Implementation of Yang residual power series method to solve fractional non-linear systems

Azzh Saad Alshehry; Roman Ullah; Nehad Ali Shah; Rasool Shah; Kamsing Nonlaopon; Azzh Saad Alshehry; Roman Ullah; Nehad Ali Shah; Rasool Shah; Kamsing Nonlaopon

doi:10.3934/math.2023418

AIMS Mathematics

2023, Volume 8, Issue 4: 8294-8309. doi: 10.3934/math.2023418

Previous Article Next Article

Research article

Implementation of Yang residual power series method to solve fractional non-linear systems

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of General Studies, Higher Colleges of Technology, Dubai Women Campus, UAE
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea
4.
Department of Mathematics, Abdul Wali khan University Mardan 23200, Pakistan
5.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
^† These authors contributed equally to this work and are co-first authors.

Received: 07 September 2022 Revised: 30 November 2022 Accepted: 12 December 2022 Published: 01 February 2023
MSC : 32B15, 34A34, 35A22, 35A24, 45A10

In this study, we implemented the Yang residual power series (YRPS) methodology, a unique analytical treatment method, to estimate the solutions of a non-linear system of fractional partial differential equations. The RPS approach and the Yang transform are togethered in the YRPS method. The suggested approach to handle fractional systems is explained along with its application. With fewer calculations and greater accuracy, the limit idea is used to solve it in Yang space to produce the YRPS solution for the proposed systems. The benefit of the new method is that it requires less computation to get a power series form solution, whose coefficients should be established in a series of algebraic steps. Two attractive initial value problems were used to test the technique's applicability and performance. The behaviour of the approximative solutions is numerically and visually discussed, along with the effect of fraction order $\varsigma$ . It was observed that the proposed method's approximations and exact solutions were completely in good agreement. The YRPS approach results highlight and show that the approach may be utilized to a variety of fractional models of physical processes easily and with analytical efficiency.

Keywords:

Yang transform,
Caputo operator,
residual power series,
systems of fractional differential equations

Citation: Azzh Saad Alshehry, Roman Ullah, Nehad Ali Shah, Rasool Shah, Kamsing Nonlaopon. Implementation of Yang residual power series method to solve fractional non-linear systems[J]. AIMS Mathematics, 2023, 8(4): 8294-8309. doi: 10.3934/math.2023418

Related Papers:

[1]	Emad Salah, Ahmad Qazza, Rania Saadeh, Ahmad El-Ajou . A hybrid analytical technique for solving multi-dimensional time-fractional Navier-Stokes system. AIMS Mathematics, 2023, 8(1): 1713-1736. doi: 10.3934/math.2023088
[2]	M. Mossa Al-Sawalha, Khalil Hadi Hakami, Mohammad Alqudah, Qasem M. Tawhari, Hussain Gissy . Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation. AIMS Mathematics, 2025, 10(3): 7099-7126. doi: 10.3934/math.2025324
[3]	Ali Khalouta, Abdelouahab Kadem . A new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients. AIMS Mathematics, 2020, 5(1): 1-14. doi: 10.3934/math.2020001
[4]	Muhammad Imran Liaqat, Sina Etemad, Shahram Rezapour, Choonkil Park . A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients. AIMS Mathematics, 2022, 7(9): 16917-16948. doi: 10.3934/math.2022929
[5]	Azzh Saad Alshehry, Humaira Yasmin, Ali M. Mahnashi . Exploring fractional Advection-Dispersion equations with computational methods: Caputo operator and Mohand techniques. AIMS Mathematics, 2025, 10(1): 234-269. doi: 10.3934/math.2025012
[6]	Meshari Alesemi . Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform. AIMS Mathematics, 2024, 9(10): 29269-29295. doi: 10.3934/math.20241419
[7]	Tareq Eriqat, Rania Saadeh, Ahmad El-Ajou, Ahmad Qazza, Moa'ath N. Oqielat, Ahmad Ghazal . A new analytical algorithm for uncertain fractional differential equations in the fuzzy conformable sense. AIMS Mathematics, 2024, 9(4): 9641-9681. doi: 10.3934/math.2024472
[8]	Qasem M. Tawhari . Mathematical analysis of time-fractional nonlinear Kuramoto-Sivashinsky equation. AIMS Mathematics, 2025, 10(4): 9237-9255. doi: 10.3934/math.2025424
[9]	Shabir Ahmad, Aman Ullah, Ali Akgül, Fahd Jarad . A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations. AIMS Mathematics, 2022, 7(5): 9389-9404. doi: 10.3934/math.2022521
[10]	Nader Al-Rashidi . Innovative approaches to fractional modeling: Aboodh transform for the Keller-Segel equation. AIMS Mathematics, 2024, 9(6): 14949-14981. doi: 10.3934/math.2024724

Abstract

1. Introduction

In the modelling of numerous real-world events in control theory, chemistry, physics and other branches of engineering and science, fractional derivatives can be used successfully ^[1,2,3,4,5]. The basic reason for this is that fractional calculus can be used to model real-world issues successfully because realistic models not only depend on the present time but also on the past historical time ^[6,7,8]. As a result, the fractional differential equations (FDEs) have drawn the interest of numerous scientific and engineering academics due to their significant role in understanding a number of real-life process that happen in the natural sciences, such as mechanical systems, wave propagation phenomena, earthquake modelling, image processing and control theory. Fractional calculus can be used to describe and reformulate these processes as FDEs. The use of FDEs in the aforementioned and other phenomena is notable for their nonlocality feature. Similarly, the differential operators give an excellent way to describe the memory and inherited characteristics of distinct processes and materials ^[9,10].

In the framework of fractional derivatives, partial differential equations (PDEs) are regarded as a potential tool in mathematical modelling to explain and understand some physical processes structures that are complicated due to outside influences. Because of this reason, researchers utilized them to both construct a natural problem that is easily accessible and to simplify the regulating plan without losing some inherited material or memorial impact ^[1,2,3]. The development of trustworthy numerical methods to handle the fraction PDEs of physical concern has also been the subject of countless efforts, many of which have been successful ^{[14,15,16,17,18,19]}. Many real-world issues, such as earthquakes, gas dynamics, traffic flow and oscillation, can be formulated as nonlinear PDEs in the framework of fractional derivative, and the solutions of fractional PDEs provide remarkable insight into the behaviour of particular dynamic systems ^[4,5]. Therefore, it is important to develop a practical and useful method for identifying analytical solutions to these and other issues. To examine and develop analytical-approximate solutions of fractional FDEs and PDEs, researchers have recently used a variety of analytical and numerical methodologies, such as the Homotopy perturbation transform method ^[22], residual power series (RPS) method ^[23], Adomian decomposition method ^[24], variational iteration transform method ^[25,26], Iterative laplace transform method ^[27], Elzaki transform decomposition method ^[28,29] and Natural transform decomposition method ^[30,31].

A numerical analytic method for solving many forms of ordinary, partial, integro-differential equations and fractional fuzzy differential equations is known as the residual power series (RPS) technique. The RPSM was introduced by Jordanian mathematician Omar Abu Arqub in 2013 ^[32]. Due to the fact that it offers closed-form solutions of well-known functions, it is an efficient optimization strategy ^[33,34]. With the use of the fractional residual power series (FRPS) technique, fuzzy FDEs, a variety of FDEs, and integral equations having fractional-order are solved. Such as, Coupled fractional resonant Schrödinger equations ^[35], time-fractional Fokker-Planck models ^[36], fractional fredholm integro-differential equations ^[37], fractional Newell-hitehead-Segel equation ^[38], singular initial value problems ^[39], Fractional partial differential equations ^[40], fractional Kundu-Eckhaus and massive Thirring equations ^[41] and several classes of fractional fuzzy differential equations ^[42,43]. The Yang transform (YT) is a powerful tool for resolving numerous complex models that are appearing in various branches of the natural sciences. When analytical methods are combined with the YT operator, non-linear problems can be solved more quickly and with more precision.

Researchers introduced a new approach to solve fractional differential equations by combining two well-known approaches. Several of these groups include a combination of the homotopy analysis method and the natural transform ^[44], also homotopy perturbation approach and the Sumudu transform ^[45], the Yang transform and the Adomian decomposition method ^[46], and the Laplace transform with RPSM ^[47]. The main goal of this work is to examine the approximate solution of nonlinear systems by implementing the Yang residual power series (YRPS) method. The YT and RPS approaches are combined in the YRPS approach, which provides both approximate and accurate solutions as quickly fractional power series (FPS) solutions. The proposed system is converted to Yang space, and then the solutions in the form of algebraic equations are created. Lastly, the Yang inverse is applied to the proposed problem results. In contrast to the FRPS approach, which depends on the fractional derivative and consume time to compute the various derivatives of fractional-order in steps of determining the solutions, the unknown coefficients in a modified Yang expansion can be identified by employing the limit idea. The YRPS approach take less time and provide higher accuracy with minor computational requirements.

In this article, we consider fractional nonlinear systems as:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}+{v}_{{\vartheta}}{w}_{{\psi}}-{v}_{{\psi}}{w}_{{\vartheta}}+{u} = 0, \ \ \ 0 < \varsigma\leq1, \\ &D_{t}^{\varsigma}{v}+{w}_{{\vartheta}}{u}_{{\psi}}-{u}_{{\psi}}{w}_{{\vartheta}}-{v} = 0, \\ &D_{t}^{\varsigma}{w}+{u}_{{\vartheta}}{v}_{{\psi}}-{u}_{{\psi}}{v}_{{\vartheta}}-{w} = 0, \\ \end{split} \end{equation}$

(1.1)

having initial sources

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = f_{0}({\vartheta}, {\psi}), \\ &{v}({\vartheta}, {\psi}, 0) = g_{0}({\vartheta}, {\psi}), \\ &{w}({\vartheta}, {\psi}, 0) = h_{0}({\vartheta}, {\psi}), \\ \end{split} \end{equation}$

(1.2)

and

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}-{u}_{{\vartheta}{\vartheta}}-2{u}{u}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \ \ \ 0 < \varsigma\leq1, \\ &D_{t}^{\varsigma}{v}-{v}_{{\vartheta}{\vartheta}}-2{v}{v}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \\ \end{split} \end{equation}$

(1.3)

having initial sources

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = f_{0}({\vartheta}, {\psi}), \\ &{v}({\vartheta}, {\psi}, 0) = g_{0}({\vartheta}, {\psi}).\\ \end{split} \end{equation}$

(1.4)

Following is the breakdown of our study. Section 2 reviews the YT, as well as some fundamental definitions and theorem pertaining to fractional calculus. Section 3 describes the idea behind the suggested technique for creating the approximation of the fractional model taken into account in Eq (1.1). In Section 4, the YRPS methodology is applied to solve fractional nonlinear systems in order to show the applicability and efficacy of the method in analysing the solutions of time-PDEs of fractional order. Section 5 concludes with a summary of our results.

2. Preliminaries

In this part, we review some definitions and fractional derivatives theorems in Caputo manner along with YT properties.

Definition 2.1. In Caputo sense the fractional derivative of a function ${u}({\vartheta}, {t})$ is given as ^[48]

$\begin{equation} ^{C}D_{{t}}^{{\varsigma}}{u}({\vartheta}, {t}) = J_{{t}}^{m-{\varsigma}}{u}^{{m}}({\vartheta}, {t}), \ \ m-1 < \varsigma\leq m, \ \ t > 0, \end{equation}$

(2.1)

with $m\in N$ and $J_{{t}}^{\varsigma}$ represents the fractional integral in Riemann-Liouville manner of ${u}({\vartheta}, {t})$ as

$\begin{equation} J_{t}^{\varsigma} u(\vartheta, t) = \frac{1}{\Gamma(\varsigma)} \int_{0}^{t}(t-\tau)^{\varsigma-1} u(\vartheta, \tau) d \tau. \end{equation}$

(2.2)

Definition 2.2. The YT of a function $\phi(\kappa)$ is stated by $\mathcal{Y}\{u(t)\}$ or M(s) as ^[49,50]

$\begin{equation} \mathcal{Y}\{u(t)\} = M(s) = \int_0^{\infty}e^{\frac{-t}{s}}u(t)dt, \ \ t > 0, \ \ s\in(-t_1, t_2). \end{equation}$

(2.3)

The inverse YT is given as

$\begin{equation} \mathcal{Y}^{-1}\{M(s)\} = u(t). \end{equation}$

(2.4)

Definition 2.3. The nth derivatives YT is given ^[49,50]

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

(2.5)

Definition 2.4. The YT of derivatives having order fraction is as ^[49,50]

$\begin{equation} \mathcal{Y}\{u^{\varsigma}(t)\} = \frac{M(s)}{s^{\varsigma}}-\sum\limits_{k = 0}^{n-1}\frac{u^{k}(0)}{s^{\varsigma-(k+1)}}, \ \ 0 < \varsigma \leq n. \end{equation}$

(2.6)

Theorem 2.5. The fractional power series $\sum_{m-0}^{\infty} a_m(t-\varepsilon)^{m p}$ can converge within just three of the following ways:

(1) The series converges only when the radius of convergence equals zero, or $t = \varepsilon$ .

(2) The series converges with a radius of convergence equal to $\infty$ for all $t \geq \varepsilon$ .

(3) The series diverges for $t > c+R$ and converges for $\varepsilon \leq t < \varepsilon+R$ and some real positive integer $R$ .

In this context, $R$ refers to the fractional power series radius of convergence.

3. Idea of YRPS

In this part, we will present the general implementation of YRPS for solving fractional nonlinear systems of PDEs.

By employing YT to Eq (1.1), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right] = 0, \\ &V({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right] = 0, \\ &W({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi})+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right] = 0.\\ \end{split} \end{equation}$

(3.1)

Considering that the solution of Eq (3.1) has the appropriate expansion

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = \sum\limits_{n = 0}^{\infty}s^{n\varsigma+1}h_{n}({\vartheta}, {\psi}, s).\\ \end{split} \end{equation}$

(3.2)

The $k^{th}$ -truncated series are

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}f_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = {s}g_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = {s}h_{0}({\vartheta}, {\psi}, s)+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}h_{n}({\vartheta}, {\psi}, s).\ \ \ k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(3.3)

By Yang residual functions (YRFs)

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u}({\vartheta}, {\psi}, s) = &U({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right], \\ \mathcal{Y}_{{t}}Res_{v}({\vartheta}, {\psi}, s) = &V({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right], \\ \mathcal{Y}_{{t}}Res_{\rho}({\vartheta}, {\psi}, s) = &W({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right].\\ \end{split} \end{equation}$

(3.4)

And the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s) = &U_{k}({\vartheta}, {\psi}, s)-{s}f_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{U}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s) = &V_{k}({\vartheta}, {\psi}, s)-{s}g_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s) = &W_{k}({\vartheta}, {\psi}, s)-{s}h_{0}({\vartheta}, {\psi}, s)+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{W}_{k}]\right].\\ \end{split} \end{equation}$

(3.5)

To highlight some facts, the YRPSM contains the following characteristics:

● $\mathcal{Y}_{{t}}Res({\vartheta}, {\psi}, s) = 0$ and $\lim_{{j}\rightarrow \infty}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = \mathcal{Y}_{{{t}}}Res_{{u}}({\vartheta}, {\psi}, s)$ for each $s > 0.$

● $\lim_{s\rightarrow \infty}s\mathcal{Y}_{{{t}}}Res_{{u}}({\vartheta}, {\psi}, s) = 0\Rightarrow \lim_{s\rightarrow \infty}s\mathcal{Y}_{{{t}}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0.$

● $\lim_{s\rightarrow \infty}s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = \lim_{s\rightarrow \infty}s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ 0 < {\varsigma}\leq 1, \ \ {k} = 1, 2, 3, \cdots$ .

Now to determine the coefficients $f_{n}({\vartheta}, {\psi}, s)$ , $g_{n}({\vartheta}, {\psi}, s)$ , $h_{n}({\vartheta}, {\psi}, s)$ and $l_{n}({\vartheta}, {\psi}, s)$ , we resolve the below system recursively as

$\begin{equation} \begin{split} &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{u}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots, \\ &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{v}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots, \\ &\lim\limits_{s\rightarrow \infty}s^{{k}{\varsigma}+1}\mathcal{Y}_{{t}}Res_{{\rho}, {k}}({\vartheta}, {\psi}, s) = 0, \ \ {k} = 1, 2, \cdots.\\ \end{split} \end{equation}$

(3.6)

Finally by employing inverse YT to Eq (3.3), to obtain the ${k}^{th}$ analytical solutions of ${u}_{{k}}({\vartheta}, {\psi}, {t})$ , ${v}_{{k}}({\vartheta}, {\psi}, {t})$ and ${\rho}_{{k}}({\vartheta}, {\psi}, {t})$

4. Applications

In this part, we find the solution of fractional nonlinear systems by implementing the suggested approach.

4.1. Problem

Let us assume the system of fractional PDEs:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}+{v}_{{\vartheta}}{w}_{{\psi}}-{v}_{{\psi}}{w}_{{\vartheta}}+{u} = 0, \\ &D_{t}^{\varsigma}{v}+{w}_{{\vartheta}}{u}_{{\psi}}-{u}_{{\psi}}{w}_{{\vartheta}}-{v} = 0, \\ &D_{t}^{\varsigma}{w}+{u}_{{\vartheta}}{v}_{{\psi}}-{u}_{{\psi}}{v}_{{\vartheta}}-{w} = 0.\\ \end{split} \end{equation}$

(4.1)

By considering Eq (4.1), having below initial sources:

$\begin{equation} \begin{split} &{u}({\vartheta}, {\psi}, 0) = e^{{\vartheta}+{\psi}}, \\ &{v}({\vartheta}, {\psi}, 0) = e^{{\vartheta}-{\psi}}, \\ &{w}({\vartheta}, {\psi}, 0) = e^{{\psi}-{\vartheta}}.\\ \end{split} \end{equation}$

(4.2)

By employing YT to Eq (4.1) and using Eq (4.2), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}+{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{U}]\right] = 0, \\ &V({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}-{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{V}]\right] = 0, \\ &W({\vartheta}, {\psi}, s)-{s}e^{{\psi}-{\vartheta}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]-\mathcal{Y}_{t}^{-1}[{W}]\right] = 0.\\ \end{split} \end{equation}$

(4.3)

The $k^{th}$ -truncated term series are

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}+{\psi}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, {\psi}, s), \ \ V({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}-{\psi}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, {\psi}, s), \\ &W({\vartheta}, {\psi}, s) = {s}e^{{\psi}-{\vartheta}}+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}h_{n}({\vartheta}, {\psi}, s), \ \ \ k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(4.4)

and the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s) = &U_{k}({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}+{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{U}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s) = &V_{k}({\vartheta}, {\psi}, s)-{s}e^{{\vartheta}-{\psi}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{W}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{V}_{k}]\right], \\ \mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s) = &W_{k}({\vartheta}, {\psi}, s)-{s}e^{{\psi}-{\vartheta}}+{s^{\varsigma}}\mathcal{Y}_{t}\left[\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\psi}, {k}}]-\mathcal{Y}_{t}^{-1}[{U}_{{\psi}, {k}}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]-\mathcal{Y}_{t}^{-1}[{W}_{k}]\right].\\ \end{split} \end{equation}$

(4.5)

To find $f_{k}({\vartheta}, {\psi}, s)$ , $g_{k}({\vartheta}, {\psi}, s)$ and $h_{k}({\vartheta}, {\psi}, s)$ $k = 1, 2, 3, \cdots,$ the $kth$ -truncated series equation Eq (4.4) will be inserted into the $kth$ -Yang residual function equation Eq (4.5), which will then be multiplied by $s^{k{\varsigma}+1}$ to solve the relation recursively $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, {\psi}, s)) = 0$ , $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, {\psi}, s)) = 0$ , and $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{w, k}({\vartheta}, {\psi}, s)) = 0$ , $k = 1, 2, 3, \cdots$ .

Few terms are as:

$\begin{equation} \begin{split} f_{1}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{1}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{1}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{2}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{2}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{2}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{3}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{3}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{3}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{4}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{4}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{4}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{5}({\vartheta}, {\psi}, s) = &e^{{\vartheta}+{\psi}}, \ \ g_{5}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{5}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \\ f_{6}({\vartheta}, {\psi}, s) = &-e^{{\vartheta}+{\psi}}, \ \ g_{6}({\vartheta}, {\psi}, s) = e^{{\vartheta}-{\psi}}, \ \ h_{6}({\vartheta}, {\psi}, s) = e^{{\psi}-{\vartheta}}, \end{split} \end{equation}$

(4.6)

and go on.

By substituting the values of $f_{k}({\vartheta}, {\psi}, s)$ , $g_{k}({\vartheta}, {\psi}, s)$ and $h_{k}({\vartheta}, {\psi}, s)$ , $k = 1, 2, 3, \cdots,$ in Eq (4.4), we have

$\begin{equation} \begin{split} &U({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}+{\psi}}-{s^{\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{2\varsigma+1}}e^{{\vartheta}+{\psi}}-{s^{3\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{4\varsigma+1}}e^{{\vartheta}+{\psi}}-{s^{5\varsigma+1}}e^{{\vartheta}+{\psi}}+{s^{6\varsigma+1}}e^{{\vartheta}+{\psi}}+\cdots, \\ &V({\vartheta}, {\psi}, s) = {s}e^{{\vartheta}-{\psi}}+{s^{\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{2\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{3\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{4\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{5\varsigma+1}}e^{{\vartheta}-{\psi}}+{s^{6\varsigma+1}}e^{{\vartheta}-{\psi}}+\cdots, \\ &W({\vartheta}, {\psi}, s) = {s}e^{{\psi}-{\vartheta}}+{s^{\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{2\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{3\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{4\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{5\varsigma+1}}e^{{\psi}-{\vartheta}}+{s^{6\varsigma+1}}e^{{\psi}-{\vartheta}}+\cdots.\\ \end{split} \end{equation}$

(4.7)

By employing inverse YT, we have

$\begin{equation*} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}}\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}}\left(1+\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}+\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}+\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &\rho({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}}\left(1+\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}+\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}+\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right).\\ \end{split} \end{equation*}$

Putting $\varsigma = 1$

$\begin{equation} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}}\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right), \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}}\left(1+{t}+\frac{{t}^{2}}{2!}+\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}+\frac{{t}^{5}}{5!}+\cdots\right), \\ &w({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}}\left(1+{t}+\frac{{t}^{2}}{2!}+\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}+\frac{{t}^{5}}{5!}+\cdots\right).\\ \end{split} \end{equation}$

(4.8)

Thus we get exact solutions as

$\begin{equation} \begin{split} &u({\vartheta}, {\psi}, {t}) = e^{{\vartheta}+{\psi}-{t}}, \\ &v({\vartheta}, {\psi}, {t}) = e^{{\vartheta}-{\psi}+{t}}, \\ &\rho({\vartheta}, {\psi}, {t}) = e^{{\psi}-{\vartheta}+{t}}.\\ \end{split} \end{equation}$

(4.9)

4.2. Problem

Let us assume the system of fractional Burger's equations:

$\begin{equation} \begin{split} &D_{t}^{\varsigma}{u}-{u}_{{\vartheta}{\vartheta}}-2{u}{u}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0, \\ &D_{t}^{\varsigma}{v}-{v}_{{\vartheta}{\vartheta}}-2{v}{v}_{{\vartheta}}+({u}{v})_{{\vartheta}} = 0.\\ \end{split} \end{equation}$

(4.10)

By considering Eq (4.10), having below initial sources:

$\begin{equation} \begin{split} &{u}({\vartheta}, 0) = \sin({{\vartheta}}), \\ &{v}({\vartheta}, 0) = \sin({{\vartheta}}).\\ \end{split} \end{equation}$

(4.11)

By employing YT to Eq (4.10) and using Eq (4.11), we get

$\begin{equation} \begin{split} &U({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}{\vartheta}}]-2\mathcal{Y}_{t}^{-1}[{U}]\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}}]\right] = 0, \\ &V({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}{\vartheta}}]-2\mathcal{Y}_{t}^{-1}[{V}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}}]\right] = 0.\\ \end{split} \end{equation}$

(4.12)

The $k^{th}$ -truncated series are

$\begin{equation} \begin{split} &U({\vartheta}, s) = {s}\sin({{\vartheta}})+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}f_{n}({\vartheta}, s), V({\vartheta}, s) = {s}\sin({{\vartheta}})+\sum\limits_{n = 1}^{k}{s^{n\varsigma+1}}g_{n}({\vartheta}, s), k = 1, 2, 3, 4\cdots \end{split} \end{equation}$

(4.13)

and the ${k}^{th}$ -YRFs as:

$\begin{equation} \begin{split} \mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, s) = &U_{k}({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}{\vartheta}, {k}}]-2\mathcal{Y}_{t}^{-1}[{U}_{k}]\mathcal{Y}_{t}^{-1}[{U}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}, {k}}]\right], \\ \mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, s) = &V_{k}({\vartheta}, s)-{s}\sin({{\vartheta}})+{s^{\varsigma}}\mathcal{Y}_{t}\left[-\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}{\vartheta}, {k}}]-2\mathcal{Y}_{t}^{-1}[{V}_{k}]\mathcal{Y}_{t}^{-1}[{V}_{{\vartheta}, {k}}]+\mathcal{Y}_{t}^{-1}[{(UV)}_{{\vartheta}, {k}}]\right].\\ \end{split} \end{equation}$

(4.14)

To find $f_{k}({\vartheta}, s)$ and $g_{k}({\vartheta}, s)$ , $k = 1, 2, 3, \cdots,$ the $kth$ -truncated series equation Eq (4.13) will be inserted into the $kth$ -Yang residual function equation Eq (4.14), which will then be multiplied by $s^{k{\varsigma}+1}$ to solve the relation recursively $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{u, k}({\vartheta}, s)) = 0$ and $\lim_{s\rightarrow \infty}(s^{k{\varsigma}+1}\mathcal{Y}_{{t}}Res_{v, k}({\vartheta}, s)) = 0$ , $k = 1, 2, 3, \cdots$ .

Few terms are as:

$\begin{equation} \begin{split} f_{1}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{1}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{2}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{2}({\vartheta}, s) = -\sin({{\vartheta}}), \\ f_{3}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{3}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{4}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{4}({\vartheta}, s) = -\sin({{\vartheta}}), \\ f_{5}({\vartheta}, s) = &\sin({{\vartheta}}), \ \ g_{5}({\vartheta}, s) = \sin({{\vartheta}}), \\ f_{6}({\vartheta}, s) = &-\sin({{\vartheta}}), \ \ g_{6}({\vartheta}, s) = -\sin({{\vartheta}}), \end{split} \end{equation}$

(4.15)

and so on.

By substituting the values of $f_{k}({\vartheta}, s)$ and $g_{k}({\vartheta}, s)$ , $k = 1, 2, 3, \cdots,$ in Eq (4.13), we have

$\begin{equation} \begin{split} &U({\vartheta}, s) = {s}\sin({{\vartheta}})-{s^{\varsigma+1}}\sin({{\vartheta}})+{s^{2\varsigma+1}}\sin({{\vartheta}})-{s^{3\varsigma+1}}\sin({{\vartheta}})+{s^{4\varsigma+1}}\sin({{\vartheta}})-{s^{5\varsigma+1}}\sin({{\vartheta}})+{s^{6\varsigma+1}}\sin({{\vartheta}})+\cdots, \\ &V({\vartheta}, s) = {s}\sin({{\vartheta}})-{s^{\varsigma+1}}\sin({{\vartheta}})+{s^{2\varsigma+1}}\sin({{\vartheta}})-{s^{3\varsigma+1}}\sin({{\vartheta}})+{s^{4\varsigma+1}}\sin({{\vartheta}})-{s^{5\varsigma+1}}\sin({{\vartheta}})+{s^{6\varsigma+1}}\sin({{\vartheta}})+\cdots.\\ \end{split} \end{equation}$

(4.16)

By employing inverse YT, we have

$\begin{equation*} \begin{split} &u({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right), \\ &v({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-\frac{{t}^{\varsigma}}{\Gamma(\varsigma+1)}+\frac{{t}^{2\varsigma}}{\Gamma(2\varsigma+1)}-\frac{{t}^{3\varsigma}}{\Gamma(3\varsigma+1)}+\frac{{t}^{4\varsigma}}{\Gamma(4\varsigma+1)}-\frac{{t}^{5\varsigma}}{\Gamma(5\varsigma+1)}+\cdots\right).\\ \end{split} \end{equation*}$

Putting $\varsigma = 1$

$\begin{equation} \begin{split} &u({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right), \\ &v({\vartheta}, {t}) = \sin({{\vartheta}})\left(1-{t}+\frac{{t}^{2}}{2!}-\frac{{t}^{3}}{3!}+\frac{{t}^{4}}{4!}-\frac{{t}^{5}}{5!}+\cdots\right).\\ \end{split} \end{equation}$

(4.17)

Thus we get accurate solutions as

$\begin{equation} \begin{split} &u({\vartheta}, {t}) = e^{-{t}}\sin({{\vartheta}}), \\ &v({\vartheta}, {t}) = e^{-{t}}\sin({{\vartheta}}).\\ \end{split} \end{equation}$

(4.18)

4.3. Results and discussion

In the present study, we obtained the closed form solution of two nonlinear fractional systems by means of YRPSM upto sixth order. The obtained results are describes with the help of graphs and table. The YRPSM results for system $1$ on the basis of error is illustrated in Table 1. Figure 1a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${u}({\vartheta}, {\psi}, {t})$ . Figure 2a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${v}({\vartheta}, {\psi}, {t})$ . Figure 3a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${w}({\vartheta}, {\psi}, {t})$ of system 1. Similarly, Figure 4a, b shows the exact and analytical behavior whereas , shows the 3-D and 2-D behavior of approximate solution at different fractional-orders for ${u}({\vartheta}, {t})$ and ${v}({\vartheta}, {t})$ of system 2. The behavior of Table and Figures shows that our solution is in good agreement with the exact solutions of the problems.

Table 1. Behavior on the basis of Error by implementing YRPSM for the suggested problem at different values of t and

${\vartheta}$ with

${\varsigma = 1}$ ,

${\psi} = 1$ .

${t}$	${\vartheta}$	$\|{u}_{exact}-{u}_{YRPSM}\|$	$\|{v}_{exact}-{v}_{YRPSM}\|$	$\|{w}_{exact}-{w}_{YRPSM}\|$
0.2	0.2	2.671 $\times10^{-8}$	9.927 $\times10^{-8}$	1.8080 $\times10^{-8}$
	0.4	3.263 $\times10^{-8}$	1.212 $\times10^{-8}$	1.4810 $\times10^{-8}$
	0.6	3.986 $\times10^{-7}$	1.481 $\times10^{-8}$	1.2126 $\times10^{-7}$
	0.8	4.868 $\times10^{-7}$	1.808 $\times10^{-7}$	9.9270 $\times10^{-7}$
	1	5.945 $\times10^{-6}$	2.209 $\times10^{-7}$	8.1280 $\times10^{-6}$
0.4	0.2	2.1267 $\times10^{-7}$	7.981 $\times10^{-8}$	1.4544 $\times10^{-8}$
	0.4	2.5975 $\times10^{-7}$	9.748 $\times10^{-7}$	1.1908 $\times10^{-7}$
	0.6	3.1727 $\times10^{-6}$	1.190 $\times10^{-7}$	9.7489 $\times10^{-7}$
	0.8	3.8752 $\times10^{-6}$	1.454 $\times10^{-6}$	7.9817 $\times10^{-6}$
	1	4.7330 $\times10^{-5}$	1.776 $\times10^{-6}$	6.5349 $\times10^{-6}$
0.6	0.2	7.142 $\times10^{-6}$	2.707 $\times10^{-7}$	4.9333 $\times10^{-6}$
	0.4	8.7233 $\times10^{-6}$	3.3068 $\times10^{-6}$	4.0390 $\times10^{-6}$
	0.6	1.0654 $\times10^{-5}$	4.0390 $\times10^{-5}$	3.3068 $\times10^{-5}$
	0.8	1.3013 $\times10^{-5}$	4.9333 $\times10^{-5}$	2.7074 $\times10^{-5}$
	1	1.5894 $\times10^{-4}$	6.0254 $\times10^{-5}$	2.2166 $\times10^{-5}$

| Show Table

DownLoad: CSV

Figure 1. 3-D and 2-D behavior for

${u}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. 3-D and 2-D behavior for

${v}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. 3-D and 2-D behavior for

${w}({\vartheta}, {\psi}, t)$ of Problem

$1$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. 3-D and 2-D behavior for

${u}({\vartheta}, t)$ and

${v}({\vartheta}, t)$ of Problem

$2$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusions

To find the solutions of DEs and FDEs, a variety of numerical and analytical techniques are employed, some of which are regarded by the prospect of non-exact solutions. This work emphasised that the suggested strategy, YRPS, is an easy-to-use analytical technique for developing accurate and approximative solutions for certain systems of FDEs with suitable initial sources. The aforementioned method gave us the solutions in the Yang transform space by making it simple to calculate the expansion series constants with the aid of the limit concept at infinity. In contrast to the RPS approach, the YRPS method requires minimal calculations to obtain the series coefficients since it uses the limit idea rather than the fractional derivative. The calculated results demonstrate how closely the approximative solutions approach the accurate solution. This proves that the aforementioned method is an appropriate and extraordinarily effective method for obtaining the approximative and analytical solutions to a large number of linear and non-linear fractional problems that arise in engineering and applied physics.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Preface, North-Holl. Math. Stud., 204 (2006). https://doi.org/10.1016/s0304-0208(06)80001-0 doi: 10.1016/s0304-0208(06)80001-0
[2]	H. Dutta, A. O. Akdemir, A. Atangana, Fractional Order Analysis: Theory, Methods and Applications, Hoboken: Wiley, 2020.
[3]	B. Bira, T. Raja Sekhar, D. Zeidan, Exact solutions for some time-fractional evolution equations using lie group theory, Math. Meth. Appl. Sci., 41 (2018), 6717–6725. https://doi.org/10.1002/mma.5186 doi: 10.1002/mma.5186
[4]	M. Alabedalhadi, M. Al-Smadi, S. Al-Omari, D. Baleanu, S. Momani, Structure of optical soliton solution for nonlinear resonant space-time Schrödinger equation in conformable sense with full nonlinearity term, Phys. Scr., 95 (2020), 105215. https://doi.org/10.1088/1402-4896/abb739 doi: 10.1088/1402-4896/abb739
[5]	M. Al-Smadi, O. Abu Arqub, Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates, Appl. Math. Comput., 342 (2019), 280–294. https://doi.org/10.1016/j.amc.2018.09.020 doi: 10.1016/j.amc.2018.09.020
[6]	D. Baleanu, A. K. Golmankhaneh, A. K. Golmankhaneh, M. C. Baleanu, Fractional electromagnetic equations using fractional forms, Int. J. Theor. Phys., 48 (2009), 3114–3123. https://doi.org/10.1007/s10773-009-0109-8 doi: 10.1007/s10773-009-0109-8
[7]	M. H. Al-Smadi, G. N. Gumah, On the homotopy analysis method for fractional SEIR epidemic model, Res. J. Appl. Sci. Eng. Technol., 7 (2014), 3809–3820. https://doi.org/10.19026/rjaset.7.738 doi: 10.19026/rjaset.7.738
[8]	S. Kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alex. Eng. J., 52 (2013), 813–819. https://doi.org/10.1016/j.aej.2013.09.005 doi: 10.1016/j.aej.2013.09.005
[9]	M. H. Al-Smadi, A. Freihat, O. Abu Arqub, N. Shawagfeh, A novel multistep generalized differential transform method for solving fractional-order Lü chaotic and hyperchaotic systems, J. Comput. Anal. Appl., 19 (2015), 713–724.
[10]	K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, New York: Academic Press, 1974.
[11]	V. N. Kovalnogov, R. V. Fedorov, A. V. Chukalin, T. E. Simos, C. Tsitouras, Eighth order two-step methods trained to perform better on Keplerian-type orbits, Mathematics, 9 (2021), 3071. https://doi.org/10.3390/math9233071 doi: 10.3390/math9233071
[12]	L. J. Sun, J. Hou, C. J. Xing, Z. W. Fang, A robust Hammerstein-Wiener model identification method for highly nonlinear systems, Processes, 10 (2022), 2664. https://doi.org/10.3390/pr10122664 doi: 10.3390/pr10122664
[13]	F. W. Meng, A. P. Pang, X. F. Dong, C. Han, X. P. Sha, H-infinity optimal performance design of an unstable plant under bode integral constraint, Complexity, 2018 (2018), 4942906. https://doi.org/10.1155/2018/4942906 doi: 10.1155/2018/4942906
[14]	S. M. Ali, W. Shatanawi, M. D. Kassim, M. S. Abdo, S. Saleh, Investigating a class of generalized Caputo-type fractional integro-differential equations, J. Funct. Spaces, 2022 (2022), 8103046. https://doi.org/10.1155/2022/8103046 doi: 10.1155/2022/8103046
[15]	S. Etemad, M. M. Matar, M. A. Ragusa, S. Rezapour, Tripled fixed points and existence study to a tripled impulsive fractional differential system via measures of noncompactness, Mathematics, 10 (2022), 25. https://doi.org/10.3390/math10010025 doi: 10.3390/math10010025
[16]	P. Kumar, V. S. Erturk, M. Vellappandi, H. Trinh, V. Govindaraja, A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in caputo sense, Chaos Solitons Fractals, 158 (2022), 112067. https://doi.org/10.1016/j.chaos.2022.112067 doi: 10.1016/j.chaos.2022.112067
[17]	Z. Odibat, V. S. Erturk, P. Kumar, A. B. Makhlouf, V. Govindaraj, An implementation of the generalized differential transform scheme for simulating impulsive fractional differential equations, Math. Probl. Eng., 2022 (2022), 8280203. https://doi.org/10.1155/2022/8280203 doi: 10.1155/2022/8280203
[18]	L. Akinyemi, H. Rezazadeh, Q. H. Shi, M. Inc, M. M. A. Khater, H. Ahmad, et al., New optical solitons of perturbed nonlinear Schrodinger-Hirota equation with spatio-temporal dispersion, Results Phys., 29 (2021), 104656. https://doi.org/10.1016/j.rinp.2021.104656 doi: 10.1016/j.rinp.2021.104656
[19]	D. Ntiamoah, W. Ofori-Atta, L. Akinyemi, The higher-order modified Korteweg-de Vries equation: Its soliton, breather and approximate solutions, J. Ocean Eng. Sci., 2022. https://doi.org/10.1016/j.joes.2022.06.042 doi: 10.1016/j.joes.2022.06.042
[20]	P. Liu, J. P. Shi, Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Continuous Dyn. Syst. B, 18 (2013), 2597–2625. https://doi.org/10.3934/dcdsb.2013.18.2597 doi: 10.3934/dcdsb.2013.18.2597
[21]	S. G. Li, Z. M. Liu, Scheduling uniform machines with restricted assignment, Math. Biosci. Eng., 19 (2022), 9697–9708. https://doi.org/10.3934/mbe.2022450 doi: 10.3934/mbe.2022450
[22]	J. S. Duan, R. Rach, D. Baleanu, A. M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc., 3 (2012), 73–99.
[23]	A. Kumar, S. Kumar, S. P. Yan, Residual power series method for fractional diffusion equations, Fundam. Inform., 151 (2017), 213–230. https://doi.org/10.3233/fi-2017-1488 doi: 10.3233/fi-2017-1488
[24]	Y. Qin, A. Khan, I. Ali, M. Al Qurashi, H. Khan, R. Shah, et al., An efficient analytical approach for the solution of certain fractional-order dynamical systems, Energies, 13 (2020), 2725. https://doi.org/10.3390/en13112725 doi: 10.3390/en13112725
[25]	A. A. Alderremy, S. Aly, R. Fayyaz, A. Khan, R. Shah, N. Wyal, The analysis of fractional-order nonlinear systems of third order kdv and burgers equations via a novel transform, Complexity, 2022 (2022), 4935809. https://doi.org/10.1155/2022/4935809 doi: 10.1155/2022/4935809
[26]	P. Sunthrayuth, H. A. Alyousef, S. A. El-Tantawy, A. Khan, N. Wyal, Solving fractional-order diffusion equations in a plasma and fluids via a novel transform, J. Funct. Spaces, 2022 (2022), 1899130. https://doi.org/10.1155/2022/1899130 doi: 10.1155/2022/1899130
[27]	H. Khan, A. Khan, M. Al-Qurashi, R. Shah, D. Baleanu, Modified modelling for heat like equations within Caputo operator, Energies, 13 (2020), 2002. https://doi.org/10.3390/en13082002 doi: 10.3390/en13082002
[28]	K. Nonlaopon, A. M. Alsharif, A. M. Zidan, A. Khan, Y. S. Hamed, R. Shah, Numerical investigation of fractional-order Swift-Hohenberg equations via a novel transform, Symmetry, 13 (2021), 1263. https://doi.org/10.3390/sym13071263 doi: 10.3390/sym13071263
[29]	Hajira, H. Khan, A. Khan, P. Kumam, D. Baleanu, M. Arif, An approximate analytical solution of the Navier-Stokes equations within Caputo operator and Elzaki transform decomposition method, Adv. Differ. Equ., 2020 (2020), 622. https://doi.org/10.1186/s13662-020-03058-1 doi: 10.1186/s13662-020-03058-1
[30]	M. K. Alaoui, K. Nonlaopon, A. M. Zidan, A. Khan, R. Shah, Analytical investigation of fractional-order Cahn-Hilliard and Gardner equations using two novel techniques, Mathematics, 10 (2022), 1643. https://doi.org/10.3390/math10101643 doi: 10.3390/math10101643
[31]	T. Botmart, R. P. Agarwal, M. Naeem, A. Khan, R. Shah, On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators, AIMS Mathematics, 7 (2022), 12483–12513. https://doi.org/10.3934/math.2022693 doi: 10.3934/math.2022693
[32]	A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52. https://doi.org/10.5373/jaram.1447.051912 doi: 10.5373/jaram.1447.051912
[33]	E. A. Az-Zobi, A. Yildirim, W. A. AlZoubi, The residual power series method for the one-dimensional unsteady flow of a van der waals gas, Physica A, 517 (2019), 188–196. https://doi.org/10.1016/j.physa.2018.11.030 doi: 10.1016/j.physa.2018.11.030
[34]	M. Khader, M. H. DarAssi, Residual power series method for solving nonlinear reaction-diffusion-convection problems, Bol. da Soc. Parana. de Mat., 39 (2021), 177–188. https://doi.org/10.5269/bspm.41741 doi: 10.5269/bspm.41741
[35]	R. Saadeh, M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. Salma Din, Application of fractional residual power series algorithm to solve Newell-Whitehead-Segel equation of fractional order, Symmetry, 11 (2019), 1431. https://doi.org/10.3390/sym11121431 doi: 10.3390/sym11121431
[36]	M. Al-Smadi, O. Abu Arqub, S. Momani, Numerical computations of coupled fractional resonant schrodinger equations arising in quantum mechanics under conformable fractional derivative sense, Phys. Scr., 95 (2020), 075218. https://doi.org/10.1088/1402-4896/ab96e0 doi: 10.1088/1402-4896/ab96e0
[37]	M. Al-Smadi, O. Abu Arqub, S. Hadid, Approximate solutions of nonlinear fractional Kundu-Eckhaus and coupled fractional massive Thirring equations emerging in quantum field theory using conformable residual power series method, Phys. Scr., 95 (2020), 105205. https://doi.org/10.1088/1402-4896/abb420 doi: 10.1088/1402-4896/abb420
[38]	R. Saadeh, M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Application of fractional residual power series algorithm to solve Newell-Whitehead-segel equation of fractional order, Symmetry, 11 (2019), 1431. https://doi.org/10.3390/sym11121431 doi: 10.3390/sym11121431
[39]	M. Al-Smadi, O. Abu Arqub, S. Hadid, An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative, Commun. Theor. Phys., 72 (2020), 085001. https://doi.org/10.1088/1572-9494/ab8a29 doi: 10.1088/1572-9494/ab8a29
[40]	I. Komashynska, M. Al-Smadi, O. A. Arqub, S. Momani, An efficient analytical method for solving singular initial value problems of nonlinear systems, Appl. Math. Inform. Sci., 10 (2016), 647–656. https://doi.org/10.18576/amis/100224 doi: 10.18576/amis/100224
[41]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Numerical computation of fractional Fredholm integrodifferential equation of order 2b arising in natural sciences, J. Phys. Conf. Ser., 1212 (2019), 012022. https://doi.org/10.1088/1742-6596/1212/1/012022 doi: 10.1088/1742-6596/1212/1/012022
[42]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, An analytical numerical method for solving fuzzy fractional Volterra Integro-differential equations, Symmetry, 11 (2019), 205. https://doi.org/10.3390/sym11020205 doi: 10.3390/sym11020205
[43]	M. Alaroud, M. Al-Smadi, R. R. Ahmad, U. K. S. Din, Computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations, Int. J. Differ. Equ., 2018 (2018), 8686502. https://doi.org/10.1155/2018/8686502 doi: 10.1155/2018/8686502
[44]	A. Khan, M. Junaid, I. Khan, F. Ali, K. Shah, D. Khan, Application of Homotopy analysis natural transform method to the solution of nonlinear partial differential equations, Sci. Int., 29 (2017), 297–303.
[45]	M. F. Zhang, Y. Q. Liu, X. S. Zhou, Efficient homotopy perturbation method for fractional non-linear equations using Sumudu transform, Therm. Sci., 19 (2015), 1167–1171. https://doi.org/10.2298/tsci1504167z doi: 10.2298/tsci1504167z
[46]	A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, W, Weera, Evaluation of time-fractional Fisher's equations with the help of analytical methods, AIMS Mathematics, 7 (2022), 18746–18766. https://doi.org/10.3934/math.20221031 doi: 10.3934/math.20221031
[47]	M. Alquran, M. Ali, M. Alsukhour, I. Jaradat, Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics, Results Phys., 19 (2020), 103667. https://doi.org/10.1016/j.rinp.2020.103667 doi: 10.1016/j.rinp.2020.103667
[48]	A. El-Ajou, Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, Eur. Phys. J. Plus, 136 (2021), 229. https://doi.org/10.1140/epjp/s13360-020-01061-9 doi: 10.1140/epjp/s13360-020-01061-9
[49]	M. Areshi, A. Khan, R. Shah, K. Nonlaopon, Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform, AIMS Mathematics, 7 (2022), 6936–6958. https://doi.org/10.3934/math.2022385 doi: 10.3934/math.2022385
[50]	M. K. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. S. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 3248376. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376

This article has been cited by:

1.	Laila F. Seddek, Essam R. El-Zahar, Jae Dong Chung, Nehad Ali Shah, A Novel Approach to Solving Fractional-Order Kolmogorov and Rosenau–Hyman Models through the q-Homotopy Analysis Transform Method, 2023, 11, 2227-7390, 1321, 10.3390/math11061321
2.	Meshari Alesemi, An Innovative Approach to Nonlinear Fractional Shock Wave Equations Using Two Numerical Methods, 2023, 11, 2227-7390, 1253, 10.3390/math11051253
3.	Ma’mon Abu Hammad, Albandari W. Alrowaily, Rasool Shah, Sherif M. E. Ismaeel, Samir A. El-Tantawy, Analytical analysis of fractional nonlinear Jaulent-Miodek system with energy-dependent Schrödinger potential, 2023, 11, 2296-424X, 10.3389/fphy.2023.1148306
4.	Mamta Kapoor, Simran Kour, An analytical approach for Yang transform on fractional-order heat and wave equation, 2024, 99, 0031-8949, 035222, 10.1088/1402-4896/ad24ab
5.	Kottakkaran Sooppy Nisar, A constructive numerical approach to solve the Fractional Modified Camassa–Holm equation, 2024, 106, 11100168, 19, 10.1016/j.aej.2024.06.076
6.	R. Jagatheeshwari, C. Ravichandran, P. Veeresha, Kottakkaran Sooppy Nisar, A strong technique for solving the fractional model of multi-dimensional Schnakenberg reaction-diffusion system, 2025, 0217-9849, 10.1142/S0217984925500952

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)

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(2063) PDF downloads(148) Cited by(6)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(4) / Tables(1)

AIMS Mathematics

Implementation of Yang residual power series method to solve fractional non-linear systems

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Idea of YRPS

4. Applications

4.1. Problem

4.2. Problem

4.3. Results and discussion

5. Conclusions

Funding

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Implementation of Yang residual power series method to solve fractional non-linear systems

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Idea of YRPS

4. Applications

4.1. Problem

4.2. Problem

4.3. Results and discussion

5. Conclusions

Funding

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog