On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $ p $

Wenpeng Zhang; Yuanyuan Meng; Wenpeng Zhang; Yuanyuan Meng

doi:10.3934/math.2021408

AIMS Mathematics

2021, Volume 6, Issue 7: 6961-6974. doi: 10.3934/math.2021408

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On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$

Wenpeng Zhang ,
Yuanyuan Meng ^,

School of Mathematics, Northwest University, Xi'an 710127, China

Received: 03 February 2021 Accepted: 19 April 2021 Published: 25 April 2021
MSC : 11L03, 11L05

The main purpose of this article is using the elementary methods and the properties of the character sums of the polynomials to study the calculating problem of one kind sixth power mean of the two-term exponential sums weighted by Legendre's symbol modulo $p$ , an odd prime, and give an interesting calculating formula for it.

Keywords:

Citation: Wenpeng Zhang, Yuanyuan Meng. On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$ [J]. AIMS Mathematics, 2021, 6(7): 6961-6974. doi: 10.3934/math.2021408

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Abstract

1. Introduction

In the past decade, many researchers have studied many fractional partial differential equations (FPDEs) types. Since the beginning of fractional calculus history in 1695, when L'Hospital raised the question: what is the meaning of $\frac{{d}^{n}y}{{dx}^{n}}$ if $n = \frac{1}{2}$ ? That is, what if n is fractional? ^[1] Even for new researchers, the fractional derivatives were complicated-although it appears in many parts of sciences such as physics, engineering, bioengineering, COVID-19 studies, and many other branches of sciences ^{[2,3,4,5,6,7,8,9]}. In addition, many definitions of fractional derivatives have been given ^[10,11]. Fractional order derivatives of a given function involve the entire function history where the following state of a fractional order system is dependent on its current state and all its historical states ^{[4,5,6,7,8,9,10,11]}.

There are several analytical and numerical techniques for handling fractional problems, B-spline functions, Bernoulli polynomials, Adomian decomposition, variational iteration, Homotopy analysis, and many others ^{[12,13,14,15,16,17]}. On top of that, some applicable analytical methods are developed to address nonlinear problems with fractional derivatives. One of these approaches is the Laplace residual power series method (LRPSM) ^{[18,19,20,21,22,23,24,25,26,27,28,29]}, which shows its efficiency and applicability in solving nonlinear problems.

The LRPSM is a very modern technique, and it is a hybrid method of two approaches, the Laplace transform (LT) and the idea of the residual power series method (RPSM) ^{[30,31,32,33]}. In 2020, the authors in the article ^[18] were able to adapt the LT to solve nonlinear neutral fractional pantograph equations using the residual function and the RPSM idea. The LT, usually, is implemented to solve linear equations only, but the LRPSM can overcome this disadvantage and thus adapts it to solve nonlinear equations of different types. The LRPSM presents an approximate analytical solution with a series form using the concept of the Laurent series and the power series ^{[18,19,20,21,22,23,24,25]}. What distinguishes LRPSM from RPSM is the use of the idea of limit at infinity in getting the coefficients of a series solution rather than the concept of a fractional derivative as in RPSM. Many articles used the proposed method to treat several types of differential equations of fractional orders. In 2021, El-Ajou adapted LRPSM to establish solitary solutions of nonlinear dispersive FPDEs ^[19] and to present series solutions for systems of Caputo FPDEs with variable coefficients ^[20]. Newly, the LRPSM is used for solving Fuzzy Quadratic Riccati Differential Equations ^[21], time-fractional nonlinear water wave PDE ^[22], fractional Lane-Emden equations ^[23], Fisher's equation and logistic system model ^[24], and nonlinear fractional reaction-diffusion for bacteria growth model ^[25].

Claude Louis Navier and Gabriel Stokes have created the so-called Navier-Stokes equations (NSEs). A French mechanical engineer Claude was affiliated in continuum mechanics with a physicist specializing and the French government, whose main contribution was the Navier-Stokes equations (1822). This famous equation made his name among the several names incised on the Eiffel Tower. Moreover, Newton's second law for fluid substance, which is central to fluid mechanics has been used in describing many physical phenomena in many applied sciences ^{[34,35,36,37]}. For example, the study of airflow around a wing and water flow in pipes and used as one of the continuity equations needed to build microscopic models in 1985 and also as a special case considered to establish the relationship between external and pressure forces on the fluid to the responses of fluid flow ^[38].

The motivation of this work is to adapt the LRPSM to provide analytical solutions for a multi-dimensional time-fractional Navier-Stokes (M-DT-FNS) system which takes the following form ^[39]:

${D}_{t}^{\alpha }u+\left(u\cdot \nabla \right)u = \upsilon {\nabla }^{2}u-\frac{1}{\rho }\nabla p, \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }0 < \alpha \le 1,$

(1.1)

where ${D}_{t}^{\alpha }$ is the Caputo fractional-derivative operator of order $\alpha$ , $p = p\left(\chi, \varsigma, \zeta, t\right)$ is the pressure, $\rho$ is the density, $u$ is a vector field that represents the flow velocity vector, $\upsilon = \frac{\mu }{\rho }$ is the kinematic viscosity ( $\mu$ is the dynamic viscosity), and $\nabla$ & ${\nabla }^{2}$ are the gradient and Laplacian operators, respectively, subject to the initial conditions (ICs) at the initial velocity:

$u = \varphi .$

(1.2)

If the density is constant throughout the fluid domain, then the vector Eq (1.1) is an incompressible NSEs.

The vector Eqs (1.1) and (1.2) can be separated in a system form as follows ^[39,40]:

$\begin{array}{l} {D}_{t}^{\alpha }{u}_{1}+{u}_{1}\frac{\partial {u}_{1}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{1}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{1}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial p}{\partial \chi }, \\ {D}_{t}^{\alpha }{u}_{2}+{u}_{1}\frac{\partial {u}_{2}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{2}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{2}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{2}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial p}{\partial \varsigma }, \\ {D}_{t}^{\alpha }{u}_{3}+{u}_{1}\frac{\partial {u}_{3}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{3}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{3}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{3}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial p}{\partial \zeta }, \end{array}$

(1.3)

subject to the initial conditions (ICs):

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , \zeta , 0\right) = f\left(\chi , \varsigma , \zeta \right) = f, \\ {u}_{2}\left(\chi , \varsigma , \zeta , 0\right) = h\left(\chi , \varsigma , \zeta \right) = h, \\ {u}_{3}\left(\chi , \varsigma , \zeta , 0\right) = g\left(\chi , \varsigma , \zeta \right) = g, \end{array}$

(1.4)

where $u = \left\langle{{u}_{1}, {u}_{2}, {u}_{3}}\right\rangle$ and $\varphi = \left\langle{f, h, g}\right\rangle$ such that ${u}_{1}$ , ${u}_{2}$ , ${u}_{3}$ , and $p$ are analytical functions of four variables $\chi, \varsigma, \zeta \ \&\ t$ .

In this equation, the solution represents the fluid velocity and pressure. It is commonly used to describe the motion of fluids in models relevant to weather, ocean currents, water flow in pipes, etc.

The novelty of this study is obvious in the proposed method when dealing with the Navier-Stokes problem, we show the simplicity and the applicability of the method, and we mention also that the method needs no differentiation, linearization, or discretization, the only mathematical step we need after taking the LT and defining the residual functions, is taking the limit at infinity which is much easier compared to other analytical techniques. Moreover, in this research, we obtain a general formula of the solution that neither researcher has, allowing us to compute as many possible terms of the series solution directly.

This study is prepared as follows: After the introduction section, a few fundamental principles and theories are reviewed for constructing an analytic series solution to the M-DT-FNS system using LRPSM. In Section 3, we constructed a Laplace residual power series (LRPS) solution to the goal problem. Three interesting examples are presented to explain the technique's simplicity and accuracy, which are displayed in Section 4. Finally, some conclusions are made about the features of the method used and its applicability in solving other types of problems.

2. Basic concepts

This part presents fundamental definitions and properties of fractional operators and power series.

Definition 2.1. ^[1] The time Caputo fractional-derivative of order $\mathrm{\alpha }$ of the multivariable function $u\left(\chi, \varsigma, \zeta, t\right)$ , is defined by

${D}_{t}^{\alpha }\left[u\left(\chi , \varsigma , \zeta , t\right)\right] = \left\{\begin{array}{c}\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\int }_{0}^{t}\frac{\frac{{\partial }^{n}}{\partial {\xi }^{n}}u\left(\chi , \varsigma , \zeta , \xi \right)}{{\left(t-\xi \right)}^{\alpha +1-n}}d\xi ,\ \ n-1 < \alpha < n, \\ \frac{{\partial }^{n}}{\partial {t}^{n}}u\left(\chi , \varsigma , \zeta , t\right), \ \ \alpha = n.\end{array}\right.$

(2.1)

Lemma 2.1. ^[1] The operator ${D}_{t}^{\alpha }$ , $t\ge 0$ , $n-1 < \alpha \le n$ satisfies the following properties:

1) ${D}_{t}^{\alpha }c = 0$ , $c\in \mathbb{R}.$

2) ${D}_{t}^{\alpha }{t}^{q} = \left\{\begin{array}{c}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ , \ \ q < \alpha, q\ {\rm{is}} \ {\rm{integer}}\\ \frac{\mathrm{\Gamma }\left(q+1\right)}{\mathrm{\Gamma }\left(q-\alpha +1\right)}{t}^{q-\alpha },\ \ {\rm{otherwise}}\end{array}, \right.$ for $q > -1.$

Definition 2.2. ^[18] The time-LT for the multivariable function $u(\chi, \varsigma, \zeta, t)$ is defined by:

$U\left(\chi , \varsigma , \zeta , s\right) = \mathcal{L}\left[u\left(\chi , \varsigma , \zeta , t\right)\right] = \int_{0}^{\infty } {e}^{-st}u\left(\chi , \varsigma , \zeta , t\right)dt, s > \delta .$

(2.2)

We denote the inverse LT of the function $U(\chi, \upsilon, \zeta, s)$ and define it as

$u\left(\chi , \varsigma , \zeta , t\right) = {\mathcal{L}}^{-1}\left[U\left(\chi , \varsigma , \zeta , s\right)\right] = \int_{{z-i\mathrm{\infty }}}^{{{z+i\mathrm{\infty }}}}{e}^{st}U\left(\chi , \varsigma , \zeta , s\right)ds;z = Re\left(s\right) > {z}_{0}.$

(2.3)

The most popular properties of the LT are mentioned below.

Lemma 2.2. ^[18,19] Assume that $U\left(\chi, \varsigma, \zeta, s\right) = \mathcal{L}\left[u\left(\chi, \varsigma, \zeta, t\right)\right]$ and $V\left(\chi, \varsigma, \zeta, s\right) = \mathcal{L}\left[v\left(\chi, \varsigma, \zeta, t\right)\right]$ . Then

1) $\underset{s\to \mathrm{\infty }}{\mathrm{lim}}sU\left(\chi, \varsigma, \zeta, s\right) = u\left(\chi, \varsigma, \zeta, 0\right).$

2) $\mathcal{L}\left[a\frac{{t}^{i\alpha }}{\mathrm{\Gamma }\left(i\alpha +1\right)}\right] = \frac{a}{{s}^{i\alpha +1}}, a\in \mathbb{R}, i = 0, \mathrm{ }1, \mathrm{ }\cdots.$

3) $\mathcal{L}\left[{D}_{t}^{m\alpha }u\left(\chi, \varsigma, \zeta, t\right)\right] = {s}^{m\alpha }U\left(\chi, \varsigma, \zeta, s\right)-\sum _{n = 0}^{m-1}{s}^{\left(m-n\right)\alpha -1}{D}_{t}^{n\alpha }u\left(\chi, \varsigma, \zeta, 0\right), \mathrm{ }0 < \alpha \le 1,$

where ${D}_{t}^{m\alpha } = {D}_{t}^{\alpha }{D}_{t}^{\alpha }\dots {D}_{t}^{\alpha }$ ( $m$ -times).

Theorem 2.1. ^[19] The power series expansion of the multivariable function $u(\chi, \varsigma, \zeta, t)$ can be expressed as

$u\left(\chi , \varsigma , \zeta , t\right) = \sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){t}^{i\alpha }}{\mathrm{\Gamma }\left(i\alpha +1\right)},$

(2.4)

where ${f}_{i}\left(\chi, \varsigma, \zeta \right) = {D}_{t}^{i\alpha }u(\chi, \varsigma, \zeta, 0)$ , $i = 0, \mathrm{ }1, \mathrm{ }\cdots$ .

Thus, the LT of Eq (2.4) is a Laurent expansion in the Laplace space of the following form:

$u\left(\chi , \varsigma , \zeta , t\right) = \sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){t}^{i\alpha }}{\mathrm{\Gamma }\left(i\alpha +1\right)}.$

(2.5)

Theorem 2.2. ^[19] Assume that

$\left|s\mathcal{L}\left[{{D}_{t}}^{(m+1)\alpha }u\left(\chi , \varsigma , \zeta , t\right)\right]\right|\le M,$

on $0\le s\le q, \mathrm{ }0 < \alpha \le 1$ and $M = M\left(\chi, \varsigma, \zeta \right)$ for some $\chi$ , $\varsigma$ , and $\zeta \in I$ . Then the remainder ${R}_{m}(\chi, \varsigma, \zeta, s)$ of the new fractional Laurent series (2.5) satisfies the following inequality

$\left|{R}_{m}(\chi , \varsigma , \zeta , s)\right|\le \frac{M}{{s}^{\left(m+1\right)\alpha +1}}.$

It is known that the LT cannot be distributed in the case of multiplication. Therefore, the following Lemma is introduced to simplify the calculations at the application of LRPSM, based on the characteristics of the powers of the power series.

Lemma 2.3. Assume that $U\left(\chi, \varsigma, \zeta, s\right) = \mathcal{L}\left[u\left(\chi, \varsigma, \zeta, t\right)\right]\left(s\right)$ and $V\left(\chi, \varsigma, \zeta, s\right) = \mathcal{L}\left[v\left(\chi, \varsigma, \zeta, t\right)\right]\left(s\right)$ . Assume that the functions $U\left(\chi, \varsigma, \zeta, s\right)$ and $V\left(\chi, \varsigma, \zeta, s\right)$ have Laurent expansions as:

$\begin{array}{l} \left(\chi , \upsilon , \zeta , s\right) = \sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \upsilon , \zeta \right)}{{s}^{i\alpha +1}}, \\ V\left(\chi , \varsigma , \zeta , s\right) = \sum _{j = 0}^{\mathrm{\infty }}\frac{{g}_{j}\left(\chi , \varsigma , \zeta \right)}{{s}^{j\alpha +1}}. \end{array}$

(2.6)

Then, $\mathcal{L}\left[u\left(\chi, \varsigma, \zeta, t\right)v\left(\chi, \varsigma, \zeta, t\right)\right]\left(s\right)$ can be expanded in a Laurent series form, as follows

$\mathcal{L}\left[u\left(\chi , \varsigma , \zeta , t\right)v\left(\chi , \varsigma , \zeta , t\right)\right]\left(s\right) = \sum _{i = 0}^{\mathrm{\infty }}\sum _{j = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){g}_{j}\left(\chi , \varsigma , \zeta \right)\mathrm{\Gamma }(\left(i+j\right)\alpha +1)}{\mathrm{\Gamma }(i\alpha +1)\mathrm{\Gamma }(j\alpha +1){s}^{(i+j)\alpha +1}}.$

(2.7)

Proof. Directly, we can prove the Lemma as follows:

$\mathcal{L}\left[u\left(\chi , \varsigma , \zeta , t\right)v\left(\chi , \varsigma , \zeta , t\right)\right]\left(s\right) = \mathcal{L}\left[{\mathcal{L}}^{-1}\left[U\left(\chi , \varsigma , \zeta , s\right)\right]{\mathcal{L}}^{-1}\left[V\left(\chi , \varsigma , \zeta , s\right)\right]\right]\left(s\right)$

$= \mathcal{L}\left[{\mathcal{L}}^{-1}\left[\sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right)}{{s}^{i\alpha +1}}\right]{\mathcal{L}}^{-1}\left[\sum _{j = 0}^{\mathrm{\infty }}\frac{{g}_{i}\left(\chi , \varsigma , \zeta \right)}{{s}^{j\alpha +1}}\right]\right]\left(s\right)$

$= \mathcal{L}\left[\sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){t}^{i\alpha }}{\mathrm{\Gamma }\left(i\alpha +1\right)}\sum _{j = 0}^{\mathrm{\infty }}\frac{{g}_{i}\left(\chi , \varsigma , \zeta \right){t}^{j\alpha }}{\mathrm{\Gamma }\left(j\alpha +1\right)}\right]\left(s\right)$

$= \mathcal{L}\left[\sum _{i = 0}^{\mathrm{\infty }}\sum _{j = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){g}_{j}\left(\chi , \varsigma , \zeta \right){t}^{\left(i+j\right)\mathrm{\alpha }}}{\mathrm{\Gamma }\left(i\alpha +1\right)\mathrm{\Gamma }\left(j\alpha +1\right)}\right]\left(s\right)$

$= \sum _{i = 0}^{\mathrm{\infty }}\sum _{j = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){g}_{j}\left(\chi , \varsigma , \zeta \right)\mathrm{\Gamma }(\left(i+j\right)\alpha +1)}{\mathrm{\Gamma }(i\alpha +1)\mathrm{\Gamma }(j\alpha +1){s}^{(i+j)\alpha +1}}.$

It is noteworthy, that we can express the $k$ th-truncated series of the Laurent series (2.7) as follows:

$\mathcal{L}\left[{u}_{k}\left(\chi , \varsigma , \zeta , t\right){v}_{k}\left(\chi , \varsigma , \zeta , t\right)\right]\left(s\right) = \sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right){g}_{j}\left(\chi , \varsigma , \zeta \right)\mathrm{\Gamma }(\left(i+j\right)\alpha +1)}{\mathrm{\Gamma }(i\alpha +1)\mathrm{\Gamma }(j\alpha +1){s}^{(i+j)\alpha +1}},$

(2.8)

which we will use extensively throughout our work on the next pages.

3. The Laplace residual power series method

We employ the LRPSM to establish a series solution for the M-DT-FNS system (1.1) in this part of this article. This technique is mainly based on applying the LT on the target equations, assuming solutions of the generated equations have Laurent expansions, and then using the idea of the limit at infinity with the residual functions to get the unknown coefficients in expansions. Finally, we run the inverse LT to obtain the solution of the given equations in the original space.

To get the LRPS solution of the system (1.1), we first apply the LT to each equation in the system (1.1) and use the third part of Lemma 2.2 with the ICs (1.2). Then, after some simplification, we get the following algebraic system in Laplace:

$\begin{array}{l} {U}_{1}-\frac{f}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{1}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{1}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{1}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{1}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \chi } = 0, \\ {U}_{2}-\frac{h}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{2}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{2}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{2}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{2}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \varsigma } = 0, \\ {U}_{3}-\frac{g}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{3}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{3}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{3}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{3}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \zeta } = 0, \end{array}$

(3.1)

where ${U}_{1} = \mathcal{L}\left[{u}_{1}\right]$ , ${U}_{2} = \mathcal{L}\left[{u}_{2}\right]$ , ${U}_{3} = \mathcal{L}\left[{u}_{3}\right]$ and $P = \mathcal{L}\left[p\right]$ .

Suppose that the solution of the system (3.1), ${U}_{1}$ , ${U}_{2}$ , and ${U}_{3}$ , has the following fractional Laurent expansions as follows:

$\begin{array}{l} {U}_{1} = \sum _{i = 0}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right)}{{s}^{i\alpha +1}}, \\ {U}_{2} = \sum _{j = 0}^{\mathrm{\infty }}\frac{{h}_{j}\left(\chi , \varsigma , \zeta \right)}{{s}^{j\alpha +1}}, \\ {U}_{3} = \sum _{r = 0}^{\mathrm{\infty }}\frac{{g}_{r}\left(\chi , \varsigma , \zeta \right)}{{s}^{r\alpha +1}}. \end{array}$

(3.2)

Assume that also $P$ has the following Laurent expansion:

$P = \frac{\varphi }{s}+\sum _{i = 1}^{\mathrm{\infty }}\frac{{\varphi }_{i}}{{s}^{i\alpha +1}}.$

(3.3)

According to Theorem 2.1 and using the ICs (1.2), the expansions in Eq (3.2) can be rewritten as follows:

$\begin{array}{l} {U}_{1} = \frac{f}{s}+\sum _{i = 1}^{\mathrm{\infty }}\frac{{f}_{i}\left(\chi , \varsigma , \zeta \right)}{{s}^{i\alpha +1}}, \\ {U}_{2} = \frac{h}{s}+\sum _{i = 1}^{\mathrm{\infty }}\frac{{h}_{i}\left(\chi , \varsigma , \zeta \right)}{{s}^{i\alpha +1}}, \\ {U}_{3} = \frac{g}{s}+\sum _{r = 1}^{\mathrm{\infty }}\frac{{g}_{r}\left(\chi , \varsigma , \zeta \right)}{{s}^{r\alpha +1}}. \end{array}$

(3.4)

The $k$ th-truncated series of the expansions in Eqs (3.3) and (3.4) are given by:

$\begin{array}{l} {U}_{1, k} = \frac{f}{s}+\sum _{i = 1}^{k}\frac{{f}_{i}}{{s}^{i\alpha +1}}, \\ {U}_{2, k} = \frac{h}{s}+\sum _{j = 1}^{k}\frac{{h}_{j}}{{s}^{j\alpha +1}}, \\ {U}_{3, k} = \frac{g}{s}+\sum _{r = 1}^{k}\frac{{g}_{r}}{{s}^{r\alpha +1}}, \\ {P}_{k} = \frac{\varphi }{s}+\sum _{i = 1}^{k}\frac{{\varphi }_{i}}{{s}^{i\alpha +1}}. \end{array}$

(3.5)

To find the coefficients in the series expansions of Eq (3.4), we establish the Laplace residual functions ( $\mathcal{L}$ RF) of the equations in the system (3.1) as follows:

$\begin{array}{l} \mathcal{L}Re{s}_{1}\left(s\right) = {U}_{1}-\frac{f}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{1}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{1}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{1}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{1}}{\partial \chi }+\frac{{\partial }^{2}{U}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{1}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \chi }, \\ \mathcal{L}Re{s}_{2}\left(s\right) = {U}_{2}-\frac{h}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{2}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{2}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{2}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{2}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \varsigma }, \\ \mathcal{L}Re{s}_{3}\left(s\right) = {U}_{3}-\frac{g}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1}\frac{\partial {u}_{3}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2}\frac{\partial {u}_{3}}{\partial \varsigma }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3}\frac{\partial {u}_{3}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{3}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial P}{\partial \zeta }, \end{array}$

(3.6)

and the kth- $\mathcal{L}$ RF for $k = \mathrm{1, 2}, \dots$ , as follows:

$\begin{array}{l} \mathcal{L}Re{s}_{1, k}\left(s\right) = {U}_{1, k}-\frac{f}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1, k}\frac{\partial {u}_{1, k}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2, k}\frac{\partial {u}_{1, k}}{\partial \varsigma }\right] \\ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3, k}\frac{\partial {u}_{1, k}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{1, k}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{1, k}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{1, k}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial {P}_{k}}{\partial \chi }, \\ \mathcal{L}Re{s}_{2, k}\left(s\right) = {U}_{2, k}-\frac{h}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1, k}\frac{\partial {u}_{2, k}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2, k}\frac{\partial {u}_{2, k}}{\partial \varsigma }\right]\\ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3, k}\frac{\partial {u}_{2, k}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{2, k}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{2, k}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{2, k}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial {P}_{k}}{\partial \varsigma }, \\ \mathcal{L}Re{s}_{3, k}\left(s\right) = {U}_{3, k}-\frac{g}{s}+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{1, k}\frac{\partial {u}_{3, k}}{\partial \chi }\right]+\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{2, k}\frac{\partial {u}_{3, k}}{\partial \varsigma }\right] \\ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{{s}^{\alpha }}\mathcal{L}\left[{u}_{3, k}\frac{\partial {u}_{3, k}}{\partial \zeta }\right]-\frac{\upsilon }{{s}^{\alpha }}\left(\frac{{\partial }^{2}{U}_{3, k}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{U}_{3, k}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{U}_{3, k}}{\partial {\zeta }^{2}}\right)+\frac{1}{{\rho s}^{\alpha }}\frac{\partial {P}_{k}}{\partial \zeta }. \end{array}$

(3.7)

Using Lemma 2.3 and substituting the expansions in Eq (3.5) into Eq (3.7) gives the Laurent series form to the kth- $\mathcal{L}$ RFs as follows:

$\begin{array}{l} \mathcal{L}Re{s}_{1, \mathrm{k}}\left(s\right) = \sum _{i = 1}^{k}\frac{{f}_{i}}{{s}^{i\alpha +1}}+\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){f}_{i}\frac{\partial {f}_{j}}{\partial \chi }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}}\\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){h}_{i}\frac{\partial {f}_{j}}{\partial \varsigma }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}} \\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){g}_{i}\frac{\partial {f}_{j}}{\partial \zeta }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}f}{\partial {\chi }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{f}_{i}}{\partial {\chi }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}\\ -\frac{\upsilon \frac{{\partial }^{2}f}{\partial {\varsigma }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{f}_{i}}{\partial {\varsigma }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}f}{\partial {\zeta }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{f}_{i}}{\partial {\zeta }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}+\frac{\frac{\partial \varphi }{\partial \chi }}{{\rho s}^{\alpha +1}}\\ +\sum _{i = 1}^{k}\frac{\frac{\partial {\varphi }_{i}}{\partial \chi }}{{\rho s}^{\left(i+1\right)\alpha +1}}, \\ \mathcal{L}Re{s}_{2, \mathrm{k}}\left(s\right) = \sum _{i = 1}^{k}\frac{{h}_{i}}{{s}^{i\alpha +1}}+\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){f}_{i}\frac{\partial {h}_{j}}{\partial \chi }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}} \\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){h}_{i}\frac{\partial {h}_{j}}{\partial \varsigma }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}} \\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){g}_{i}\frac{\partial {h}_{j}}{\partial \zeta }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}h}{\partial {\chi }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{h}_{i}}{\partial {\chi }^{2}}}{{s}^{\left(i+1\right)\alpha +1}} \\ -\frac{\upsilon \frac{{\partial }^{2}h}{\partial {\varsigma }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{h}_{i}}{\partial {\varsigma }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}h}{\partial {\zeta }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{h}_{i}}{\partial {\zeta }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}+\frac{\frac{\partial }{\partial \varsigma }\varphi }{{\rho s}^{\alpha +1}} \\ +\sum _{i = 1}^{k}\frac{\frac{\partial {\varphi }_{i}}{\partial \varsigma }}{{\rho s}^{\left(i+1\right)\alpha +1}}, \\ \mathcal{L}Re{s}_{3, \mathrm{k}}\left(s\right) = \sum _{i = 1}^{k}\frac{{g}_{i}}{{s}^{i\alpha +1}}+\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){f}_{i}\frac{\partial {g}_{j}}{\partial \chi }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}}\\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){h}_{i}\frac{\partial {g}_{j}}{\partial \varsigma }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}} \\ +\sum _{i = 0}^{k}\sum _{j = 0}^{k}\frac{\mathit{\Gamma } \left(\left(i+j\right)\alpha +1\right){g}_{i}\frac{\partial {g}_{j}}{\partial \zeta }}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(j\alpha +1\right){s}^{\left(i+j+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}h}{\partial {\chi }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{g}_{i}}{\partial {\chi }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}\\ -\frac{\upsilon \frac{{\partial }^{2}g}{\partial {\varsigma }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{g}_{i}}{\partial {\varsigma }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}-\frac{\upsilon \frac{{\partial }^{2}g}{\partial {\zeta }^{2}}}{{s}^{\alpha +1}}-\sum _{i = 1}^{k}\frac{\upsilon \frac{{\partial }^{2}{g}_{i}}{\partial {\zeta }^{2}}}{{s}^{\left(i+1\right)\alpha +1}}+\frac{\frac{\partial \varphi }{\partial \zeta }}{{\rho s}^{\alpha +1}}\\ +\sum _{i = 1}^{k}\frac{\frac{\partial {\varphi }_{i}}{\partial \zeta }}{{\rho s}^{(i+1)\alpha +1}}. \end{array}$

(3.8)

El-Ajou in ^[19] proved the validity of the following formula, which is considered the essential tool of LRPSM:

$\underset{\mathrm{s}\to \mathrm{\infty }}{\mathrm{lim}}{s}^{k\alpha +1}\mathcal{L}Re\mathrm{s}\left(s\right) = \underset{\mathrm{s}\to \mathrm{\infty }}{\mathrm{lim}}{s}^{k\alpha +1}\mathcal{L}Re{s}_{\mathrm{k}}\left(s\right) = 0, k = \mathrm{1, 2}, 3, \dots .$

(3.9)

To find the first coefficients in the expansions in Eq (3.4), consider the first truncated series of the Laurent expansions of Eq (3.5) as follows:

$\begin{array}{l} {U}_{\mathrm{1, 1}} = \frac{f}{s}+\frac{{f}_{1}}{{s}^{\alpha +1}}, \\ {U}_{\mathrm{2, 1}} = \frac{h}{s}+\frac{{h}_{1}}{{s}^{\alpha +1}}, \\ {U}_{\mathrm{3, 1}} = \frac{g}{s}+\frac{{g}_{1}}{{s}^{\alpha +1}}, \\ {P}_{1} = \frac{\varphi }{s}+\frac{{\varphi }_{1}}{{s}^{\alpha +1}}. \end{array}$

(3.10)

Substituting Eq (3.10) into the first- $\mathcal{L}$ Res, multiplying each equation by ${s}^{\alpha +1}$ , and taking the limit as $s\to \mathrm{\infty },$ we have truth in Eq (3.9):

$\begin{array}{l} \underset{\mathrm{s}\to \mathrm{\infty }}{\mathrm{lim}}{s}^{\alpha +1}\mathcal{L}Re{s}_{\mathrm{1, 1}}\left(s\right) = {f}_{1}+f\frac{\partial f}{\partial \chi }+h\frac{\partial f}{\partial \varsigma }+g\frac{\partial f}{\partial \zeta }-\upsilon \frac{{\partial }^{2}f}{\partial {\chi }^{2}}-\upsilon \frac{{\partial }^{2}f}{\partial {\varsigma }^{2}}-\upsilon \frac{{\partial }^{2}f}{\partial {\zeta }^{2}}+\frac{1}{\rho }\frac{\partial \varphi }{\partial \chi } = 0, \\ \underset{s\to \mathrm{\infty }}{\mathrm{lim}}{s}^{\alpha +1}\mathcal{L}Re{s}_{\mathrm{2, 1}}\left(s\right) = {h}_{1}+f\frac{\partial h}{\partial \chi }+h\frac{\partial h}{\partial \varsigma }+g\frac{\partial h}{\partial \zeta }-\upsilon \frac{{\partial }^{2}h}{\partial {\chi }^{2}}-\upsilon \frac{{\partial }^{2}h}{\partial {\varsigma }^{2}}-\upsilon \frac{{\partial }^{2}h}{\partial {\zeta }^{2}}+\frac{1}{\rho }\frac{\partial \varphi }{\partial \varsigma } = 0, \\ \underset{s\to \mathrm{\infty }}{\mathrm{lim}}{s}^{\alpha +1}\mathcal{L}Re{s}_{\mathrm{3, 1}}\left(\mathrm{s}\right) = {g}_{1}+f\frac{\partial g}{\partial \chi }+h\frac{\partial g}{\partial \varsigma }+g\frac{\partial g}{\partial \zeta }-\rho \frac{{\partial }^{2}g}{\partial {\chi }^{2}}-\rho \frac{{\partial }^{2}g}{\partial {\varsigma }^{2}}-\rho \frac{{\partial }^{2}g}{\partial {\zeta }^{2}}+\frac{1}{\rho }\frac{\partial \varphi }{\partial \zeta } = 0. \end{array}$

(3.11)

Therefore, the first coefficients of the expansions of Eq (3.4) become known and take the following forms:

$\begin{array}{l} {f}_{1} = \upsilon \left(\frac{\partial f}{\partial {\chi }^{2}}+\frac{\partial f}{\partial {\varsigma }^{2}}+\frac{\partial f}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial \varphi }{\partial \chi }-\left(f\frac{\partial f}{\partial \chi }+h\frac{\partial f}{\partial \varsigma }+g\frac{\partial f}{\partial \zeta }\right), \\ {h}_{1} = \upsilon \left(\frac{\partial h}{\partial {\chi }^{2}}+\frac{\partial h}{\partial {\varsigma }^{2}}+\frac{\partial h}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial \varphi }{\partial \varsigma }-\left(f\frac{\partial h}{\partial \chi }+h\frac{\partial h}{\partial \varsigma }+g\frac{\partial h}{\partial \zeta }\right), \\ {g}_{1} = \upsilon \left(\frac{\partial g}{\partial {\chi }^{2}}+\frac{\partial g}{\partial {\varsigma }^{2}}+\frac{\partial g}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial \varphi }{\partial \zeta }-\left(f\frac{\partial g}{\partial \chi }+h\frac{\partial g}{\partial \varsigma }+g\frac{\partial g}{\partial \zeta }\right). \end{array}$

(3.12)

Now, to obtain the second coefficients of the series expansions in Eq (3.4), we substitute the following second truncated series:

$\begin{array}{l} {U}_{\mathrm{1, 2}} = \frac{f}{s}+\frac{{f}_{1}}{{s}^{\alpha +1}}+\frac{{f}_{2}}{{s}^{2\alpha +1}}, \\ {U}_{\mathrm{2, 2}} = \frac{h}{s}+\frac{{h}_{1}}{{s}^{\alpha +1}}+\frac{{h}_{2}}{{s}^{2\alpha +1}}, \\ {U}_{\mathrm{3, 2}} = \frac{g}{s}+\frac{{g}_{1}}{{s}^{\alpha +1}}+\frac{{g}_{2}}{{s}^{2\alpha +1}}, \\ {P}_{2} = \frac{\varphi }{s}+\frac{{\varphi }_{1}}{{s}^{\alpha +1}}+\frac{{\varphi }_{2}}{{s}^{2\alpha +1}}, \end{array}$

(3.13)

into the 2nd- $\mathcal{L}$ RF, multiply each equation by ${s}^{2\alpha +1}$ , and use Eq (3.9), we obtain the forms of required coefficients as follows:

$\begin{array}{l} {f}_{2} = \upsilon \left(\frac{{\partial }^{2}{f}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{f}_{1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{1}}{\partial \chi }-\left({f}_{1}\frac{\partial f}{\partial \chi }+f\frac{\partial {f}_{1}}{\partial \chi }+{h}_{1}\frac{\partial f}{\partial \varsigma }+h\frac{\partial {f}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial f}{\partial \zeta }+g\frac{\partial {f}_{1}}{\partial \zeta }\right), \\ {h}_{2} = \upsilon \left(\frac{{\partial }^{2}{h}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{h}_{1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{1}}{\partial \varsigma }-\left({f}_{1}\frac{\partial h}{\partial \chi }+f\frac{\partial {h}_{1}}{\partial \chi }+{h}_{1}\frac{\partial h}{\partial \varsigma }+h\frac{\partial {h}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial h}{\partial \zeta }+g\frac{\partial {h}_{1}}{\partial \zeta }\right), \\ {g}_{2} = \upsilon \left(\frac{{\partial }^{2}{g}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{g}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{g}_{1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{1}}{\partial \zeta }-\left({f}_{1}\frac{\partial g}{\partial \chi }+f\frac{\partial {g}_{1}}{\partial \chi }+{h}_{1}\frac{\partial g}{\partial \varsigma }+h\frac{\partial {g}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial g}{\partial \zeta }+g\frac{\partial {g}_{1}}{\partial \zeta }\right). \end{array}$

(3.14)

We can repeat the previous steps to find the third and the fourth coefficients of the expansions (3.4) as:

$\begin{array}{l} {f}_{3} = \upsilon \left(\frac{{\partial }^{2}{f}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{f}_{2}}{\partial {\zeta }^{2}}\right)+\frac{1}{\rho }\frac{\partial {\varphi }_{2}}{\partial \chi }\\ \ \ \ \ \ -\left({f}_{2}\frac{\partial f}{\partial \chi }+f\frac{\partial {f}_{2}}{\partial \chi }+{h}_{2}\frac{\partial f}{\partial \varsigma }+h\frac{\partial {f}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial f}{\partial \zeta }+g\frac{\partial {f}_{2}}{\partial \zeta }\right) \\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }}^{2}\left(\alpha +1\right)}\left({f}_{1}\frac{\partial {f}_{1}}{\partial \chi }+{h}_{1}\frac{\partial {f}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial {f}_{1}}{\partial \zeta }\right), \\ {h}_{3} = \upsilon \left(\frac{{\partial }^{2}{h}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{h}_{2}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{2}}{\partial \varsigma } \\ \ \ \ \ \ -\left({f}_{2}\frac{\partial h}{\partial \chi }+f\frac{\partial {h}_{2}}{\partial \chi }+{h}_{2}\frac{\partial h}{\partial \varsigma }+h\frac{\partial {h}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial h}{\partial \zeta }+g\frac{\partial {h}_{2}}{\partial \zeta }\right)\\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }}^{2}\left(\alpha +1\right)}\left({f}_{1}\frac{\partial {h}_{1}}{\partial \chi }+{h}_{1}\frac{\partial {h}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial {h}_{1}}{\partial \zeta }\right), \\ {g}_{3} = \upsilon \left(\frac{{\partial }^{2}{g}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{g}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{g}_{2}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{2}}{\partial \zeta } \\ \ \ \ \ \ -\left({f}_{2}\frac{\partial g}{\partial \chi }+f\frac{\partial {g}_{2}}{\partial \chi }+{h}_{2}\frac{\partial g}{\partial \varsigma }+h\frac{\partial {g}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial g}{\partial \zeta }+g\frac{\partial {g}_{2}}{\partial \zeta }\right) \\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }}^{2}(\alpha +1)}\left({f}_{1}\frac{\partial {g}_{1}}{\partial \chi }+{h}_{1}\frac{\partial {g}_{1}}{\partial \varsigma }+{g}_{1}\frac{\partial {g}_{1}}{\partial \zeta }\right). \end{array}$

(3.15)

Also, one can get

$\begin{array}{l} {f}_{4} = \upsilon \left(\frac{{\partial }^{2}{f}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{f}_{3}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{3}}{\partial \chi } \\ \ \ \ \ \ -\left({f}_{3}\frac{\partial f}{\partial \chi }+f\frac{\partial {f}_{3}}{\partial \chi }+{h}_{3}\frac{\partial f}{\partial \varsigma }+h\frac{\partial {f}_{3}}{\partial \varsigma }+{g}_{3}\frac{\partial f}{\partial \zeta }+g\frac{\partial {f}_{3}}{\partial \zeta }\right) \\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(3\alpha +1\right)}{\mathrm{\Gamma }\left(2\mathrm{\alpha }+1\right)\mathrm{\Gamma }\left(\alpha +1\right)}\left({f}_{2}\frac{\partial {f}_{1}}{\partial \chi }+{f}_{1}\frac{\partial {f}_{2}}{\partial \chi }+{h}_{2}\frac{\partial {f}_{1}}{\partial \varsigma }+{h}_{1}\frac{\partial {f}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial {f}_{1}}{\partial \zeta }+{g}_{1}\frac{\partial {f}_{2}}{\partial \zeta }\right), \\ {h}_{4} = \upsilon \left(\frac{{\partial }^{2}{h}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{h}_{3}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{3}}{\partial \varsigma } \\ \ \ \ \ \ -\left({f}_{3}\frac{\partial h}{\partial \chi }+f\frac{\partial {h}_{3}}{\partial \chi }+{h}_{3}\frac{\partial h}{\partial \varsigma }+h\frac{\partial {h}_{3}}{\partial \varsigma }+{g}_{3}\frac{\partial h}{\partial \zeta }+g\frac{\partial {h}_{3}}{\partial \zeta }\right) \\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(3\alpha +1\right)}{\mathrm{\Gamma }\left(2\mathrm{\alpha }+1\right)\mathrm{\Gamma }\left(\alpha +1\right)}\left({f}_{2}\frac{\partial {h}_{1}}{\partial \chi }+{f}_{1}\frac{\partial {h}_{2}}{\partial \chi }+{h}_{2}\frac{\partial {h}_{1}}{\partial \varsigma }+{h}_{1}\frac{\partial {h}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial {h}_{1}}{\partial \zeta }+{g}_{1}\frac{\partial {h}_{2}}{\partial \zeta }\right), \\ {g}_{4} = \upsilon \left(\frac{{\partial }^{2}{g}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{g}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{g}_{3}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{3}}{\partial \zeta } \\ \ \ \ \ \ -\left({f}_{3}\frac{\partial g}{\partial \chi }+f\frac{\partial {g}_{3}}{\partial \chi }+{h}_{3}\frac{\partial g}{\partial \varsigma }+h\frac{\partial {g}_{3}}{\partial \varsigma }+{g}_{3}\frac{\partial g}{\partial \zeta }+g\frac{\partial {g}_{3}}{\partial \zeta }\right) \\ \ \ \ \ \ -\frac{\mathrm{\Gamma }\left(3\alpha +1\right)}{\mathrm{\Gamma }\left(2\mathrm{\alpha }+1\right)\mathrm{\Gamma }\left(\alpha +1\right)}\left({f}_{2}\frac{\partial {g}_{1}}{\partial \chi }+{f}_{1}\frac{\partial {g}_{2}}{\partial \chi }+{h}_{2}\frac{\partial {g}_{1}}{\partial \varsigma }+{h}_{1}\frac{\partial {g}_{2}}{\partial \varsigma }+{g}_{2}\frac{\partial {g}_{1}}{\partial \zeta }+{g}_{1}\frac{\partial {g}_{2}}{\partial \zeta }\right) \end{array}$

(3.16)

Now, we can get the form of the unknown $k$ th-coefficients in the expansions of Eq (3.4) as follows:

$\begin{array}{l} {f}_{k} = \upsilon \left(\frac{{\partial }^{2}{f}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{k-1}}{\partial \chi } \\ \ \ \ \ \ \ -\sum _{i = 0}^{k-1}\frac{\mathrm{\Gamma }\left(\left(k-1\right)\alpha +1\right)}{\mathrm{\Gamma }\left(i\alpha +1\right)\mathrm{\Gamma }\left(\left(k-i-1\right)\alpha +1\right)}\left({f}_{i}\frac{\partial {f}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {f}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {f}_{k-i-1}}{\partial \zeta }\right), k = \mathrm{1, 2}, \cdots , \\ {h}_{k} = \upsilon \left(\frac{{\partial }^{2}{h}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{k-1}}{\partial \varsigma } \\ \ \ \ \ \ \ -\sum _{i = 0}^{k-1}\frac{\mathrm{\Gamma }\left(\left(k-1\right)\alpha +1\right)}{\mathrm{\Gamma }\left(i\alpha +1\right)\mathrm{\Gamma }\left(\left(k-i-1\right)\alpha +1\right)}\left({f}_{i}\frac{\partial {h}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {h}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {h}_{\mathrm{k}-i-1}}{\partial \zeta }\right), k = \mathrm{1, 2}, \cdots , \\ {g}_{k} = \upsilon \left(\frac{{\partial }^{2}{g}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{g}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{g}_{k-1}}{\partial {\zeta }^{2}}\right)-\frac{1}{\rho }\frac{\partial {\varphi }_{k-1}}{\partial \zeta }\\ \ \ \ \ \ \ -\sum _{i = 0}^{k-1}\frac{\mathrm{\Gamma }(\left(k-1\right)\alpha +1)}{\mathrm{\Gamma }\left(i\alpha +1\right)\mathrm{\Gamma }\left(\left(k-i-1\right)\alpha +1\right)}\left({f}_{i}\frac{\partial {g}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {g}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {g}_{k-i-1}}{\partial \zeta }\right), k = \mathrm{1, 2}, \cdots . \end{array}$

(3.17)

Thus, the solution of the algebraic system (3.1) can be expressed as:

$\begin{array}{l} {U}_{1}\left(\chi , \varsigma , \zeta , s\right) = \frac{f}{s}+\frac{{f}_{1}}{{s}^{\alpha +1}}+\frac{{f}_{2}}{{s}^{2\alpha +1}}+\dots , \\ {U}_{1}\left(\chi , \varsigma , \zeta , s\right) = \frac{h}{s}+\frac{{h}_{1}}{{s}^{\alpha +1}}+\frac{{h}_{2}}{{s}^{2\alpha +1}}+\dots , \\ {U}_{3}\left(\chi , \varsigma , \zeta , s\right) = \frac{g}{s}+\frac{{g}_{1}}{{s}^{\alpha +1}}+\frac{{g}_{2}}{{s}^{2\alpha +1}}+\dots . \end{array}$

(3.18)

Finally, to get the LRPS solution of the M-DT-FNS systems (1.1) and (1.2) in the original space, we apply the inverse LT on the solution in Eq (3.18), to get

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , \zeta , t\right) = f+\frac{{f}_{1}{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}+\frac{{f}_{2}{t}^{2\alpha }}{\mathrm{\Gamma }\left(2\alpha +1\right)}+\dots , \\ {u}_{2}\left(\chi , \varsigma , \zeta , t\right) = h+\frac{{h}_{1}{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}+\frac{{h}_{2}{t}^{2\alpha }}{\mathrm{\Gamma }\left(2\alpha +1\right)}+\dots , \\ {u}_{3}\left(\chi , \varsigma , \zeta , t\right) = g+\frac{{g}_{1}{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}+\frac{{g}_{2}{t}^{2\alpha }}{\mathrm{\Gamma }\left(2\alpha +1\right)}+\dots . \end{array}$

(3.19)

4. Numerical application

In this section, we present some numerical examples that explain the working mechanism of the LRPSM. Comparisons and graphical illustrations are made to demonstrate the accuracy and efficiency of the technique.

Example 4.1. ^[35,36] Consider the following two-DT-FNSEs:

$\begin{array}{l} {D}_{t}^{\alpha }{u}_{1}+{u}_{1}\frac{\partial {u}_{1}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{1}}{\partial \varsigma } = \upsilon \left(\frac{{\partial }^{2}{u}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{1}}{\partial {\varsigma }^{2}}\right), \\ {D}_{t}^{\alpha }{u}_{2}+{u}_{1}\frac{\partial {u}_{2}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{2}}{\partial \varsigma } = \upsilon \left(\frac{{\partial }^{2}{u}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{2}}{\partial {\varsigma }^{2}}\right), \end{array}$

(4.1)

with the ICs:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , 0\right) = -\mathrm{sin}\left(\chi +\varsigma \right), \\ {u}_{2}\left(\chi , \varsigma , 0\right) = \mathrm{sin}\left(\chi +\varsigma \right), \end{array}$

(4.2)

where $\upsilon \in \mathbb{R}$ , and ${u}_{1}$ and ${u}_{2}$ are two functions of three variables $\chi, \varsigma$ , and $t$ .

Note that, when $\alpha = 1$ the exact solution of the systems (4.1) and (4.2) is

$\begin{array}{l} {u}_{1} = -{e}^{-2\upsilon t}\mathrm{sin}\left(\chi +\varsigma \right), \\ {u}_{2} = {e}^{-2\upsilon t}\mathrm{sin}\left(\chi +\varsigma \right). \end{array}$

(4.3)

Based on the algorithm of the solution obtained in Section 3 and the result in Eq (3.19), we can obtain the LRPS solution of the systems (4.1) and (4.2) as follows:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , t\right) = f+\frac{{f}_{1}{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}+\frac{{f}_{2}{t}^{2\alpha }}{\mathrm{\Gamma }\left(2\alpha +1\right)}+\dots , \\ {u}_{2}\left(\chi , \varsigma , t\right) = h+\frac{{h}_{1}{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}+\frac{{h}_{2}{t}^{2\alpha }}{\mathrm{\Gamma }\left(2\alpha +1\right)}+\dots , \end{array}$

(4.4)

where the coefficients ${f}_{k}$ , ${h}_{k}$ , $k = \mathrm{1, 2}, \dots$ , are given by the general formula obtained in Eq (3.17) which in the two dimensions will be as follows:

$\begin{array}{l} {f}_{k} = \upsilon \left(\frac{{\partial }^{2}{f}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial \varsigma }\right)-\sum _{i = 0}^{k-1}\frac{\mathit{\Gamma } \left(\left(k-1\right)\alpha +1\right)\left({f}_{i}\frac{\partial {f}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {f}_{k-i-1}}{\partial \varsigma }\right)}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(\left(k-i-1\right)\alpha +1\right)}, \\ {h}_{k} = \upsilon \left(\frac{{\partial }^{2}{h}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\varsigma }^{2}}\right)-\sum _{i = 0}^{k-1}\frac{\mathit{\Gamma } \left(\left(k-1\right)\alpha +1\right)\left({f}_{i}\frac{\partial {h}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {fh}_{k-i-1}}{\partial \varsigma }\right)}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(\left(k-i-1\right)\alpha +1\right)}. \end{array}$

(4.5)

Thus, we conclude that

${f}_{1} = 2\upsilon \mathrm{sin}\left(\chi +\varsigma \right), \ \ \ \ \ \ \ {h}_{1} = -2\upsilon \mathrm{sin}\left(\chi +\varsigma \right)$

${f}_{2} = -{\left(2\upsilon \right)}^{2}\mathrm{sin}\left(\chi +\varsigma \right), \ \ \ \ \ \ \ {h}_{2} = {\left(2\upsilon \right)}^{2}\mathrm{sin}\left(\chi +\varsigma \right)$

${f}_{3} = {\left(2\upsilon \right)}^{3}\mathrm{sin}\left(\chi +\varsigma \right), \ \ \ \ \ \ \ {h}_{3} = {-\left(2\upsilon \right)}^{3}\mathrm{sin}\left(\chi +\varsigma \right)$

${f}_{4} = -{\left(2\upsilon \right)}^{4}\mathrm{sin}\left(\chi +\varsigma \right), \ \ \ \ \ \ \ {h}_{4} = {\left(2\upsilon \right)}^{4}\mathrm{sin}\left(\chi +\varsigma \right)$

$\ \ \ \ \ \ \ \ \ \ \ ⋮\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ⋮$

${f}_{k} = {\left(-1\right)}^{k+1}{\left(2\upsilon \right)}^{k}\mathrm{sin}\left(\chi +\varsigma \right), \ \ \ \ \ \ \ {h}_{k} = {\left(-2\upsilon \right)}^{k}\mathrm{sin}\left(\chi +\varsigma \right),\ \ \ \ \ \ \ k = \mathrm{1, 2}, \cdots .$

Hence, we can express the LRPS solution of Eqs (4.1) and (4.2) as:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , t\right) = -\mathrm{sin}\left(\chi +\varsigma \right)\sum _{n = 0}^{\mathrm{\infty }}\frac{{\left(-2\upsilon \right)}^{n}{t}^{n\alpha }}{\mathrm{\Gamma }\left(n\alpha +1\right)}, \\ {u}_{2}\left(\chi , \varsigma , t\right) = \mathrm{sin}\left(\chi +\varsigma \right)\sum _{n = 0}^{\mathrm{\infty }}\frac{{\left(-2\upsilon \right)}^{n}{t}^{n\alpha }}{\mathrm{\Gamma }\left(n\alpha +1\right)}. \end{array}$

(4.6)

shows the stability of the results obtained, where the solution is stable with various values of $t$ .

Table 1. Numerical results for Example 1 at

$\alpha = 1, \chi = 0.1, \varsigma = 0.5$ .

$t$	Numerical results of ${u}_{1}\left(\chi, \varsigma, t\right)$
	${u}_{\mathrm{1, 6}}\left(\chi, \varsigma, t\right)$	${u}_{\mathrm{1, 8}}\left(\chi, \varsigma, t\right)$	${u}_{\mathrm{1, 10}}\left(\chi, \varsigma, t\right)$	${u}_{1}\left(\chi, \varsigma, t\right)$	$\left\|{u}_{1}\left(\chi, \varsigma, t\right)-{u}_{1, 10}\left(\chi, \varsigma, t\right)\right\|$
$0.16$	$\begin{array}{c}-0.481157\end{array}$	$\begin{array}{c}-0.481157\end{array}$	$\begin{array}{c}-0.481157\end{array}$	$-0.481157$	$5.551115\times {10}^{-17}$
$0.32$	$\begin{array}{c}-0.410015\end{array}$	$\begin{array}{c}-0.410015\end{array}$	$\begin{array}{c}-0.410015\end{array}$	$-0.410015$	$4.957146\times {10}^{-14}$
$0.48$	$\begin{array}{c}-0.349392\end{array}$	$\begin{array}{c}-0.349391\end{array}$	$\begin{array}{c}-0.349391\end{array}$	$-0.349391$	$4.238276\times {10}^{-12}$
$0.64$	$\begin{array}{c}-0.297736\end{array}$	$\begin{array}{c}-0.297732\end{array}$	$\begin{array}{c}-0.297731\end{array}$	$-0.297732$	$9.907064\times {10}^{-11}$
$0.80$	$\begin{array}{c}-0.253732\end{array}$	$\begin{array}{c}-0.253710\end{array}$	$\begin{array}{c}-0.253710\end{array}$	$-0.253710$	$1.138797\times {10}^{-9}$
$0.96$	$\begin{array}{c}-0.216273\end{array}$	$\begin{array}{c}-0.216199\end{array}$	$\begin{array}{c}-0.216198\end{array}$	$-0.216198$	$8.355848\times {10}^{-9}$
$t$	Numerical results of ${u}_{2}\left(\chi, \varsigma, t\right)$
	${u}_{\mathrm{2, 6}}\left(\chi, \varsigma, t\right)$	${u}_{\mathrm{2, 8}}\left(\chi, \varsigma, t\right)$	${u}_{\mathrm{2, 10}}\left(\chi, \varsigma, t\right)$	${u}_{2}\left(\chi, \varsigma, t\right)$	$\left\|{u}_{2}\left(\chi, \varsigma, t\right)-{u}_{2, 10}\left(\chi, \varsigma, t\right)\right\|$
$0.16$	$\begin{array}{c}0.481157\end{array}$	$\begin{array}{c}0.481157\end{array}$	$\begin{array}{c}0.481157\end{array}$	$0.481157$	$5.551115\times {10}^{-17}$
$0.32$	$\begin{array}{c}0.410015\end{array}$	$\begin{array}{c}0.410015\end{array}$	$\begin{array}{c}0.410015\end{array}$	$0.410015$	$4.957146\times {10}^{-14}$
$0.48$	$\begin{array}{c}0.349392\end{array}$	$\begin{array}{c}0.349391\end{array}$	$\begin{array}{c}0.349391\end{array}$	$0.349391$	$4.238276\times {10}^{-12}$
$0.64$	$\begin{array}{c}0.297736\end{array}$	$\begin{array}{c}0.297732\end{array}$	$\begin{array}{c}0.297731\end{array}$	$0.297732$	$9.907064\times {10}^{-11}$
$0.80$	$\begin{array}{c}0.253732\end{array}$	$\begin{array}{c}0.253710\end{array}$	$\begin{array}{c}0.253710\end{array}$	$0.253710$	$1.138797\times {10}^{-9}$
$0.96$	$\begin{array}{c}0.216273\end{array}$	$\begin{array}{c}0.216199\end{array}$	$\begin{array}{c}0.216198\end{array}$	$0.216198$	$8.355848\times {10}^{-9}$

| Show Table

DownLoad: CSV

This solution is the same as that obtained by the Laplace decomposition method ^[36] and the variational iteration transform method ^[35]. In a special case, taking $\alpha = 1$ gives the exact solution in terms of elementary functions as follows:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , t\right) = -{e}^{-2\upsilon t}\mathrm{sin}\left(\chi +\varsigma \right), \\ {u}_{2}\left(\chi , \varsigma , t\right) = {e}^{-2\upsilon t}\mathrm{sin}\left(\chi +\varsigma \right). \end{array}$

(4.7)

The behavior of the velocity field of the two-DT-FNSEs (4.1) and (4.2) is depicted in for various values of $\alpha$ at $t = 0.5$ and $\upsilon = 0.5$ . The 10th-truncated series of Eq (4.6) is plotted in - for $\alpha = 0.6$ , $\alpha = 0.8$ , and $\alpha = 1$ , respectively, whereas, the exact solution at $\alpha = 1$ is plotted in . The graphics indicate consistency in the behavior of the solution at various values of $\alpha$ , as well as the convention of the exact solution with the obtained solution in Figure 1(c, d).

Figure 1. The 3D surface plot of the 10th approximate solutions of

${u}_{1}$ and

${u}_{2}$ at different values of

$\alpha$ and

$t = 0.5$ &

$\upsilon = 0.5$ for the problem in Example 4.1. (a)

$\alpha = 0.6$ , (b)

$\alpha = 0.8$ , (c)

$\alpha = 1$ , (d)

$\alpha = 1$ (Exact solutions).

DownLoad: Full-Size Img PowerPoint

shows the action of the 10th approximate analytical solution of the initial value problems (IVP) (4.1) and (4.2) along the line $\varsigma = \chi$ and in the region $D = \{\left(\chi, t\right):-3\le \chi \le 3, \mathrm{ }0\le t < 1\}$ for distinct values of $\alpha$ , and at $\upsilon = 0.5$ . The 10th approximate solution is plotted in - for $\alpha = 0.6$ , $\alpha = 0.8$ , and $\alpha = 1$ , respectively, whereas, the exact solution at $\alpha = 1$ is plotted in (d). Also, the graphics indicate consistency in the action of the solution at distinct values of $\alpha$ , the accord nation of the exact solution with the approximate solution in Figure 2(c, d) as well as the region of convergence of the series solution.

Figure 2. The graph of the 3D surface of the 10th approximate solutions of

${u}_{1}$ and

${u}_{2}$ along the line

$\varsigma = \chi$ and at various values of

$\alpha$ and

$\upsilon = 0.5$ for the problem in Example 4.1. (a)

$\alpha = 0.6$ , (b)

$\alpha = 0.8$ , (c)

$\alpha = 1$ , (d)

$\alpha = 1$ (Exact).

DownLoad: Full-Size Img PowerPoint

Example 4.2. ^[35,36] Consider the following two-DT-FNSEs:

(4.8)

with the ICs:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , 0\right) = -{e}^{\chi +\varsigma }, \\{u}_{2}\left(\chi , \varsigma , 0\right) = {e}^{\chi +\varsigma }, \end{array}$

(4.9)

where $\upsilon \in \mathbb{R}$ and ${u}_{1}$ and ${u}_{2}$ are two functions of three variables $\chi, \varsigma,$ and $t$ .

The exact solution to problems (4.8) and (4.9) can be obtained, when putting $\alpha = 1$ , to be ${u}_{1} = -{e}^{-2\upsilon t+\chi +\varsigma }$ and ${u}_{2} = {e}^{-2\upsilon t+\chi +\varsigma }$ . Applying the same procedure in Example 4.1, one can obtain the following recurrence relations:

$\begin{array}{l} {f}_{k} = \upsilon \left(\frac{{\partial }^{2}{f}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial {\varsigma }^{2}}\right)-\sum _{i = 0}^{k-1}\frac{\mathit{\Gamma } \left((k-1)\alpha +1\right)\left({f}_{i}\frac{\partial {f}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {f}_{k-i-1}}{\partial \varsigma }\right)}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(\left(k-i-1\right)\alpha +1\right)}, \\{h}_{k} = \upsilon \left(\frac{{\partial }^{2}{h}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\varsigma }^{2}}\right)-\sum _{i = 0}^{k-1}\frac{\mathit{\Gamma } \left(\left(k-1\right)\alpha +1\right)\left({f}_{i}\frac{\partial {h}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {fh}_{k-i-1}}{\partial \varsigma }\right)}{\mathit{\Gamma } \left(i\alpha +1\right)\mathit{\Gamma } \left(\left(k-i-1\right)\alpha +1\right)}. \end{array}$

(4.10)

Thus, we conclude that

${f}_{1} = -2\upsilon {e}^{\chi +\varsigma },\ \ \ \ \ \ {h}_{1} = 2\upsilon {e}^{\chi +\varsigma }$

${f}_{2} = -{\left(2\upsilon \right)}^{2}{e}^{\chi +\varsigma }, \ \ \ \ \ \ {h}_{2} = {\left(2\upsilon \right)}^{2}{e}^{\chi +\varsigma }$

${f}_{3} = -{\left(2\upsilon \right)}^{3}{e}^{\chi +\varsigma }, \ \ \ \ \ \ {h}_{3} = {\left(2\upsilon \right)}^{3}{e}^{\chi +\varsigma },$

${f}_{4} = -{\left(2\upsilon \right)}^{4}{e}^{\chi +\varsigma },\ \ \ \ \ \ {h}_{3} = {\left(2\upsilon \right)}^{3}{e}^{\chi +\varsigma }$

$\ \ \ \ \ \ \ \ \ \ \ \ ⋮\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ⋮$

${f}_{k} = -{\left(2\upsilon \right)}^{k}{e}^{\chi +\varsigma }, \ \ \ \ \ \ {h}_{k} = {\left(2\upsilon \right)}^{k}{e}^{\chi +\varsigma }, \ \ \ \ \ \ k = \mathrm{1, 2}, \cdots .$

Thus, the LRPS solution of the systems (4.8) and (4.9) can be expressed as follows:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , t\right) = -{e}^{\chi +\varsigma }\sum _{i = 0}^{\mathrm{\infty }}\frac{{\left(-2\upsilon \right)}^{i}{t}^{i\alpha }}{\mathrm{\Gamma }\left(i\alpha +1\right)}, \\ {u}_{2}\left(\chi , \varsigma , t\right) = {e}^{\chi +\varsigma }\sum _{j = 0}^{\mathrm{\infty }}\frac{{\left(-2\upsilon \right)}^{j}{t}^{j\alpha }}{\mathrm{\Gamma }\left(j\alpha +1\right)}. \end{array}$

(4.11)

For $\alpha = 1$ , the solution in Eq (4.11) has the form:

${u}_{1}\left(\chi , \varsigma , t\right) = -{e}^{-2\upsilon t+\chi +\varsigma },$

${u}_{2}\left(\chi , \varsigma , t\right) = {e}^{-2\upsilon t+\chi +\varsigma }.$

shows the velocity field behavior of the two-DT-FNSEs (4.8) and (4.9) for distinct values of $\alpha$ at $t = 0.5$ and $\upsilon = 0.5$ . The 10th LRPS approximate analytical solution of the IVP (4.8) and (4.8) plotted in - for $\alpha = 0.6$ , $\alpha = 0.8,$ and $\alpha = 1$ respectively, while the exact solution at $\alpha = 1$ is plotted in (d). The graphics indicate the consistency in the solution behavior at various values of $\alpha$ , as well as the exact solution agreement with the proposed analytical solution in Figure 3(c, d).

Figure 3. The 3D surface plot of the 10th approximate solutions of

${u}_{1}$ and

${u}_{2}$ at distinct values of

$\alpha$ and

$t = 0.5$ &

$\upsilon = 0.5$ for the problem in Example 4.2. (a)

$\alpha = 0.6$ , (b)

$\alpha = 0.8$ , (c)

$\alpha = 1$ , (d)

$\alpha = 1$ (Exact solutions).

DownLoad: Full-Size Img PowerPoint

illustrates the behavior of the 10th approximate solution of the IVP (4.8) and (4.9) along the line $\varsigma = \chi$ and in the region $D = \{\left(\chi, t\right):-1\le \chi \le \mathrm{1, 0}\le t < 1\}$ for different values of $\alpha$ , and at $\upsilon = 0.5$ . The 10th approximate solution is plotted in - for $\alpha = 0.6$ , $\alpha = 0.8$ , and $\alpha = 1$ , respectively, whereas, the exact solution at $\alpha = 1$ is plotted in (d). Also, the graphics indicate consistency in the action of the solution at distinct values of $\alpha$ , the coordination between the exact solution and the approximate analytical solution as illustrated in Figure 4(c, d) as well as the determination of the region of convergence for the series solution, is clear.

Figure 4.

${u}_{1}$ and

${u}_{2}$ along the line

$\varsigma = \chi$ and at various of

$\alpha$ and

$\upsilon = 0.5$ for the problem in Example 4.2. (a)

$\alpha = 0.6$ , (b)

$\alpha = 0.8$ , (c)

$\alpha = 1$ , (d)

$\alpha = 1$ (Exact).

DownLoad: Full-Size Img PowerPoint

Example 4.3. ^[36,41] Consider the following three-DT-FNSEs:

$\begin{array}{l} {D}_{t}^{\alpha }{u}_{1}+{u}_{1}\frac{\partial {u}_{1}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{1}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{1}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{1}}{\partial {\zeta }^{2}}\right), \\ {D}_{t}^{\alpha }{u}_{2}+{u}_{1}\frac{\partial {u}_{2}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{2}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{2}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{2}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{2}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{2}}{\partial {\zeta }^{2}}\right), \\ {D}_{t}^{\alpha }{u}_{3}+{u}_{1}\frac{\partial {u}_{3}}{\partial \chi }+{u}_{2}\frac{\partial {u}_{3}}{\partial \varsigma }+{u}_{3}\frac{\partial {u}_{3}}{\partial \zeta } = \upsilon \left(\frac{{\partial }^{2}{u}_{3}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{u}_{3}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{u}_{3}}{\partial {\zeta }^{2}}\right), \end{array}$

(4.12)

where $v\in \mathbb{R}$ , and with the ICs:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , \zeta , 0\right) = -0.5\chi +\varsigma +\zeta , \\ {u}_{2}\left(\chi , \varsigma , \zeta , 0\right) = \chi -0.5\varsigma +\zeta , \\ {u}_{3}\left(\chi , \varsigma , \zeta , 0\right) = \chi +\varsigma -0.5\zeta , \end{array}$

(4.13)

and the exact solution, when $\alpha = 1$ , is

${u}_{1} = \frac{-0.5\chi +\varsigma +\zeta -2.25\chi t}{1-2.25{t}^{2}}, {u}_{2} = \frac{\chi -0.5\varsigma +\zeta -2.25\varsigma t}{1-2.25{t}^{2}}, {u}_{3} = \frac{\chi +\varsigma -0.5\zeta -2.25\zeta t}{1-2.25{t}^{2}}.$

The general formula (3.12) for the systems (4.12) and (4.13) will be as follows:

${f}_{k} = \upsilon \left(\frac{{\partial }^{2}{f}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{f}_{k-1}}{\partial {\zeta }^{2}}\right)$

$-\sum\limits_{i = 0}^{k-1}\frac{\varGamma \left((k-1)\alpha +1\right)\left({f}_{i}\frac{\partial {f}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {f}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {f}_{k-i-1}}{\partial \zeta }\right)}{\varGamma \left(i\alpha +1\right)\varGamma \left(\left(k-i-1\right)\alpha +1\right)},$

${h}_{k} = \upsilon \left(\frac{{\partial }^{2}{h}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{h}_{k-1}}{\partial {\zeta }^{2}}\right)$

$-\sum\limits_{i = 0}^{k-1}\frac{\varGamma \left(\left(k-1\right)\alpha +1\right)\left({f}_{i}\frac{\partial {h}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {h}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {h}_{k-i-1}}{\partial \zeta }\right)}{\varGamma \left(i\alpha +1\right)\varGamma \left(\left(k-i-1\right)\alpha +1\right)},$

${g}_{k} = \upsilon \left(\frac{{\partial }^{2}{g}_{k-1}}{\partial {\chi }^{2}}+\frac{{\partial }^{2}{g}_{k-1}}{\partial {\varsigma }^{2}}+\frac{{\partial }^{2}{g}_{k-1}}{\partial {\zeta }^{2}}\right)$

$-\sum\limits_{i = 0}^{k-1}\frac{\varGamma \left(\left(k-1\right)\alpha +1\right)\left({f}_{i}\frac{\partial {g}_{k-i-1}}{\partial \chi }+{h}_{i}\frac{\partial {g}_{k-i-1}}{\partial \varsigma }+{g}_{i}\frac{\partial {g}_{k-i-1}}{\partial \zeta }\right)}{\varGamma \left(i\alpha +1\right)\varGamma \left(\left(k-i-1\right)\alpha +1\right)}.$

Thus, we conclude that

${f}_{1} = -2.25\chi , \ \ \ \ \ \ \ {h}_{1} = -2.25\varsigma ,\ \ \ \ \ \ \ {g}_{1} = -2.25\zeta ,$

${f}_{2} = -2.25\chi +4.5\varsigma +4.5\zeta ,\ \ \ \ \ \ \ {h}_{2} = 4.5\chi -2.25\varsigma +4.5\zeta ,\ \ \ \ \ \ \ {g}_{2} = 4.5\chi +4.5\varsigma -2.25\zeta ,$

${f}_{3} = -\chi \left(\frac{5.0625\varGamma \left(2\alpha +1\right)}{{\varGamma \left(\alpha +1\right)}^{2}}+20.25\right),$

${h}_{3} = -\varsigma \left(\frac{5.0625\varGamma \left(2\alpha +1\right)}{{\varGamma \left(\alpha +1\right)}^{2}}+20.25\right),$

${g}_{3} = -\zeta \left(\frac{5.0625\varGamma \left(2\alpha +1\right)}{{\varGamma \left(\alpha +1\right)}^{2}}+20.25\right),$

${f}_{4} = \frac{\varGamma \left(2\alpha +1\right)\left(-5.0625\chi +10.125\varsigma +10.125\zeta \right)}{{\varGamma \left(\alpha +1\right)}^{2}}+\frac{\varGamma \left(3\alpha +1\right)\left(-10.125\chi +20.25\varsigma +20.25\zeta \right)}{\varGamma \left(\alpha +1\right)\varGamma \left(2\alpha +1\right)}$

$-20.25\chi +40.5\varsigma +40.5\zeta ,$

${h}_{4} = \frac{\varGamma \left(2\alpha +1\right)\left(10.125\chi -5.0625\varsigma +10.125\zeta \right)}{{\varGamma \left(\alpha +1\right)}^{2}}+\frac{\varGamma \left(3\alpha +1\right)\left(20.25\chi -10.125\varsigma +20.25\zeta \right)}{\varGamma \left(\alpha +1\right)\varGamma \left(2\alpha +1\right)}$

$+40.5\chi -20.25\varsigma +40.5\zeta ,$

${g}_{4} = \frac{\mathrm{\Gamma }\left(2\alpha +1\right)\left(10.125\chi +10.125\varsigma -5.0625\zeta \right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{2}}+\frac{\mathrm{\Gamma }\left(3\alpha +1\right)\left(20.25\chi +20.25\varsigma -10.125\zeta \right)}{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(2\alpha +1\right)}$

$+40.5\chi +40.5\varsigma -20.25\zeta , \text{}$

${f}_{5} = \chi \left(\begin{array}{l}-182.25-\frac{22.78125\mathrm{\Gamma }\left(4\alpha +1\right)\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{3}\mathrm{\Gamma }\left(3\alpha +1\right)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{2}}\\ +\frac{-\frac{91.125\mathrm{ }\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}-\frac{91.125\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{\mathrm{\Gamma }(3\alpha +1)}}{\mathrm{\Gamma }(\alpha +1)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{{\mathrm{\Gamma }(2\alpha +1)}^{2}}\end{array}\right),$

${h}_{5} = \varsigma \left(\begin{array}{l}-182.25-\frac{22.78125\mathrm{\Gamma }\left(4\alpha +1\right)\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{3}\mathrm{\Gamma }\left(3\alpha +1\right)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{2}}\\ +\frac{-\frac{91.125\mathrm{ }\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}-\frac{91.125\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{\mathrm{\Gamma }(3\alpha +1)}}{\mathrm{\Gamma }(\alpha +1)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{{\mathrm{\Gamma }(2\alpha +1)}^{2}}\end{array}\right),$

${g}_{5} = \zeta \left(\begin{array}{l}-182.25-\frac{22.78125\mathrm{\Gamma }\left(4\alpha +1\right)\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{3}\mathrm{\Gamma }\left(3\alpha +1\right)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }\left(2\alpha +1\right)}{{\mathrm{\Gamma }\left(\alpha +1\right)}^{2}}\\ +\frac{-\frac{91.125\mathrm{ }\mathrm{\Gamma }(3\alpha +1)}{\mathrm{\Gamma }(2\alpha +1)}-\frac{91.125\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{\mathrm{\Gamma }(3\alpha +1)}}{\mathrm{\Gamma }(\alpha +1)}-\frac{45.5625\mathrm{ }\mathrm{\Gamma }(4\alpha +1)}{{\mathrm{\Gamma }(2\alpha +1)}^{2}}\end{array}\right).$

So, the LRPS solution of the systems (4.12) and (4.13) has the following series form:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , \zeta , t\right) = -0.5\chi +\varsigma +\zeta -\frac{2.25}{\mathrm{\Gamma }\left(1+\alpha \right)}\chi {t}^{\alpha }+\frac{2\left(2.25\right)}{\mathrm{\Gamma }\left(1+2\alpha \right)}\left(-0.5\chi +\varsigma +\zeta \right){t}^{2\alpha } \\ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+3\alpha \right)}\left(4+\frac{\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right)\chi {t}^{3\alpha }+\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+4\alpha \right)}\left(8+\frac{2\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \left.+\frac{4\mathrm{\Gamma }\left(1+3\alpha \right)}{\mathrm{\Gamma }\left(1+\alpha \right)\mathrm{\Gamma }\left(1+2\alpha \right)}\right)\left(-0.5\chi +\varsigma +\zeta \right){t}^{4\alpha }+\cdots \mathrm{ }, \\ {u}_{2}\left(\chi , \varsigma , \zeta , t\right) = \chi -0.5\varsigma +\zeta -\frac{2.25}{\mathrm{\Gamma }\left(1+\alpha \right)}\varsigma {t}^{\alpha }+\frac{2\left(2.25\right)}{\mathrm{\Gamma }\left(1+2\alpha \right)}\left(\chi -0.5\varsigma +\zeta \right){t}^{2\alpha }\\ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+3\alpha \right)}\left(4+\frac{\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right)\varsigma {t}^{3\alpha }+\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+4\alpha \right)}\left(8+\frac{2\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \left.+\frac{4\mathrm{\Gamma }\left(1+3\alpha \right)}{\mathrm{\Gamma }\left(1+\alpha \right)\mathrm{\Gamma }\left(1+2\alpha \right)}\right)\left(\chi -0.5\varsigma +\zeta \right){t}^{4\alpha }+\cdots , \\ {u}_{3}\left(\chi , \varsigma , \zeta , t\right) = \chi +\varsigma -0.5\zeta -\frac{2.25}{\mathrm{\Gamma }\left(1+\alpha \right)}\zeta {t}^{\alpha }+\frac{2\left(2.25\right)}{\mathrm{\Gamma }\left(1+2\alpha \right)}\left(\chi +\varsigma -0.5\zeta \right){t}^{2\alpha }\\ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+3\alpha \right)}\left(4+\frac{\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right)\zeta {t}^{3\alpha }+\frac{{\left(2.25\right)}^{2}}{\mathrm{\Gamma }\left(1+4\alpha \right)}\left(8+\frac{2\mathrm{\Gamma }\left(1+2\alpha \right)}{{\mathrm{\Gamma }}^{2}\left(1+\alpha \right)}\right.\\ \ \ \ \ \ \ \ \ \ \ \ \ \left.+\frac{4\mathrm{\Gamma }\left(1+3\alpha \right)}{\mathrm{\Gamma }\left(1+\alpha \right)\mathrm{\Gamma }\left(1+2\alpha \right)}\right)\left(\chi +\varsigma -0.5\zeta \right){t}^{4\alpha }+\cdots . \end{array}$

(4.14)

The sum of the series in Eq (4.14) at $\alpha = 1$ has the following rational forms:

$\begin{array}{l} {u}_{1}\left(\chi , \varsigma , \zeta , t\right) = \frac{-0.5\chi +\varsigma +\zeta -2.25\chi t}{1-2.25{t}^{2}}, \\ {u}_{2}\left(\chi , \varsigma , \zeta , t\right) = \frac{\chi -0.5\varsigma +\zeta -2.25\varsigma t}{1-2.25{t}^{2}}, \\ {u}_{3}\left(\chi , \varsigma , \zeta , t\right) = \frac{\chi +\varsigma -0.5\zeta -2.25\zeta t}{1-2.25{t}^{2}}. \end{array}$

(4.15)

The behavior of the velocity field of the Three-DT-FNSEs (4.12) and (4.13) is depicted in Figure 5 for various values of at and. The th-truncated series of Eq (4.10) is plotted in Figure 5(a-c) for, and, respectively, whereas, the exact solution at is plotted in (d). The graphics indicate consistency in the behavior of the solution at various values of, as well as the agreement of the exact solution with the approximate solution in Figure 5(c, d).

Figure 5. The 3D surfaces plot of the 10th approximate solutions of

${u}_{1}, {u}_{2}$ , and

${u}_{3}$ at various values of

$\alpha$ and

$t = 0.5$ &

$\zeta = 3$ for the problem in Example 4.3. (a)

$\alpha = 0.6$ , (b)

$\alpha = 0.8$ , (c)

$\alpha = 1$ , (d)

$\alpha = 1$ (Exact solutions).

DownLoad: Full-Size Img PowerPoint

5. Mathematical reviews

This article presents the LRPSM in a new scheme. We proposed the method and used it to solve the M-DT-FNS system. In the following, we state the advantages of using the presented method and the disadvantages of treating the M-DT-FNS.

5.1. Advantages of the method

1) The method is simple to apply to solve linear and non-linear FPDE compared to other techniques, other power series methods are based on finding derivatives and the calculations are usually complex, but LRPSM mainly depends on computing the limit at infinity which is much easier.

2) The proposed method is applicable in finding approximate solutions for physical applications, and in finding many terms of the analytical series solutions.

3) The method is accurate and gives approximate solutions close to the exact ones.

5.2. Disadvantages of the method

LRPSM needs first to find the LT of the target equations and finally to run the inverse LT to obtain the solution in the original space. So, if we have nonhomogeneous equations, the source functions need to be piecewise continuous and of exponential order, and after the computations, the inverse LT must exist.

6. Conclusions

In this article, we have introduced the LRPSM in a new scheme and simplified the technique to present series solutions for the M-DT-FNS system in the sense of the Caputo derivative. It is worth noting here that we obtained a general formula for an analytic solution of M-DT-FN, which other researchers have not previously obtained by other methods. We tested three examples by solving them in the proposed technique and then analyzing the results. In the future, we will use LRPSM to solve more problems and make new modifications to address the flaws of the presented technique.

Acknowledgments

The authors express their gratitude to the dear referees, who wish to remain anonymous, and the editor for their helpful suggestions, which improved the final version of this paper.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	T. M. Apostol, Introduction to analytic number theory, New York: Springer-Verlag, 1976.
[2]	L. Chen, J. Y. Hu, A linear recurrence formula involving cubic Gauss Sums and Kloosterman Sums, Acta Math. Sin (Chinese Series), 61 (2018), 67–72.
[3]	L. Chen, X. Wang, A new fourth power mean of two-term exponential sums, Open Math., 17 (2019), 407–414. doi: 10.1515/math-2019-0034
[4]	L. Chen, Z. Y. Chen, Some new hybrid power mean formulae of trigonometric sums, Adv. Differ. Equ., 2020 (2020), 220–228. doi: 10.1186/s13662-020-02660-7
[5]	S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502–506. doi: 10.1016/0022-314X(77)90010-5
[6]	Z. Y. Chen, W. P. Zhang, On the fourth-order linear recurrence formula related to classical Gauss sums, Open Math., 15 (2017), 1251–1255. doi: 10.1515/math-2017-0104
[7]	D. Han, A Hybrid mean value involving two-term exponential sums and polynomial character sums, Czech. Math. J., 64 (2014), 53–62. doi: 10.1007/s10587-014-0082-0
[8]	K. Ireland, M. Rosen, A classical introduction to modern number theory, New York: Springer-Verlag, 1982.
[9]	X. X. Li, J. Y. Hu, The hybrid power mean quartic Gauss sums and Kloosterman sums, Open Math., 15 (2017), 151–156. doi: 10.1515/math-2017-0014
[10]	X. Y. Liu, W. P. Zhang, On the high-power mean of the generalized Gauss sums and Kloosterman sums, Mathematics, 7 (2019), 907–915. doi: 10.3390/math7100907
[11]	J. Z. Wang, Y. K. Ma, The hybrid power mean of the $k$ -th Gauss sums and Kloosterman sums, Journal of Shaanxi Normal University (Natural Science Edition), 45 (2017), 5–7.
[12]	A. Weil, Basic number theory, New York: Springer-Verlag, 1974.
[13]	H. Zhang, W. P. Zhang, The fourth power mean of two-term exponential sums and its application, Math. Rep., 19 (2017), 75–81.
[14]	J. Zhang, W. P. Zhang, A certain two-term exponential sum and its fourth power means, AIMS Mathematics, 5 (2020), 7500–7509. doi: 10.3934/math.2020480
[15]	W. P. Zhang, D. Han, On the sixth power mean of the two-term exponential sums, J. Number Theory, 136 (2014), 403–413. doi: 10.1016/j.jnt.2013.10.022

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AIMS Mathematics

On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$

Related Papers:

Abstract

1. Introduction

2. Basic concepts

3. The Laplace residual power series method

4. Numerical application

5. Mathematical reviews

5.1. Advantages of the method

5.2. Disadvantages of the method

6. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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AIMS Mathematics

On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo p p

Related Papers:

Abstract

1. Introduction

2. Basic concepts

3. The Laplace residual power series method

4. Numerical application

5. Mathematical reviews

5.1. Advantages of the method

5.2. Disadvantages of the method

6. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$