On sequential fractional Caputo $ (p, q) $-integrodifference equations via three-point fractional Riemann-Liouville $ (p, q) $-difference boundary condition

Jarunee Soontharanon; Thanin Sitthiwirattham; Jarunee Soontharanon; Thanin Sitthiwirattham

doi:10.3934/math.2022044

AIMS Mathematics

2022, Volume 7, Issue 1: 704-722. doi: 10.3934/math.2022044

Previous Article Next Article

Research article

On sequential fractional Caputo $(p, q)$ -integrodifference equations via three-point fractional Riemann-Liouville $(p, q)$ -difference boundary condition

Jarunee Soontharanon ¹,
Thanin Sitthiwirattham ^{2
,
,}

1.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
2.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand

Received: 09 September 2021 Accepted: 14 October 2021 Published: 15 October 2021
MSC : 39A10, 39A13, 39A70

In this paper, we aim to study the problem of a sequential fractional Caputo $(p, q)$ -integrodifference equation with three-point fractional Riemann-Liouville $(p, q)$ -difference boundary condition. We use some properties of $(p, q)$ -integral in this study and employ Banach fixed point theorems and Schauder's fixed point theorems to prove existence results of this problem.

Keywords:

Citation: Jarunee Soontharanon, Thanin Sitthiwirattham. On sequential fractional Caputo $(p, q)$ -integrodifference equations via three-point fractional Riemann-Liouville $(p, q)$ -difference boundary condition[J]. AIMS Mathematics, 2022, 7(1): 704-722. doi: 10.3934/math.2022044

Related Papers:

[1]	Dongdong Gao, Daipeng Kuang, Jianli Li . Some results on the existence and stability of impulsive delayed stochastic differential equations with Poisson jumps. AIMS Mathematics, 2023, 8(7): 15269-15284. doi: 10.3934/math.2023780
[2]	Miao Zhang, Bole Li, Weiqiang Gong, Shuo Ma, Qiang Li . Matrix measure-based exponential stability and synchronization of Markovian jumping QVNNs with time-varying delays and delayed impulses. AIMS Mathematics, 2024, 9(12): 33930-33955. doi: 10.3934/math.20241618
[3]	Yanshou Dong, Junfang Zhao, Xu Miao, Ming Kang . Piecewise pseudo almost periodic solutions of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. AIMS Mathematics, 2023, 8(9): 21828-21855. doi: 10.3934/math.20231113
[4]	Shuo Ma, Jiangman Li, Qiang Li, Ruonan Liu . Adaptive exponential synchronization of impulsive coupled neutral stochastic neural networks with Lévy noise and probabilistic delays under non-Lipschitz conditions. AIMS Mathematics, 2024, 9(9): 24912-24933. doi: 10.3934/math.20241214
[5]	Qinghua Zhou, Li Wan, Hongbo Fu, Qunjiao Zhang . Exponential stability of stochastic Hopfield neural network with mixed multiple delays. AIMS Mathematics, 2021, 6(4): 4142-4155. doi: 10.3934/math.2021245
[6]	Zhigang Zhou, Li Wan, Qunjiao Zhang, Hongbo Fu, Huizhen Li, Qinghua Zhou . Exponential stability of periodic solution for stochastic neural networks involving multiple time-varying delays. AIMS Mathematics, 2024, 9(6): 14932-14948. doi: 10.3934/math.2024723
[7]	Yuanfu Shao . Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps. AIMS Mathematics, 2022, 7(3): 4068-4093. doi: 10.3934/math.2022225
[8]	Biwen Li, Qiaoping Huang . Synchronization issue of uncertain time-delay systems based on flexible impulsive control. AIMS Mathematics, 2024, 9(10): 26538-26556. doi: 10.3934/math.20241291
[9]	Zhifeng Lu, Fei Wang, Yujuan Tian, Yaping Li . Lag synchronization of complex-valued interval neural networks via distributed delayed impulsive control. AIMS Mathematics, 2023, 8(3): 5502-5521. doi: 10.3934/math.2023277
[10]	Li Wan, Qinghua Zhou, Hongbo Fu, Qunjiao Zhang . Exponential stability of Hopfield neural networks of neutral type with multiple time-varying delays. AIMS Mathematics, 2021, 6(8): 8030-8043. doi: 10.3934/math.2021466

Abstract

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.

DownLoad: Full-Size Img PowerPoint

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

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

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

DownLoad: Full-Size Img PowerPoint

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]	F. H. Jackson, On $q$ -difference equations, Am. J. Math., 32 (1910), 305–314.
[2]	F. H. Jackson, On $q$ -difference integrals, Q. J. Pure. Appl. Math., 41 (1910), 193–203.
[3]	R. D. Carmichael, The general theory of linear $q$ -difference equations, Am. J. Math., 34 (1912), 147–168. doi: 10.2307/2369887. doi: 10.2307/2369887
[4]	T. E. Mason, On properties of the solutions of linear $q$ -difference equations with entire function coefficients, Am. J. Math., 37 (1915), 439–444. doi: 10.2307/2370216. doi: 10.2307/2370216
[5]	W. J. Trjitzinsky, Analytic theory of linear $q$ -differece equations, Acta Math., 62 (1933), 167–226. doi: 10.1007/BF02547785. doi: 10.1007/BF02547785
[6]	V. Kac, P. Cheung, Quantum Calculus, Springer, New York, 2002. doi: 10.2307/3647765.
[7]	T. Ernst, A new notation for $q$ -calculus and a new $q$ -Taylor formula, Department of Mathematics, Uppsala University, 1999.
[8]	R. Floreanini, L. Vinet, $q$ -gamma and $q$ -beta functions in quantum algebra representation theory, J. Comput. Appl. Math., 68 (1996), 57–68. doi: 10.1016/0377-0427(95)00253-7. doi: 10.1016/0377-0427(95)00253-7
[9]	K. N. Ilinski, G. V. Kalinin, A. S. Stepanenko, $q$ -functional field theory for particles with exotic statistics, Phys. Lett. A, 232 (1997), 399–408. doi: 10.1016/S0375-9601(97)00402-7. doi: 10.1016/S0375-9601(97)00402-7
[10]	R. J. Finkelstein, $q$ -field theory, Lett. Math. Phys., 34 (1995), 169–176. doi: 10.1007/BF00739095. doi: 10.1007/BF00739095
[11]	R. J. Finkelstein, The $q$ -Coulomb problem, J. Math. Phys., 37 (1996), 2628–2636. doi: 10.1063/1.531532. doi: 10.1063/1.531532
[12]	J. Feigenbaum, P. G. Freund, A $q$ -deformation of the Coulomb problem, J. Math. Phys., 37 (1996), 1602–1616. doi: 10.1063/1.531471. doi: 10.1063/1.531471
[13]	R. J. Finkelstein, $q$ -Gravity, Lett. Math. Phys., 38 (1996), 53–62. doi: 10.1007/BF00398298. doi: 10.1007/BF00398298
[14]	L. P. Marinova, P. P. Raychev, J. Maruani, Molecular backbending in AgH and its description in terms of $q$ -algebras, Mol. Phys., 82 (1994), 1115–1129. doi: 10.1080/00268979400100794. doi: 10.1080/00268979400100794
[15]	R. Chakrabarti, R. Jagannathan, A $(p, q)$ -oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), 5683–5701. doi: 10.1088/0305-4470/24/13/002. doi: 10.1088/0305-4470/24/13/002
[16]	V. Sahai, S. Yadav, Representations of two parameter quantum algebras and $(p, q)$ -special functions, J. Math. Anal. Appl., 335 (2007), 268–279. doi: 10.1016/j.jmaa.2007.01.072. doi: 10.1016/j.jmaa.2007.01.072
[17]	I. Burban, Two-parameter deformation of the oscillator algebra and $(p, q)$ -analog of two-dimensional conformal field theory, J. Nonlinear Math. Phy., 2 (1995), 384–391. doi: 10.2991/jnmp.1995.2.3-4.18. doi: 10.2991/jnmp.1995.2.3-4.18
[18]	P. N. Sadjang, On the fundamental theorem of $(p, q)$ -calculus and some $(p, q)$ -Taylor formulas, Results Math., 73 (2018). doi: 10.1007/s00025-018-0783-z. doi: 10.1007/s00025-018-0783-z
[19]	M. Mursaleen, K. J. Ansari, A. Khan, On $(p, q)$ -analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), 874–882. doi: 10.1016/j.amc.2015.04.090. doi: 10.1016/j.amc.2015.04.090
[20]	A. Khan, V. Sharma, Statistical approximation by $(p, q)$ -analogue of Bernstein-Stancu operators, Azerbaijan J. Math., 8 (2018), 100–121.
[21]	K. Khan, D. K. Lobiyal, Bezier curves based on Lupas $(p, q)$ -analogue of Bernstein functions in CAGD, J. Comput. Appl. Math., 317 (2017), 458–477. doi: 10.1016/j.cam.2016.12.016. doi: 10.1016/j.cam.2016.12.016
[22]	G. V. Milovanovic, V. Gupta, N. Malik, $(p, q)$ -Beta functions and applications in approximation, Bol. Soc. Mat. Mex., 24 (2018), 219–237. doi: 10.1007/s40590-016-0139-1. doi: 10.1007/s40590-016-0139-1
[23]	W. T. Cheng, W. H. Zhang, Q. B. Cai, $(p, q)$ -gamma operators which preserve $x^2$ , J. Inequal. Appl., 2019 (2019), 108. doi: 10.1186/s13660-019-2053-3. doi: 10.1186/s13660-019-2053-3
[24]	M. Nasiruzzaman, A. Mukheimer, M. Mursaleen, Some Opial-type integral inequalities via $(p, q)$ -calculus, J. Inequal. Appl., 2019 (2019), 295. doi: 10.1186/s13660-019-2247-8. doi: 10.1186/s13660-019-2247-8
[25]	P. Jain, C. Basu, V. Panwar, On the $(p, q)$ -Melin transform and its applications, Acta Math. Sci., 41 (2021), 1719–1732. doi:10.1007/s10473-021-0519-0. doi: 10.1007/s10473-021-0519-0
[26]	A. Aral, E. Deniz, H. Erbay, The Picard and Gauss-Weierstrass singular integral in $(p, q)$ -calculus, B. Malays. Math. Sci. So., 43 (2020), 1569–1583. doi: 10.1007/s40840-019-00759-z. doi: 10.1007/s40840-019-00759-z
[27]	N. Kamsrisuk, C. Promsakon, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for $(p, q)$ -difference equations, Differ. Equat. Appl., 10 (2018), 183–195. doi: 10.7153/dea-2018-10-11. doi: 10.7153/dea-2018-10-11
[28]	C. Promsakon, N. Kamsrisuk, S. K. Ntouyas, J. Tariboon, On the second-order quantum $(p, q)$ -difference equations with separated boundary conditions, Adv. Math. Phys., 2018, Article ID 9089865, 9 pages. doi: 10.1155/2018/9089865.
[29]	T. Nuntigrangjana, S. Putjuso, S. K. Ntouyas, J. Tariboon, Impulsive quantum $(p, q)$ -difference equations, Adv. Differ. Equ., 2020 (2020), 98. doi: 10.1186/s13662-020-02555-7. doi: 10.1186/s13662-020-02555-7
[30]	R. P. Agarwal, Certain fractional $q$ -integrals and $q$ -derivatives, Proc. Cambridge. Philos. Soc., 66 (1969), 365–370. doi: 10.1017/S0305004100045060. doi: 10.1017/S0305004100045060
[31]	W. A. Al-Salam, Some fractional $q$ -integrals and $q$ -derivatives, Proc. Edinburgh Math. Soc., 15 (1966), 135–140. doi: 10.1017/S0013091500011469. doi: 10.1017/S0013091500011469
[32]	J. B. Díaz, T. J. Osler, Differences of fractional order, Math. Comput., 28 (1974), 185–202. doi: 10.1090/S0025-5718-1974-0346352-5. doi: 10.1090/S0025-5718-1974-0346352-5
[33]	T. Brikshavana, T. Sitthiwirattham, On fractional Hahn calculus, Adv. Differ. Equ., 2017 (2017), 354. doi: 10.1186/s13662-017-1412-y. doi: 10.1186/s13662-017-1412-y
[34]	N. Patanarapeelert, T. Sitthiwirattham, On fractional symmetric Hahn calculus, Mathematics, 7 (2019), 873. doi: 10.3390/math7100873. doi: 10.3390/math7100873
[35]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, USA, 1999.
[36]	A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[37]	J. Sabatier, O. P. Agrawal, J. A. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
[38]	D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, Boston, 2012.
[39]	M. Tariq, H. Ahmad, S. K. Sahoo, The Hermite-Hadamard type inequality and its estimations via generalized convex functions of Raina type, Math. Model. Num. Simu. Appl., 1 (2021), 32–43. doi: 10.53391/mmnsa.2021.01.004. doi: 10.53391/mmnsa.2021.01.004
[40]	A. Keten, M. Yavuz, D. Baleanu, Nonlocal Cauchy problem via a fractional operator involving power kernel in Banach spaces, Fractal Fract., 3 (2019), 27. doi: 10.3390/fractalfract3020027. doi: 10.3390/fractalfract3020027
[41]	Z. Hammouch, M. Yavuz, N. Özdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Math. Model. Num. Simu. Appl., 1 (2021), 11–23. doi: 10.53391/mmnsa.2021.01.002. doi: 10.53391/mmnsa.2021.01.002
[42]	J. Soontharanon, T. Sitthiwirattham, On fractional $(p, q)$ -calculus, Adv. Differ. Equ., 2020 (2020), 35. doi: 10.1186/s13662-020-2512-7. doi: 10.1186/s13662-020-2512-7
[43]	N. Pheak, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, Fractional $(p, q)$ -calculus on finite intervals and some integral inequalities, Symmetry, 13 (2021), 504. doi: 10.3390/sym13030504. doi: 10.3390/sym13030504
[44]	N. Pheak, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, Praveen Agarwal, Some trapezoid and midpoint type inequalities via fractional $(p, q)$ -calculus, Adv. Differ. Equ., 2021 (2021), 333. doi: 10.1186/s13662-021-03487-6. doi: 10.1186/s13662-021-03487-6
[45]	J. Soontharanon, T. Sitthiwirattham, Existence results of nonlocal Robin boundary value problems for fractional $(p, q)$ -integrodifference equations, Adv. Differ. Equ., 2020 (2020), 342. doi: 10.1186/s13662-020-02806-7. doi: 10.1186/s13662-020-02806-7
[46]	Z. Qin, S. Sun, Positive solutions for fractional $(p, q)$ -difference boundary value problems, J. Appl. Math. Comput., 2021. doi: 10.1007/s12190-021-01630-w. doi: 10.1007/s12190-021-01630-w
[47]	Z. Qin, S. Sun, On a nonlinear fractional $(p, q)$ -difference Schr $\ddot{o}$ dinger equation, J. Appl. Math. Comput., 2021. doi: 10.1007/s12190-021-01586-x. doi: 10.1007/s12190-021-01586-x
[48]	D. H. Griffel, Applied Functional Analysis, Ellis Horwood Publishers, Chichester, 1981.
[49]	D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, Orlando, 1988.

This article has been cited by:

1.	Yao Lu, Quanxin Zhu, Exponential stability of impulsive random delayed nonlinear systems with average-delay impulses, 2024, 361, 00160032, 106813, 10.1016/j.jfranklin.2024.106813
2.	Chenhui Wu, Runzhang Zhang, Fang Gao, Dissipative control for linear time-delay systems based a modified equivalent-input-disturbance approach, 2024, 12, 2195-268X, 3472, 10.1007/s40435-024-01445-0
3.	Yuxiao Zhao, Hui Wang, Dongxu Wang, The Dynamic Behavior of a Stochastic SEIRM Model of COVID-19 with Standard Incidence Rate, 2024, 12, 2227-7390, 2966, 10.3390/math12192966
4.	Haowen Gong, Huijun Xiang, Yifei Wang, Huaijin Gao, Xinzhu Meng, Strategy evolution of a novel cooperative game model induced by reward feedback and a time delay, 2024, 9, 2473-6988, 33161, 10.3934/math.20241583
5.	Turki D. Alharbi, Md Rifat Hasan, Global stability and sensitivity analysis of vector-host dengue mathematical model, 2024, 9, 2473-6988, 32797, 10.3934/math.20241569
6.	Qiaofeng Li, Jianli Li, pth Moment Exponential Stability of Impulsive Stochastic Functional Differential Equations, 2025, 24, 1575-5460, 10.1007/s12346-024-01170-1
7.	Linna Liu, Shuaibo Zhou, Jianyin Fang, Stability Analysis of Split‐Step θ $\theta$ ‐Milstein Scheme for Stochastic Delay Integro‐Differential Equations, 2024, 0170-4214, 10.1002/mma.10622
8.	Mengqing Zhang, Quanxin Zhu, Jing Tian, Convergence Rates of Partial Truncated Numerical Algorithm for Stochastic Age-Dependent Cooperative Lotka–Volterra System, 2024, 16, 2073-8994, 1659, 10.3390/sym16121659
9.	Yicheng Hao, Yantao Luo, Jianhua Huang, Long Zhang, Zhidong Teng, Analysis of a stochastic HIV/AIDS model with commercial heterosexual activity and Ornstein–Uhlenbeck process, 2025, 03784754, 10.1016/j.matcom.2025.02.020

Reader Comments

Your name:*

Email:*
© 2022 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2511) PDF downloads(83) Cited by(5)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

On sequential fractional Caputo $(p, q)$ -integrodifference equations via three-point fractional Riemann-Liouville $(p, q)$ -difference boundary condition

Related Papers:

Abstract

1. Introduction

2. Preliminary concepts

3. General implementation of the method

4. Applications

5. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

On sequential fractional Caputo (p,q) (p, q) -integrodifference equations via three-point fractional Riemann-Liouville (p,q) (p, q) -difference boundary condition

Related Papers:

Abstract

1. Introduction

2. Preliminary concepts

3. General implementation of the method

4. Applications

5. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

On sequential fractional Caputo $(p, q)$ -integrodifference equations via three-point fractional Riemann-Liouville $(p, q)$ -difference boundary condition