2017), 1291–1321. https://doi.org/10.1137/17M1113850 doi: 10.1137/17M1113850
[52]	R. Marklein, K. Balasubramanian, A. Qing, K. J. Langenberg, Linear and nonlinear iterative scalar inversion of multi-frequency multi-bistatic experimental electromagnetic scattering data, Inverse Probl., 17 (2001), 1597. https://doi.org/10.1088/0266-5611/17/6/304 doi: 10.1088/0266-5611/17/6/304
[53]	W. K. Park, Topological derivative strategy for one-step iteration imaging of arbitrary shaped thin, curve-like electromagnetic inclusions, J. Comput. Phys., 231 (2012), 1426–1439. https://doi.org/10.1016/j.jcp.2011.10.014 doi: 10.1016/j.jcp.2011.10.014
[54]	W. K. Park, Analysis of a multi-frequency electromagnetic imaging functional for thin, crack-like electromagnetic inclusions, Appl. Numer. Math., 77 (2014), 31–42. https://doi.org/10.1016/j.apnum.2013.11.001 doi: 10.1016/j.apnum.2013.11.001
[55]	W. K. Park, Asymptotic properties of MUSIC-type imaging in two-dimensional inverse scattering from thin electromagnetic inclusions, SIAM J. Appl. Math., 75 (2015), 209–228. https://doi.org/10.1137/140975176 doi: 10.1137/140975176
[56]	W. K. Park, Multi-frequency subspace migration for imaging of perfectly conducting, arc-like cracks in full- and limited-view inverse scattering problems, J. Comput. Phys., 283 (2015), 52–80. https://doi.org/10.1016/j.jcp.2014.11.036 doi: 10.1016/j.jcp.2014.11.036
[57]	W. K. Park, A novel study on subspace migration for imaging of a sound-hard arc, Comput. Math. Appl., 74 (2017), 3000–3007. https://doi.org/10.1016/j.camwa.2017.07.045 doi: 10.1016/j.camwa.2017.07.045
[58]	W. K. Park, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc., 153 (2021), 107501. https://doi.org/10.1016/j.ymssp.2020.107501 doi: 10.1016/j.ymssp.2020.107501
[59]	W. K. Park, Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging, Mech. Syst. Signal Proc., 171 (2022), 108937. https://doi.org/10.1016/j.ymssp.2022.108937 doi: 10.1016/j.ymssp.2022.108937
[60]	W. K. Park, On the application of orthogonality sampling method for object detection in microwave imaging, IEEE Trans. Antenn. Propag., 71 (2023), 934–946. https://doi.org/10.1109/TAP.2022.3220033 doi: 10.1109/TAP.2022.3220033
[61]	W. K. Park, D. Lesselier, Reconstruction of thin electromagnetic inclusions by a level set method, Inverse Probl., 25 (2009), 085010. https://doi.org/10.1088/0266-5611/25/8/085010 doi: 10.1088/0266-5611/25/8/085010
[62]	R. Potthast, On the convergence of a new {N}ewton-type method in inverse scattering, Inverse Probl., 17 (2001), 1419. https://doi.org/10.1088/0266-5611/17/5/312 doi: 10.1088/0266-5611/17/5/312
[63]	C. Ramananjaona, M. Lambert, D. Lesselier, Shape inversion from TM and TE real data by controlled evolution of level sets, Inverse Probl., 17 (2001), 1585–1595. https://doi.org/10.1088/0266-5611/17/6/303 doi: 10.1088/0266-5611/17/6/303
[64]	F. Santosa, A level-set approach for inverse problems involving obstacles, ESAIM: Control Optim. Calc. Var., 1 (1996), 17–33. https://doi.org/10.1051/cocv:1996101 doi: 10.1051/cocv:1996101
[65]	J. D. Shea, P. Kosmas, S. C. Hagness, B. D. V. Veen, Three-dimensional microwave imaging of realistic numerical breast phantoms via a multiple-frequency inverse scattering technique, Med. Phys., 37 (2010), 4210–4226. https://doi.org/10.1118/1.3443569 doi: 10.1118/1.3443569
[66]	S. H. Son, W. K. Park, Application of the bifocusing method in microwave imaging without background information, J. Korean Soc. Ind. Appl. Math., 27 (2023), 109–122. https://doi.org/10.12941/jksiam.2023.27.109 doi: 10.12941/jksiam.2023.27.109
[67]	M. Testorf, M. Fiddy, Imaging from real scattered field data using a linear spectral estimation technique, Inverse Probl., 17 (2001), 1645. https://doi.org/10.1088/0266-5611/17/6/308 doi: 10.1088/0266-5611/17/6/308
[68]	A. G. Tijhuis, K. Belkebir, A. Litman, B. P. de Hon, Multiple-frequency distorted-wave Born approach to 2D inverse profiling, Inverse Probl., 17 (2001), 1635. https://doi.org/10.1088/0266-5611/17/6/307 doi: 10.1088/0266-5611/17/6/307
[69]	Y. Zhong, X. Chen, Twofold subspace-based optimization method for solving inverse scattering problems, Inverse Probl., 25 (2009), 085003. https://doi.org/10.1088/0266-5611/25/8/085003 doi: 10.1088/0266-5611/25/8/085003

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)