Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator

Ekram E. Ali; Georgia Irina Oros; Rabha M. El-Ashwah; Abeer M. Albalahi; Ekram E. Ali; Georgia Irina Oros; Rabha M. El-Ashwah; Abeer M. Albalahi

doi:10.3934/math.20241031

AIMS Mathematics

2024, Volume 9, Issue 8: 21239-21254. doi: 10.3934/math.20241031

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Research article

Applications of fuzzy differential subordination theory on analytic $p$ -valent functions connected with $\mathfrak{q}$ -calculus operator

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
2.
Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt
3.
University of Oradea, Department of Mathematics, Universitatii 1, 410087 Oradea, Romania
4.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt

Received: 27 April 2024 Revised: 07 June 2024 Accepted: 18 June 2024 Published: 01 July 2024
MSC : 30C45, 30C80

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of

$p$ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy

$p$ -valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a

$\mathfrak{q}$ -calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized

$\mathfrak{q}$ -calculus operator, defined here and having the

$(p, \mathfrak{q)}$ -Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.

Keywords:

Citation: Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah, Abeer M. Albalahi. Applications of fuzzy differential subordination theory on analytic $p$ -valent functions connected with $\mathfrak{q}$ -calculus operator[J]. AIMS Mathematics, 2024, 9(8): 21239-21254. doi: 10.3934/math.20241031

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Abstract

$p$ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy

$p$ -valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a

$\mathfrak{q}$ -calculus operator, defined here and having the

1. Introduction

In recent years, fractional differential equations have gained prominence due to their proven usefulness in several unrelated scientific and engineering fields. For example, the nonlinear oscillations of an earthquake can be characterized by a fractional derivative, and the fractional derivative of the traffic fluid dynamics model can solve the insufficiency resulting from the assumption of continuous traffic flows ^[1,3]. Numerous chemical processes, mathematical biology, engineering, and scientific problems ^[4,5,6,7] are also modeled with fractional differential equations. Nonlinear partial differential equations (NPDEs) characterize various physical, biological, and chemical phenomena. Current research is focused on developing precise traveling wave solutions for such equations. Exact and explicit solutions help scientists understand the complicated physical phenomena and dynamic processes portrayed by NPDEs ^[8,9,10]. In the past four decades, numerous essential methodologies for attaining accurate solutions to NPDEs have been proposed ^[11,12].

The Helmholtz equation (HE) derives from the elliptic and wave equations. In a multi-dimensional nonhomogeneous isotropic standard with velocity c, the wave result is $\upsilon({\xi}, \psi)$ , which corresponds to a source of harmonic $({\xi}, \psi)$ vibrating at a given frequency and satisfying the Helmholtz equation in the area R. The classical order HE is

$\begin{eqnarray} D^{\mathrm{2}}_{\xi}{\upsilon}({\xi},{\psi})+D^{\mathrm{2}}_{\psi}{\upsilon}({\xi},\psi)+\varepsilon {\upsilon}({\xi},\psi) = -\upsilon ({\xi},\psi). \end{eqnarray}$

(1.1)

Here, ${\upsilon}$ is a suitable boundary differentiable term of R, is a known function, and the wave number with wavelength $2/\xi = 0$ renders Eq (1.1) homogeneous. If (1.1) is expressed as

$\begin{eqnarray} D^{\mathrm{2}}_{\xi}{\upsilon}({\xi},\psi)+D^{\mathrm{2}}_\psi{\upsilon}({\xi},\psi)-\varepsilon {\upsilon}({\xi},\psi) = -\upsilon ({\xi},\psi). \end{eqnarray}$

Then it explains mass transfer with density biochemical processes of the 1st order. Equation (1.1) is investigated using the decomposition method ^[13], the finite element approach ^[14], the differential transform method ^[15], the Trefftz method ^[16], and the spectral collocation method ^[17], among others ^[3,4,5].

The Helmholtz equation is a partial differential equation that describes wave phenomena in various fields of physics, such as electromagnetism, acoustics, and fluid mechanics. Traditionally, the Helmholtz equation has been formulated using integer-order derivatives. However, in recent years, there has been a growing interest in the use of fractional-order derivatives to describe complex phenomena more accurately. In particular, fractional-order space Helmholtz equations are derived directly from mathematical formulas that involve fractional derivatives, rather than being generalized from integer-order space derivative Helmholtz equations. These equations can provide a more accurate description of wave propagation in complex media, such as porous materials, biological tissues, and fractal structures. Fractional-order space Helmholtz equations have attracted significant attention due to their potential applications in a wide range of fields, including medical imaging, geophysics, and telecommunications. They offer a promising avenue for understanding the behavior of waves in complex media and developing new technologies for wave-based sensing and imaging ^[6,7,8].

It is advantageous to utilize fractional differential equations in physical problems due to their nonlocal features. Non-locality characterizes fractional-order derivatives, whereas locality characterizes integer-order derivatives ^{[24,25,26,27]}. It demonstrates that the future state of the physical system depends on all of its previous states in addition to its current state. Consequently, fractional models are more accurate. In fractional differential equations, the response expression has a parameter that specifies the fractional derivative of the variable order, which may vary to achieve many responses ^[9,10,11].

Standard HEs can be generalized to fractional-order Helmholtz equations by extending the Caputo fractional-order space derivative to the integer-order space derivative. The fractional Helmholtz equation in space is

$\begin{eqnarray} D^{\varrho }_{\xi}{\upsilon}({\xi},\psi)+D^{\mathrm{2}}_\psi{\upsilon}({\xi},\psi)+\varepsilon {\upsilon}({\xi},\psi) = -\psi({\xi},\psi), \end{eqnarray}$

with ${\upsilon}(0, \psi) = g (\psi)$ as the initial condition (IC). Gupta et al. ^[31] solved the multi-dimensional fractional Helmholtz equation using the homotopy perturbation approach. In contrast, Abuasad et al. ^[14] recently solved a fractional model of the Helmholtz problem using the reduced differential transform method.

2. Preliminary concepts

This section describes the properties of the fractional derivatives and a few essential details concerning the Yang transform.

Definition 2.1. The fractional derivative in terms of Caputo is as follows

$\begin{equation} \ \begin{split} &D_{\psi}^{\varrho}{\upsilon}(\xi,\psi) = \frac{1}{\Gamma(k-\varrho)}\int_{0}^{\psi}(\psi-\varrho)^{k-\varrho-1}{\upsilon}^{(k)}(\xi,\varrho)d\varrho,\quad k-1 < \varrho\leq k,\quad k\in N. \end{split} \end{equation}$

(2.1)

Definition 2.2. The YT is represented as follows

$\begin{equation} Y\{{\upsilon}(\psi)\} = M(u) = \int_0^{\infty}e^{\frac{-\psi}{u}}{\upsilon}(\psi)d\psi,\ \ \psi > 0, \ \ u\in(-\psi_1,\psi_2), \end{equation}$

(2.2)

having inverse YT as follows

$\begin{equation} Y^{-1}\{M(u)\} = {\upsilon}(\psi). \end{equation}$

(2.3)

Definition 2.3. The nth derivative YT is stated as follows

$\begin{equation} Y\{{\upsilon}^{n}(\psi)\} = \frac{M(u)}{u^{n}}-\sum\limits_{k = 0}^{n-1}\frac{{\upsilon}^{k}(0)}{u^{n-k-1}}, \ \ \forall \ \ n = 1,2,3,\cdots \end{equation}$

(2.4)

Definition 2.4. The YT of derivative having fractional-order is stated as follows

$\begin{equation} Y\{{\upsilon}^{\varrho}(\psi)\} = \frac{M(u)}{u^{\varrho}}-\sum\limits_{k = 0}^{n-1}\frac{{\upsilon}^{k}(0)}{u^{\varrho-(k+1)}}, \ \ 0 < \varrho \leq n. \end{equation}$

(2.5)

3. General implementation of the method

Consider the general fractional partial differential equations,

$\begin{equation} \begin{split} &D_{{{\psi}}}^{\varrho}{{\upsilon}}({\xi},{\psi})+M{{\upsilon}}({\xi},{\psi})+N{{\upsilon}}({\xi},{\psi}) = h({\xi},{\psi}),\quad {\psi} > 0,\quad 0 < \varrho\leq1,\\ &{{\upsilon}}({\xi},0) = g(\xi),\quad\nu\in \Re.\\ \end{split} \end{equation}$

(3.1)

Using Yang transform of Eq (3.1), we get

$\begin{equation} \ \begin{split} &{Y}[D_{{y}}^{\varrho}{{\upsilon}}({\xi},{\psi})+M{{\upsilon}}({\xi},{\psi})+N{{\upsilon}}({\xi},{\psi})] = {Y}[h({\xi},{\psi})], \quad {\psi} > 0,0 < \varrho\leq1,\\ &{\upsilon}({\xi},{\psi}) = {s}{g(\xi)}+{s^{\varrho}}{Y}[h({\xi},{\psi})]-{s^{\varrho}}{Y}[M{{\upsilon}}({\xi},{\psi})+N{{\upsilon}}({\xi},{\psi})].\\ \end{split} \end{equation}$

(3.2)

Now, applying inverse Yang transform, we have

$\begin{equation} {{\upsilon}}({\xi},{\psi}) = F({\xi},{\psi})-{Y}^{-1}\left[{s^{\varrho}}{Y}\{M{{\upsilon}}({\xi},{\psi})+N{{\upsilon}}({\xi},{\psi})\}\right], \end{equation}$

(3.3)

where

$\begin{equation} \ F({\xi},{\psi}) = {Y}^{-1}\left[{s}{g(\xi)}+{s^{\varrho}}{Y}[h({\xi},{\psi})]\right] = g(\nu)+{Y}^{-1}\left[{s^{\varrho}}{Y}[h({\xi},{\psi})]\right]. \end{equation}$

(3.4)

The parameter $p$ is perturbation technique and $p\in[0, 1]$ defined as

$\begin{equation} {{\upsilon}}({\xi},{\psi}) = \sum\limits_{k = 0}^{\infty}p^{k}{{\upsilon}}_{k}({\xi},{\psi}), \end{equation}$

(3.5)

The nonlinear function is expressed as

$\begin{equation} N{{\upsilon}}({\xi},{\psi}) = \sum\limits_{k = 0}^{\infty}p^{k}H_{k}({{\upsilon}}_k), \end{equation}$

(3.6)

where $H_{n}$ are He's polynomials in term of ${{\upsilon}}_{0}, {{\upsilon}}_{1}, {{\upsilon}}_{2}, \cdots, {{\upsilon}}_{n},$ and can be calculated as

$\begin{equation} H_{n}({{\upsilon}}_{0},{{\upsilon}}_{1},\cdots,{{\upsilon}}_{n}) = \frac{1}{\varrho(n+1)}D_{p}^{\imath}\left[{N}\left(\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}\right)\right]_{p = 0}, \end{equation}$

(3.7)

where $D_{p}^{\imath} = \frac{\partial^{\imath}}{\partial p^{\imath}}.$

Putting Eqs (3.6) and (3.7) in Eq (3.3), we achieved as

$\begin{equation} \ \sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi}) = F({\xi},{\psi})-p\times\left[{Y}^{-1}\left\{{s^{\varrho}}{Y}\{M\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})+ \sum\limits_{\imath = 0}^{\infty}p^{\imath}H_{\imath}({{\upsilon}}_\imath)\}\right\}\right]. \end{equation}$

(3.8)

Comparison both sides of coefficient $p$ , we get

$\begin{equation} \begin{split} &p^{0}:{{\upsilon}}_{0}({\xi},{\psi}) = F({\xi},{\psi}),\\ &p^{1}:{{\upsilon}}_{1}({\xi},{\psi}) = {Y}^{-1}\left[{s^{\varrho}}{Y}(M{{\upsilon}}_{0}({\xi},{\psi})+H_{0}({{\upsilon}}))\right],\\ &p^{2}:{{\upsilon}}_{2}({\xi},{\psi}) = {Y}^{-1}\left[{s^{\varrho}}{Y}(M{{\upsilon}}_{1}({\xi},{\psi})+H_{1}({{\upsilon}}))\right],\\ &\vdots\\ &p^{\imath}:{{\upsilon}}_{\imath}({\xi},{\psi}) = {Y}^{-1}\left[{s^{\varrho}}{Y}(M{{\upsilon}}_{\imath-1}({\xi},{\psi})+H_{\imath-1}({{\upsilon}}))\right],\quad \imath > 0,\imath\in N. \end{split} \end{equation}$

(3.9)

Finally, present the obtained solution and check it with any available analytical or numerical solutions for the given PDE. The ${{\upsilon}}_{\imath}({\xi}, {\psi})$ components can be calculated easily which quickily converges to series form. We can get $p\rightarrow1,$

$\begin{equation} \ {{\upsilon}}({\xi},{\psi}) = \lim\limits_{M\rightarrow \infty}\sum\limits_{\imath = 1}^{M}{{\upsilon}}_{\imath}({\xi},{\psi}). \end{equation}$

(3.10)

4. Applications

Problem 4.1. Consider the space fractional Helmholtz equation

$\begin{equation} \begin{split} \frac{\partial^\varrho{{\upsilon}({\xi},{\psi})}}{\partial{{\xi}^\varrho}}+\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-{{\upsilon}({\xi},{\psi})} = 0,\qquad 1 < \varrho\leq2, \end{split} \end{equation}$

(4.1)

with the ICs

$\begin{equation} \ {\upsilon}(0,{\psi}) = {\psi} \ \ and \quad {\upsilon}_{\xi}(0,{\psi}) = 0. \end{equation}$

(4.2)

Using the Yang transform of Eq (4.1), we obtained as

$\begin{equation} \ \begin{split} &\frac{1}{s^{\varrho}}{Y}[{{\upsilon}}({\xi},{\psi})] = {{\upsilon}}({0},{\psi}){s^{1-\varrho}}-{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-{{\upsilon}({\xi},{\psi})}\right\},\\ \end{split} \end{equation}$

(4.3)

$\begin{equation} \ \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = {{s}}{{\upsilon}}({0},{\psi})-s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-{{\upsilon}({\xi},{\psi})}\right\},\\ \end{split} \end{equation}$

(4.4)

Taking inverse Yang Transformation, we have

$\begin{equation} \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = {\psi}-{Y}^{-1}\left[s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-{{\upsilon}({\xi},{\psi})}\right\}\right],\\ \end{split} \end{equation}$

(4.5)

Implemented HPM in Eq (4.5), we can achieve as

$\begin{equation} \ \begin{split} \sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi}) = &{\psi}-p\left[{Y}^{-1}\left\{{s^{\varrho}}{Y}\left\{\left(\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right)_{{\psi}{\psi}}-\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right\}\right\}\right].\\ \end{split} \end{equation}$

(4.6)

On both sides comparing coefficients of $p$ , we get

$\begin{equation} \ \begin{split} &p^{0}:{{\upsilon}}_{0}({\xi},{\psi}) = {\psi},\\ &p^{1}:{{\upsilon}}_{1}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_0({\xi},{\psi})}{\partial {\psi}^2}-{{\upsilon}}_{0}({\xi},{\psi})\right\}\right] = \frac{{\xi}^{\varrho}}{\Gamma(\varrho+1)}{\psi},\\ &p^{2}:{{\upsilon}}_{2}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_1({\xi},{\psi})}{\partial {\psi}^2}-{{\upsilon}}_{1}({\xi},{\psi})\right\}\right] = \frac{{\xi}^{2\varrho}}{\Gamma(2\varrho+1)}{\psi},\\ &p^{3}:{{\upsilon}}_{3}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_2({\xi},{\psi})}{\partial {\psi}^2}-{{\upsilon}}_{2}({\xi},{\psi})\right\}\right] = \frac{{\xi}^{3\varrho}}{\Gamma(3\varrho+1)}{\psi},\\ &p^{4}:{{\upsilon}}_{4}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_3({\xi},{\psi})}{\partial {\psi}^2}-{{\upsilon}}_{3}({\xi},{\psi})\right\}\right] = \frac{{\xi}^{4\varrho}}{\Gamma(4\varrho+1)}{\psi},\\ &\vdots\\ \end{split} \end{equation}$

(4.7)

The series type result of the first problem example is

$\begin{equation} \ \begin{split} &{{\upsilon}}({\xi},{\psi}) = {{\upsilon}}_{0}({\xi},{\psi})+{{\upsilon}}_{1}({\xi},{\psi})+{{\upsilon}}_{2}({\xi},{\psi})+{{\upsilon}}_{3}({\xi},{\psi})+{{\upsilon}}_{4}({\xi},{\psi})+\cdots\\ &{\upsilon}({\xi},{\psi}) = {\psi}\left[1+\frac{{\xi}^\varrho}{\Gamma(\varrho+1)}+\frac{{\xi}^{2\varrho}}{\Gamma(2\varrho+1)}+\frac{x ^{3\varrho}}{\Gamma(3\varrho+1)}+\frac{{\xi}^{4\varrho}}{\Gamma(4\varrho+1)}+\cdots\right]. \end{split} \end{equation}$

(4.8)

The exact solution is

${\upsilon}({\xi},{\psi}) = {\psi}\cosh{\xi}.$

Similarly $y$ -space can be calculated as:

$\begin{equation} \frac{\partial^\varrho{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^\varrho}}+\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\xi}^2}}-{{\upsilon}({\xi},{\psi})} = 0, \end{equation}$

(4.9)

with the IC

$\begin{equation} {\upsilon}({\xi},0) = {\xi}. \end{equation}$

(4.10)

Thus, the solution of the above Eq (4.9) is obtain as

${\upsilon}({\xi},{\psi}) = {\xi}\left(1+\frac{{\psi}^\varrho}{\Gamma(\varrho+1)}+\frac{{\psi}^{2\varrho}}{\Gamma(2\varrho+1)}+\frac{{\psi} ^{3\varrho}}{\Gamma(3\varrho+1)}+\frac{{\psi}^{4\varrho}}{\Gamma(4\varrho+1)}+\dots \right),$

in the case when $\varrho = 2$ , then the solution through HPTM is

$\begin{equation} {\upsilon}({\xi},{\psi}) = \xi\cosh{\psi}. \end{equation}$

(4.11)

illustrates the exact and HPTM solutions in two-dimensional plots for various values of $\varrho$ ranging from 2 to 1.5, with ${\xi}$ values ranging from 0 to 1 and ${\psi}$ set to 1. The solutions are presented in Figure 1a and for the exact and HPTM methods, respectively. displays the 3-dimensional plots of the exact and HPTM solutions for $\varrho = 2$ and analyzes the point of intersection between the two solutions. and depict the HPTM solutions at $\varrho = 1.8$ and $1.6$ respectively for Problem 4.1. The fractional results were also evaluated for their convergence towards an integer-order result for each problem. Similarly, the figures for the ${\psi}$ -space can also be generated using the same approach.

Figure 1. The first graph show that the exact and approximate solution and second HPTM solution at the different fractional-order graph of Problem 4.1.

DownLoad: Full-Size Img PowerPoint

Figure 2. Exact and proposed method solution at various fractional orders of Problem 4.1.

DownLoad: Full-Size Img PowerPoint

Table 1. Exact and proposed method solution of Problem 4.1 at various fractional orders.

$({\xi},{\psi})$	${\upsilon}({\xi},{\psi})$ at ${\varrho }=$ 1.5	${\upsilon}({\xi},{\psi})$ at ${\varrho }=$ 1.75	$(HPTM)$ at ${\varrho }=$ 2	Exact result
(0.2, 0.1)	0.2102371	0.2100072	0.2100000	0.2100000
(0.4, 0.1)	0.4104630	0.4100141	0.4100000	0.4100000
(0.6, 0.1)	0.6106889	0.6100209	0.6100000	0.6100000
(0.2, 0.2)	0.2103355	0.2100121	0.2100000	0.2100000
(0.4, 0.2)	0.4106550	0.4100237	0.4100000	0.4100000
(0.6, 0.2)	0.6109746	0.6100353	0.6100000	0.6100000
(0.2, 0.3)	0.2104110	0.2100164	0.2100000	0.2100000
(0.4, 0.3)	0.4108025	0.4100321	0.4100000	0.4100000
(0.6, 0.3)	0.6111940	0.6100478	0.6100000	0.6100000
(0.2, 0.4)	0.2104747	0.2100204	0.2100000	0.2100000
(0.4, 0.4)	0.4109269	0.4100399	0.4100000	0.4100000
(0.6, 0.4)	0.6113790	0.6100593	0.6100000	0.6100000
(0.2, 0.5)	0.2105309	0.2100241	0.2100000	0.2100000
(0.4, 0.5)	0.4110365	0.4100471	0.4100000	0.4100000
(0.6, 0.5)	0.6115421	0.6100701	0.6100000	0.6100000

| Show Table

DownLoad: CSV

Problem 4.2. Consider the space-fractional HE

$\begin{equation} \begin{split} \frac{\partial^\varrho{{\upsilon}({\xi},{\psi})}}{\partial{{\xi}^\varrho}}+\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}+5{{\upsilon}({\xi},{\psi})} = 0,\qquad 1 < \varrho\leq2, \end{split} \end{equation}$

(4.12)

with the ICs

$\begin{equation} \ {\upsilon}(0,{\psi}) = {\psi} \ \ and\quad{\upsilon}_{\xi}(0,{\psi}) = 0. \end{equation}$

(4.13)

Using the Yang transform of Eq (4.12), we obtain as

$\begin{equation} \ \begin{split} &\frac{1}{s^{\varrho}}{Y}[{{\upsilon}}({\xi},{\psi})] = {{\upsilon}}({0},{\psi}){s^{1-\varrho}}-{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}+5{{\upsilon}({\xi},{\psi})}\right\},\\ \end{split} \end{equation}$

(4.14)

$\begin{equation} \ \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = {{s}}{{\upsilon}}({0},{\psi})-s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}+5{{\upsilon}({\xi},{\psi})}\right\}.\\ \end{split} \end{equation}$

(4.15)

Applying the inverse Yang Transform, we get

$\begin{equation} \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = {\psi}-{Y}^{-1}\left[s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}+5{{\upsilon}({\xi},{\psi})}\right\}\right],\\ \end{split} \end{equation}$

(4.16)

Using the HPM in Eq (4.16), we obtained as

$\begin{equation} \ \begin{split} \sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi}) = &{\psi}-p\left[{Y}^{-1}\left\{{s^{\varrho}}{Y}\left\{\left(\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right)_{{\psi}{\psi}}+5\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right\}\right\}\right].\\ \end{split} \end{equation}$

(4.17)

On both sides comparing coefficients of $p$ , we get

$\begin{equation} \ \begin{split} &p^{0}:{{\upsilon}}_{0}({\xi},{\psi}) = {\psi},\\ &p^{1}:{{\upsilon}}_{1}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_0({\xi},{\psi})}{\partial {\psi}^2}+5{{\upsilon}}_{0}({\xi},{\psi})\right\}\right] = -5{\psi}\frac{{\xi}^{\varrho}}{\Gamma(\varrho+1)},\\ &p^{2}:{{\upsilon}}_{2}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_1({\xi},{\psi})}{\partial {\psi}^2}+5{{\upsilon}}_{1}({\xi},{\psi})\right\}\right] = 25{\psi}\frac{{\xi}^{2\varrho}}{\Gamma(2\varrho+1)},\\ &p^{3}:{{\upsilon}}_{3}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_2({\xi},{\psi})}{\partial {\psi}^2}+5{{\upsilon}}_{2}({\xi},{\psi})\right\}\right] = -125\frac{{\xi}^{3\varrho}}{\Gamma(3\varrho+1)},\\ &p^{4}:{{\upsilon}}_{4}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_3({\xi},{\psi})}{\partial {\psi}^2}+5{{\upsilon}}_{3}({\xi},{\psi})\right\}\right] = 625{\psi}\frac{{\xi}^{4\varrho}}{\Gamma(4\varrho+1)},\\ &\vdots\\ \end{split} \end{equation}$

(4.18)

The series type result of second problem as

$\begin{equation} \ \begin{split} &{{\upsilon}}({\xi},{\psi}) = {{\upsilon}}_{0}({\xi},{\psi})+{{\upsilon}}_{1}({\xi},{\psi})+{{\upsilon}}_{2}({\xi},{\psi})+{{\upsilon}}_{3}({\xi},{\psi})+{{\upsilon}}_{4}({\xi},{\psi})+\cdots\\ &{\upsilon}({\xi},{\psi}) = {\psi}\left[1-\frac{5{\xi}^\varrho}{\Gamma(\varrho+1)}+\frac{25{\xi}^{2\varrho}}{\Gamma(2\varrho+1)}-\frac{125x ^{3\varrho}}{\Gamma(3\varrho+1)}+\frac{625{\xi}^{4\varrho}}{\Gamma(4\varrho+1)}+\cdots\right]. \end{split} \end{equation}$

(4.19)

The exact solution is

${\upsilon}({\xi},{\psi}) = {\psi}\cos\sqrt{5}{\xi}.$

Now similarly, the result of $y$ -space can be calculated with the help of homotopy perturbation

(4.20)

with the IC

$\begin{equation} {\upsilon}({\xi},0) = {\xi}. \end{equation}$

(4.21)

The solution of the Eq (4.20) is expressed as

$\begin{eqnarray} {\upsilon}({\xi},{\psi}) = {\xi}\left(1-\frac{5{\psi}^\varrho}{\Gamma(\varrho+1)}+\frac{25{\psi}^{2\varrho}}{\Gamma(2\varrho+1)}-\frac{125{\psi}^{3\varrho}}{\Gamma(3\varrho+1)}+\frac{625{\psi}^{4\varrho}}{\Gamma(4\varrho+1)}+\cdots \right). \end{eqnarray}$

The exact solution is

$\begin{equation} {\upsilon}({\xi},{\psi}) = {\xi}\cos\sqrt{5}{\psi}. \end{equation}$

(4.22)

Figure 3 illustrates the solutions of exact and HPTM in two-dimensional plots, as shown in and for different values of $\varrho$ , ranging from 2 to 1.5, respectively. The interval considered for ${\xi}$ is [0, 1], while ${\psi}$ is constant at 1. The results obtained from the fractional-order model converge to the integer-order solution of the problem. In , the 3-dimensional plots of exact and HPTM solutions are presented in Figures (a) and (b), respectively, for $\varrho = 2$ . The closed contact of the two solutions is analyzed. Additionally, and depict the HPTM solutions at $\varrho = 1.8$ and $1.6$ , respectively, for Problem 4.2. Similarly, graphs for $\psi$ -space can also be generated.

Figure 3. Exact and proposed method solution at various fractional orders of Problem 4.2.

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Figure 4. Exact and proposed method solution at various fractional orders of Problem 4.2.

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Problem 4.3. Consider the space-fractional HE

$\begin{equation} \begin{split} \frac{\partial^\varrho{{\upsilon}({\xi},{\psi})}}{\partial{{\xi}^\varrho}}+\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-2{{\upsilon}({\xi},{\psi})} = {(12{\xi}^2-3{\xi}^4)}{\sin{\psi}},\quad 1 < \varrho\leq2,\quad0\leq {\psi} \leq 2\pi, \end{split} \end{equation}$

(4.23)

with the ICs

$\begin{equation} \ {\upsilon}(0,{\psi}) = 0 \ \ and \quad {\upsilon}_{\xi}(0,{\psi}) = 0. \end{equation}$

(4.24)

Applying the Yang transform of Eq (4.23), we achieve

$\begin{equation} \ \begin{split} &\frac{1}{s^{\varrho}}{Y}[{{\upsilon}}({\xi},{\psi})] = {{\upsilon}}({0},{\psi}){s^{1-\varrho}}-{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-2{{\upsilon}({\xi},{\psi})}\right\},\\ \end{split} \end{equation}$

(4.25)

$\begin{equation} \ \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = {{s}}{{\upsilon}}({0},{\psi})-s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-2{{\upsilon}({\xi},{\psi})}\right\}.\\ \end{split} \end{equation}$

(4.26)

Implementing inverse Yang transform, we get

$\begin{equation} \begin{split} &{Y}[{{\upsilon}}({\xi},{\psi})] = \left({\xi}^4-\frac{{\xi}^6}{10}\right)\sin{\psi}-{Y}^{-1}\left[s^{\varrho}{Y}\left\{\frac{\partial^2{{\upsilon}({\xi},{\psi})}}{\partial{{\psi}^2}}-2{{\upsilon}({\xi},{\psi})}\right\}\right].\\ \end{split} \end{equation}$

(4.27)

Applying Homotopy perturbation method in Eq (4.27), we achieved as

$\begin{equation} \ \begin{split} \sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi}) = &{\psi}-p\left[{Y}^{-1}\left\{{s^{\varrho}}{Y}\left\{\left(\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right)_{{\psi}{\psi}}-2\sum\limits_{\imath = 0}^{\infty}p^{\imath}{{\upsilon}}_{\imath}({\xi},{\psi})\right\}\right\}\right].\\ \end{split} \end{equation}$

(4.28)

Both sides on comparison coefficients of $p$ , we obtain

$\begin{equation} \ \begin{split} &p^{0}:{{\upsilon}}_{0}({\xi},{\psi}) = \left({\xi}^4-\frac{{\xi}^6}{10}\right)\sin{\psi},\\ &p^{1}:{{\upsilon}}_{1}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_0({\xi},{\psi})}{\partial {\psi}^2}-2{{\upsilon}}_{0}({\xi},{\psi})\right\}\right] = 3\left(\frac{{\xi}^{\varrho+4}}{\Gamma(\varrho+5)}-\frac{72{\xi}^{\varrho+6}}{\Gamma(\varrho+7)}\right)\sin{\psi},\\ &p^{2}:{{\upsilon}}_{2}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_1({\xi},{\psi})}{\partial {\psi}^2}-2{{\upsilon}}_{1}({\xi},{\psi})\right\}\right] = 3\left(\frac{{\xi}^{2\varrho+4}}{\Gamma(2\varrho+5)}-\frac{216{\xi}^{2\varrho+6}}{\Gamma(2\varrho+7)}\right)\sin{\psi},\\ &p^{3}:{{\upsilon}}_{3}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_2({\xi},{\psi})}{\partial {\psi}^2}-2{{\upsilon}}_{2}({\xi},{\psi})\right\}\right] = 3\left(\frac{{\xi}^{3\varrho+4}}{\Gamma(3\varrho+5)}-\frac{648{\xi}^{3\varrho+6}}{\Gamma(3\varrho+7)}\right)\sin{\psi},\\ &p^{4}:{{\upsilon}}_{4}({\xi},{\psi}) = -{Y}^{-1}\left[{s^{\varrho}}{Y}\left\{\frac{\partial^2{\upsilon}_3({\xi},{\psi})}{\partial {\psi}^2}-2{{\upsilon}}_{3}({\xi},{\psi})\right\}\right] = 3\left(\frac{{\xi}^{4\varrho+4}}{\Gamma(4\varrho+5)}-\frac{1944{\xi}^{2\varrho+6}}{\Gamma(2\varrho+7)}\right)\sin{\psi},\\ &\vdots\\ \end{split} \end{equation}$

(4.29)

The series type result of the third problem is

$\begin{equation} \ \begin{split} {{\upsilon}}({\xi},{\psi}) = &{{\upsilon}}_{0}({\xi},{\psi})+{{\upsilon}}_{1}({\xi},{\psi})+{{\upsilon}}_{2}({\xi},{\psi})+{{\upsilon}}_{3}({\xi},{\psi})+{{\upsilon}}_{4}({\xi},{\psi})+\cdots\\ {\upsilon}({\xi},{\psi}) = &\left({\xi}^4-\frac{{\xi}^6}{10}\right)\sin{\psi}+3\left(\frac{{\xi}^{\varrho+4}}{\Gamma(\varrho+5)}-\frac{72{\xi}^{\varrho+6}}{\Gamma(\varrho+7)}\right)\sin{\psi}+3\left(\frac{{\xi}^{2\varrho+4}}{\Gamma(2\varrho+5)}-\frac{216{\xi}^{2\varrho+6}}{\Gamma(2\varrho+7)}\right)\sin{\psi}\\ &+3\left(\frac{{\xi}^{3\gamma+4}}{\Gamma(3\varrho+5)}-\frac{648{\xi}^{3\varrho+6}}{\Gamma(3\varrho+7)}\right)\sin{\psi}+3\left(\frac{{\xi}^{4\varrho+4}}{\Gamma(4\varrho+5)}-\frac{1944{\xi}^{2\varrho+6}}{\Gamma(2\varrho+7)}\right)\sin{\psi}+\cdots. \end{split} \end{equation}$

(4.30)

The exact solution is

${\upsilon}({\xi},{\psi}) = {\xi}^4\sin{\psi}.$

and display the exact and HPTM solutions, respectively, in a 3-dimensional plot at $\varrho = 2$ . The closed contact between the exact and HPTM solutions is examined. depicts the exact and HPTM solutions in two-dimensional plot for various values of $\varrho = 2, 1.9, 1.8, 1.7, 1.6, 1.5$ for ${\xi}\in[0, 1]$ and ${\psi} = 1$ . The fractional results are observed to approach an integer-order solution of the problem. Similarly, the graphs for ${\psi}$ -space fractional-order derivative can also be plotted.

Figure 5. Exact and proposed method solution at various fractional orders of Problem 4.3.

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Figure 6. Exact and proposed method solution at various fractional orders of Problem 4.3.

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5. Conclusions

In this study, fractional-order Helmholtz equations were solved using the Homotopy Perturbation Yang transform method. Due to the great agreement between the generated approximative solution and the precise solution, the homotopy perturbation Yang transform method was demonstrated to be a successful method for solving partial differential equations with Caputo operators. The computation size of the approach was compared to those required by other numerical methods to demonstrate how tiny it is. Additionally, the procedure's quick convergence demonstrates its dependability and marks a notable advancement in the way linear and non-linear fractional-order partial differential equations are solved.

Acknowledgements

This research has been funded by Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha'il–Saudi Arabia through project number IFP-22 064.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
[2]	G. I. Oros, G. Oros, The notion of subordination in fuzzy sets theory, General Mathematics, 19 (2011), 97–103.
[3]	S. S. Miller, P. T. Mocanu, Second order-differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 289–305. https://doi.org/10.1016/0022-247X(78)90181-6 doi: 10.1016/0022-247X(78)90181-6
[4]	S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28 (1981), 157–172. https://doi.org/10.1307/mmj/1029002507 doi: 10.1307/mmj/1029002507
[5]	G. I. Oros, G. Oros, Fuzzy differential subordination, Acta Universitatis Apulensis, 30 (2012), 55–64.
[6]	I. Dzitac, F. G. Filip, M. J. Manolescu, Fuzzy logic is not fuzzy: world-renowned computer scientist Lotfi A. Zadeh, Int. J. Comput. Commun., 12 (2017), 748–789. https://doi.org/10.15837/ijccc.2017.6.3111 doi: 10.15837/ijccc.2017.6.3111
[7]	G. I. Oros, G. Oros, Dominants and best dominants in fuzzy differential subordinations, Stud. Univ. Babes-Bolyai Math., 57 (2012), 239–248.
[8]	E. E. Ali, M. Vivas-Cortez, R. M. El-Ashwah, New results about fuzzy $\gamma$ -convex functions connected with the $\mathfrak{q}$ -analogue multiplier-Noor integral operator, AIMS Mathematics, 9 (2024), 5451–5465. https://doi.org/10.3934/math.2024263 doi: 10.3934/math.2024263
[9]	E. E. Ali, M. Vivas-Cortez, R. M. El-Ashwah, A. M. Albalahi, Fuzzy subordination results for meromorphic functions connected with a linear operator, Fractal Fract., 8 (2024), 308. https://doi.org/10.3390/fractalfract8060308 doi: 10.3390/fractalfract8060308
[10]	G. I. Oros, Briot-Bouquet fuzzy differential subordination, Analele Universitatii Oradea Fasc. Matematica, 2 (2012), 83–97.
[11]	F. H. Jackson, On $\mathfrak{q}$ -functions and a certain difference operator, Earth Env. Sci. T. R. So., 46 (1909), 253–281. https://doi.org/10.1017/S0080456800002751 doi: 10.1017/S0080456800002751
[12]	F. H. Jackson, On $\mathfrak{q}$ -definite integrals, The Quarterly Journal of Pure and Applied Mathematics, 41 (1910), 193–203.
[13]	R. D. Carmichael, The general theory of linear $\mathfrak{q}$ -difference equations, Am. J. Math., 34 (1912), 147–168.
[14]	T. E. Mason, On properties of the solution of linear $\mathfrak{q }$ -difference equations with entire function coefficients, Am. J. Math., 37 (1915), 439–444. https://doi.org/10.2307/2370216 doi: 10.2307/2370216
[15]	W. J. Trjitzinsky, Analytic theory of linear difference equations, Acta Math., 61 (1933), 1–38. https://doi.org/10.1007/BF02547785 doi: 10.1007/BF02547785
[16]	M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables, Theory and Application, 14 (1990), 77–84. https://doi.org/10.1080/17476939008814407 doi: 10.1080/17476939008814407
[17]	H. M. Srivastava, Operators of basic (or $\mathfrak{q}$ -) calculus and fractional $\mathfrak{q}$ -calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. Sci., 44 (2020), 327–344. https://doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[18]	H. M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex A., 22 (2021), 1501–1520.
[19]	H. M. Srivastava, An introductory overview of Bessel polynomials, the generalized Bessel polynomials and the $\mathfrak{q}$ -Bessel polynomials, Symmetry, 15 (2023), 822. https://doi.org/10.3390/sym15040822 doi: 10.3390/sym15040822
[20]	E. E. Ali, T. Bulboaca, Subclasses of multivalent analytic functions associated with a $\mathfrak{q}$ -difference operator, Mathematics, 8 (2020), 2184. https://doi.org/10.3390/math8122184 doi: 10.3390/math8122184
[21]	E. E. Ali, A. M. Lashin, A. M. Albalahi, Coefficient estimates for some classes of biunivalent function associated with Jackson $\mathfrak{q}$ -difference operator, J. Funct. Space., 2022 (2022), 2365918. https://doi.org/10.1155/2022/2365918 doi: 10.1155/2022/2365918
[22]	E. E. Ali, H. M. Srivastava, A. M. Y. Lashin, A. M. Albalahi, Applications of some subclasses of meromorphic functions associated with the $\mathfrak{q}$ -derivatives of the $\mathfrak{q}$ -binomials, Mathematics, 11 (2023), 2496. https://doi.org/10.1155/2022/2365918 doi: 10.1155/2022/2365918
[23]	E. E. Ali, H. M. Srivastava, A. M. Albalahi, Subclasses of $p$ -valent $k$ -uniformly convex and starlike functions defined by the $\mathfrak{ q}$ -derivative operator, Mathematics, 11 (2023), 2578. https://doi.org/10.3390/math11112578 doi: 10.3390/math11112578
[24]	E. E. Ali, G. I. Oros, S. A. Shah, A. M. Albalahi, Applications of $\mathfrak{q}$ -calculus multiplier operators and subordination for the study of particular analytic function subclasses, Mathematics, 11 (2023), 2705. https://doi.org/10.3390/math11122705 doi: 10.3390/math11122705
[25]	W. Y. Kota, R. M. El-Ashwah, Some application of subordination theorems associated with fractional $\mathfrak{q}$ -calculus operator, Math. Bohem., 148 (2023), 131–148. http://doi.org/10.21136/MB.2022.0047-21 doi: 10.21136/MB.2022.0047-21
[26]	B. Wang, R. Srivastava, J. L. Liu, A certain subclass of multivalent analytic functions defined by the $\mathfrak{q}$ -difference operator related to the Janowski functions, Mathematics, 9 (2021), 1706. https://doi.org/10.3390/math9141706 doi: 10.3390/math9141706
[27]	S. Kanas, D. Raducanu, Some classes of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183–1196. https://doi.org/10.2478/s12175-014-0268-9 doi: 10.2478/s12175-014-0268-9
[28]	K. I. Noor, S. Riaz, M. A. Noor, On $\mathfrak{q}$ -Bernardi integral operator, TWMS J. Pure Appl. Math., 8 (2017), 3–11.
[29]	M. K. Aouf, S. M. Madian, Inclusion and properties neighbourhood for certain $p$ -valent functions associated with complex order and $\mathfrak{q}$ - $p$ -valent Cătaş operator, J. Taibah Univ. Sci., 14 (2020), 1226–1232. https://doi.org/10.1080/16583655.2020.1812923 doi: 10.1080/16583655.2020.1812923
[30]	M. Arif, H. M. Srivastava, S. Umar, Some applications of a $\mathfrak{q}$ -analogue of the Ruscheweyh type operator for multivalent functions, RACSAM, 113 (2019), 1211–1221. https://doi.org/10.1007/s13398-018-0539-3 doi: 10.1007/s13398-018-0539-3
[31]	R. M. Goel, N. S. Sohi, A new criterion for $p$ -valent functions, P. Am. Math. Soc., 78 (1980), 353–357. https://doi.org/10.2307/2042324 doi: 10.2307/2042324
[32]	S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109–115. https://doi.org/10.2307/2039801 doi: 10.2307/2039801
[33]	K. I. Noor, M. Arif, On some applications of Ruscheweyh derivative, Comput. Math. Appl., 62 (2011), 4726–4732. https://doi.org/10.1016/j.camwa.2011.10.063 doi: 10.1016/j.camwa.2011.10.063
[34]	I. Aldawish, M. Darus, Starlikeness of $q$ -differential operator involving quantum calculus, Korean J. Math., 22 (2014), 699–709. https://doi.org/10.11568/kjm.2014.22.4.699 doi: 10.11568/kjm.2014.22.4.699
[35]	H. Aldweby, M. Darus, A subclass of harmonic univalent functions associated with $\mathfrak{q}$ -analogue of Dziok-Srivastava operator, ISRN Mathematical Analysis, 2013 (2013), 382312. https://doi.org/10.1155/2013/382312 doi: 10.1155/2013/382312
[36]	M. K. Aouf, R. M. El-Ashwah, Inclusion properties of certain subclass of analytic functions defined by multiplier transformations, Annales Universitatis Mariae Curie-Sklodowska Sectio A–Mathematica, 63 (2009), 29–38. https://doi.org/10.2478/v10062-009-0003-0 doi: 10.2478/v10062-009-0003-0
[37]	R. M. El-Ashwah, M. K. Aouf, Some properties of new integral operator, Acta Universitatis Apulensis, 24 (2010), 51–61.
[38]	T. B. Jung, Y. C. Kim, H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operator, J. Math. Anal. Appl., 176 (1993), 138–147. https://doi.org/10.1006/jmaa.1993.1204 doi: 10.1006/jmaa.1993.1204
[39]	G. S. Sălăgean, Subclasses of univalent functions, In: Complex analysis—Fifth Romanian-Finnish seminar, Berlin: Springer, 1983,362–372. https://doi.org/10.1007/BFb0066543
[40]	S. A. Shah, K. I. Noor, Study on $\mathfrak{q}$ -analogue of certain family of linear operators, Turk. J. Math., 43 (2019), 2707–2714. https://doi.org/10.3906/mat-1907-41 doi: 10.3906/mat-1907-41
[41]	H. M. Srivastava, A. A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination, Integr. Transf. Spec. F., 18 (2007), 207–216. https://doi.org/10.1080/10652460701208577 doi: 10.1080/10652460701208577
[42]	H. M. Srivastava, J. Choi, Series associated with the Zeta and related functions, Dordrecht: Springer, 2001.
[43]	S. G. Gal, A. I. Ban, Elemente de matematică fuzzy, Romania: Editura Universităţii din Oradea, 1996.
[44]	S. S. Miller, P. T. Mocanu, Differential subordinations theory and applications, Boca Raton: CRC Press, 2000. https://doi.org/10.1201/9781482289817
[45]	S. A. Shah, E. E. Ali, A. A. Maitlo, T. Abdeljawad, A. M. Albalahi, Inclusion results for the class of fuzzy $\alpha$ -convex functions, AIMS Mathematics, 8 (2023), 1375–1383. https://doi.org/10.3934/math.2023069 doi: 10.3934/math.2023069
[46]	B. Kanwal, S. Hussain, A. Saliu, Fuzzy differential subordination related to strongly Janowski functions, Appl. Math. Sci. Eng., 31 (2023), 2170371. https://doi.org/10.1080/27690911.2023.2170371 doi: 10.1080/27690911.2023.2170371
[47]	S. A. Shah, E. E. Ali, A. Catas, A. M. Albalahi, On fuzzy differential subordination associated with $q$ -difference operator, AIMS Mathematics, 8 (2023), 6642–6650. https://doi.org/10.3934/math.2023336 doi: 10.3934/math.2023336
[48]	B. Kanwal, K. Sarfaraz, M. Naz, A. Saliu, Fuzzy differential subordination associated with generalized Mittag-Leffler type Poisson distribution, Arab Journal of Basic and Applied Sciences, 31 (2024), 206–212. https://doi.org/10.1080/25765299.2024.2319366 doi: 10.1080/25765299.2024.2319366
[49]	S. H. Hadi, M. Darus, A class of harmonic $(p, \mathfrak{q})$ -starlike functions involving a generalized $(p, \mathfrak{q})$ -Bernardi integral operator, Probl. Anal. Issues Anal., 12 (2023), 17–36. https://doi.org/10.15393/j3.art.2023.12850 doi: 10.15393/j3.art.2023.12850
[50]	P. H. Long, H. Tang, W. S. Wang, Functional inequalities for several classes of $\mathfrak{q}$ -starlike and $\mathfrak{q}$ -convex type analytic and multivalent functions using a generalized Bernardi integral operator, AIMS Mathematics, 6 (2021), 1191–1208. https://doi.org/10.3934/math.2021073 doi: 10.3934/math.2021073
[51]	O. A. Arqub, J. Singh, M. Alhodaly, Adaptation of kernel functions-based approach with Atangana-Baleanu-Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations, Math. Method. Appl. Sci., 46 (2023), 7807–7834. https://doi.org/10.1002/mma.7228 doi: 10.1002/mma.7228
[52]	O. A. Arqub, J. Singh, B. Maayah, M. Alhodaly, Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator, Math. Method. Appl. Sci., 46 (2023), 7965–7986. https://doi.org/10.1002/mma.7305 doi: 10.1002/mma.7305
[53]	O. A. Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput. & Applic., 28 (2017), 1591–1610. https://doi.org/10.1007/s00521-015-2110-x doi: 10.1007/s00521-015-2110-x
[54]	O. A. Arqub, S. Momani, S. Al-Mezel, M. Kutbi, Existence, Uniqueness, and characterization theorems for nonlinear fuzzy integrodifferential equations of Volterra type, Math. Probl. Eng., 2015 (2015), 835891. http://doi.org/10.1155/2015/835891 doi: 10.1155/2015/835891

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AIMS Mathematics

Applications of fuzzy differential subordination theory on analytic $p$ -valent functions connected with $\mathfrak{q}$ -calculus operator

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2. Preliminary concepts

3. General implementation of the method

4. Applications

5. Conclusions

Acknowledgements

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Applications of fuzzy differential subordination theory on analytic p p -valent functions connected with q \mathfrak{q} -calculus operator

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Applications of fuzzy differential subordination theory on analytic $p$ -valent functions connected with $\mathfrak{q}$ -calculus operator