An attractive analytical technique for coupled system of fractional partial differential equations i
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Mohammed Al-Smadi1, Omar Abu Arqub,2, Samir Hadid31Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan 2Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan 3Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates
Abstract Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations (FPDEs) corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the (n−1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached. Keywords：nonlinear coupled system;fractional partial differential equations;residual power series method;conformable fractional derivative
PDF (1940KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Mohammed Al-Smadi, Omar Abu Arqub, Samir Hadid. An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative. Communications in Theoretical Physics, 2020, 72(8): 085001- doi:10.1088/1572-9494/ab8a29
1. Introduction
Computational models of physical systems are typically triggered through computer simulations, symbolic programming, and approximate numerical algorithms for emerging nonlinear problems, which often include a set of corresponding fractional partial differential equations (FPDEs). Solutions of such nonlinear issues are great importance for describing the dynamic and asymptotic behavior of materials in many applications such as nuclear reactors, fluid mechanics, viscoelastic damping, electromagnetic, chemical reactions, and electrochemistry [1–5]. In this light, there exists no classic, precise method that yields an analytical solution within a closed-form in terms of spatial and temporal parameters to deal with these types of nonlinear evolution systems. Therefore, there is an urgent need for effective and sophisticated methods for exploring analytical solutions of these models, which gives us the motivation to search for those numerical solutions.
On the other hand, the subject of fractional calculus is not new, dates back to the late seventeenth century. It is a powerful generalization of the classical calculus that deals with ordinary and partial differentiation and integration of noninteger order. Recently, it has been extensively examined as an excellent tool in describing genetic properties, memory effort, and material transfer mechanisms in many connected areas of engineering and applied sciences such as bioengineering, rheology, communication mechanics, optics, entropy, electromagnetic, thermodynamics, etc. [6–10]. Unlike classical calculus, which has a unique definition and clear geometrical and physical interpretations, there are many definitions for the fractional operations, including Riemann–Liouville, Caputo, Riesz, and Grünwald–Letnikov [11–20]. In view of this, a novel concept of fractional calculus, namely the conformable fractional derivative, has been proposed in [21]. Since then several studies have appeared in the literature, for more details see [22–27]. Further, in the last few years, several common numerical and analytical techniques have been proposed to solve different classes of linear and nonlinear FPDEs occurring in physics and applied mathematics. Among these methods, the Sumudu transform method [28], reduced differential transform method [29], Kudryashov method [30], direct integral method [31], Bernstein method [32], reproducing kernel method [33–38], and the dynamical system method [39–41].
The real-world problems can be fully described theoretically through FPDEs, which is in nature affected by various external forces that affect its behavior to become more sophisticated and unpredictable. So, the best way to deal with such a situation is to study such real-world issues statistically or via numerical approximations to reach the model that gives an acceptable solution. In the current work, the application of the conformable residual power series (CRPS) method is extended to provide a convenient methodology to derive approximate series solutions for a class of coupled system of FPDEs arising in shallow water waves with conformable derivative. To achieve our goal, we consider the following coupled FPDEs system in the form:The nonlinear time-fractional coupled Whitham–Broer–Kaup (WBK) equations:$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }v+v{v}_{x}+{w}_{x}+\xi {v}_{xx}=0,\\ {T}_{t}^{\alpha }w+{(vw)}_{x}-\xi {w}_{xx}+\eta {v}_{xxx}=0.\end{array}\end{eqnarray}$ The nonlinear time-fractional coupled Jaulent–Miodek (JM) equations: $ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }v+{v}_{xxx}+\displaystyle \frac{3}{2}w{w}_{xxx}+\displaystyle \frac{9}{2}{w}_{x}{w}_{xx}-6v{v}_{x}\\ \,-\,6vw{w}_{x}-\displaystyle \frac{3}{2}{v}_{x}{w}^{2}=0,\\ {T}_{t}^{\alpha }w+{w}_{xxx}-6{v}_{x}w-6v{w}_{x}-\displaystyle \frac{15}{2}{w}_{x}{w}^{2}=0.\end{array}\end{eqnarray}$Hither, both WBK and JM equations are considered along with the following initial conditions:$ \begin{eqnarray}\begin{array}{rcl}v\left(x,0\right) & = & r\left(x\right),\\ w\left(x,0\right) & = & \&\left(x\right),\end{array}\end{eqnarray}$where $0\lt \alpha \leqslant 1,$$\xi ,\eta $ are real finite parameters, $\left(x,t\right)\in \left[a,b\right]\times [0,T],$$r\left(x\right),$$\&\left(x\right)$ are given analytical functions of x, $v=v\left(x,t\right),$$w=w\left(x,t\right)$ are unknown real-valued functions in terms of x and t to be determined such that v is the field of horizontal velocity and w is the height that deviates from the equilibrium position of liquid, and ${T}_{t}^{\alpha }$ indicates the conformable fractional derivative of order α. For α=1, the fractional models (1) and (2) reduce to classic nonlinear coupled WBK and JM system, respectively. As a special case, if the parameters $\eta =0,$$\kappa =\sigma =1,$ and $\xi =1/2$ in system (1), then we obtain the time-fractional coupled approximate long-wave (LW) equations that describe the proliferation of shallow water waves, while if the parameters $\eta =1$ and $\xi =0$ in system (1), then we obtain the time-fractional coupled modified Boussinesq (MB) equations that describe the fluids flow in a dynamic system.
The standard RPS method was proposed in 2013 by Abu Arqub [42] as a powerful and effective approximate algorithm to solve a class of uncertain initial value problems. Later, RPS method has been used in generating fractional power series solutions for strongly nonlinear FPDEs in the form of a rapidly convergent with a minimum size of calculations without any restrictive hypotheses. So, this adaptive can be used as an alternative technique in solving several nonlinear problems arising in engineering and physics [43–47].
This article is organized as follows. In section 2, some basic characteristics of the conformable derivative are given and the fractional power series is presented in the conformable sense. In section 3, the main steps of the extended CRPS method are discussed. In section 4, we construct the approximate series solutions of the nonlinear coupled fractional WBK system, nonlinear coupled fractional MB system, and nonlinear coupled fractional JM system associated with Schrödinger's potential energy. Section 5 is devoted to conclusions.
2. Notations and preliminaries
In the literature, a few concepts of fractional differential operators are listed, including the Riemann–Liouville, Caputo–Fabrizio, Riesz, Grunwald–Letnikov, Atangana–Baleanu, and conformable [11–20]. This section is dedicated to briefly present some definitions and properties for the conformable fractional calculus and its corresponding fractional power series.
[21] Let f be n-differentiable at $t\gt s;$ the conformable fractional derivative starting from s of a function $f:\left(s,\infty \right]\to {\mathbb{R}}$ of order $\alpha \gt 0$ is defined by
$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{{\rm{d}}}^{\alpha }f}{{\rm{d}}{t}^{\alpha }} & = & {T}^{\alpha }f\left(t\right)\\ & = & \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}\displaystyle \frac{{f}^{\left(\alpha -1\right)}\left(t+\varepsilon {\left(t-s\right)}^{\alpha -\alpha }\right)-{f}^{\left(\alpha -1\right)}(t)}{\varepsilon },\\ & & \alpha \in \left(n-1,n\right],{\rm{}}t\gt s,\end{array}\end{eqnarray}$and ${T}^{\alpha }f\left(s\right)={\mathrm{lim}}_{t\to {s}^{+}}{T}^{\alpha }f\left(t\right)$ provided $f(t)$ is α-differentiable in some $(0,s),$$s\gt 0,$ and ${\mathrm{lim}}_{t\to {s}^{+}}{T}^{\alpha }f\left(t\right)$ exists, where α is the smallest integer greater than or equal α.
It is worth noting here that f is called α-differentiable at a point t whenever f has a conformable fractional derivative of order α at a point t. Some features of the α-differentiable are provided in [21]. In the next theorem, we mention some of these features.
[21] Let $\alpha \in (0,\,1]$ and assume $f,g$ be α-differentiable at a point $t\gt s.$ Then (1) $\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}(kf\,+hg)\,=\,k{\rm{}}{f}^{\left(\alpha \right)}+h{\rm{}}{g}^{\left(\alpha \right)},k,\,h\in {\mathbb{R}}.$ (2) $\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}\left(\lambda \right)=0,\lambda \in {\mathbb{R}}.$ (3) $\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}\left(\lambda f\right)=\lambda \displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}\left(f\right).$ (4) $\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}\left({\left(t-a\right)}^{p}\right)=\,p{\left(t-a\right)}^{p-\alpha },p\in {\mathbb{R}}.$ (5) If f is differentiable, then $\displaystyle \frac{{{\rm{d}}}^{\alpha }f}{{\rm{d}}{t}^{\alpha }}(t)={\left(t-a\right)}^{1-\alpha }\displaystyle \frac{{\rm{d}}f}{{\rm{d}}t}(t).$
For $\alpha \,\in \,(n-1,n],$ if f is n-differentiable at $t\gt s,$ then $\tfrac{{{\rm{d}}}^{\alpha }f}{{\rm{d}}{t}^{\alpha }}\left(t\right)={\left(t-s\right)}^{n-\alpha }\tfrac{{{\rm{d}}}^{n}f}{{\rm{d}}{t}^{n}}\left(t\right).$ Further, for$\alpha \,\in \,(n-1,n],$ if $f:[0,\infty )\to {\mathbb{R}}$ is α-differentiable at $t\gt s,$ then f is continuous at s.
[21] The conformable fractional integral starting from s of order $\alpha \in (n-1,n]$ of f is defined as
For $\alpha =n,$ equation (5) is the Cauchy formula ${I}_{s}^{n}f\left(t\right)=\tfrac{1}{\left(n-1\right)!}\displaystyle {\int }_{s}^{t}{\left(t-\tau \right)}^{n-1}f\left(\tau \right){\rm{d}}\tau $ in which $t\gt \tau \,\geqslant s\,\geqslant 0.$ Sometimes, we can write α-fractional integral to indicate the conformable fractional integral ${I}_{s}^{\alpha }.$
Let $\alpha \in (n-1,n]$ and assume f be n-times differentiable function. Then (1) $\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}\left({I}_{s}^{\alpha }f\left(t\right)\right)=f\left(t\right).$ (2) ${I}_{s}^{\alpha }\left(\displaystyle \frac{{{\rm{d}}}^{\alpha }}{{\rm{d}}{t}^{\alpha }}f\left(t\right)\right)=f\left(t\right)-\displaystyle \sum _{k=0}^{n-1}\displaystyle \frac{{f}^{\left(k\right)}\left(s\right){\left(t-s\right)}^{k}}{k!}.$
[22] Let ${\partial }^{k}u/\partial {t}^{k}$ and ${\partial }^{k}u/\partial {x}^{k},$$k\,=1,\,2,\,\ldots ,\,n-1$ be defined on $I\times \left[s,\infty \right),$ then the conformable time-fractional differential operator of order $\alpha \in \left(n-1,n\right]$ of a function $v\left(x,t\right):I\times [s,\infty )\to {\mathbb{R}}$ is defined by
[22] The conformable fractional integral starting from s of order $\alpha \in (n-1,n]$ of a function $v\left(x,t\right):I\,\times [s,\infty )\to {\mathbb{R}}$ is defined by
[43] For $0\leqslant n-1\lt \alpha \leqslant n,$ the power series of the form
$ \begin{eqnarray}\begin{array}{lll}\displaystyle \sum _{k=0}^{\infty }{f}_{k}(x)\left(t-{t}_{0}\right){}^{k\alpha } & = & {f}_{0}\left(x\right)+{f}_{1}\left(x\right){\left(t-{t}_{0}\right)}^{\alpha }\\ & & +\,{f}_{2}\left(x\right){\left(t-{t}_{0}\right)}^{2\alpha }+\ldots ,\,x\in I,\end{array}\end{eqnarray}$is called a multiple fractional power series (MFPS) at $t={t}_{0},$ where t is a variable such that ${t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha }$ and ${f}_{k}$ are functions called the coefficients of the MFPS.
As the classical power series [42–46], it clear that all terms of the MFPS (8) are vanish as soon as $t={t}_{0}$ except the first term, which means the MFPS is convergent when $t={t}_{0}.$ Furthermore, if $t\geqslant {t}_{0},$ the MFPS is definitely convergent for $\left|t-{t}_{0}\right|\lt {R}^{1/\alpha }$ with $R\gt 0,$ where ${R}^{1/\alpha }$ is the radius of convergence of the series.
For $0\leqslant n-1\lt \alpha \leqslant n,$ assume that $v\left(x,t\right):I\times [{t}_{0},{t}_{0}+{R}^{1/\alpha })\to {\mathbb{R}}$ can be expressed as the MFPS about $t={t}_{0}$ in the form
$ \begin{eqnarray}\begin{array}{l}v\left(x,t\right)=\displaystyle \sum _{k=0}^{\infty }{f}_{k}(x)\left(t-{t}_{0}\right){}^{k\alpha },\,\\ x\in I,\,{t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha },R\gt 0.\end{array}\end{eqnarray}$If $v\left(x,t\right)$ is continuous on $I\times [{t}_{0},{t}_{0}+{R}^{1/\alpha })$ and $\tfrac{{\partial }^{k\alpha }}{\partial {t}^{k\alpha }}v\left(x,t\right)={T}_{t}^{k\alpha }v\left(x,t\right)$$\in C({t}_{0},{t}_{0}+{R}^{1/\alpha })$ for $k=1,\,2,\,\ldots ,$ then the coefficients ${f}_{k}(x)$ can be given by ${f}_{k}(x)\,=\tfrac{{T}_{t}^{k\alpha }v\left(x,{t}_{0}\right)}{{\alpha }^{k}(k)!},\,$where ${T}_{t}^{k\alpha }$ stands for sequential conformable time-fractional derivative of order kα defined by ${T}^{k\alpha }v(x,t)\,=\mathop{\underbrace{{T}^{\alpha }\cdot {T}^{\alpha }\cdots {T}^{\alpha }v(x,t)}}\limits_{k-times}.$
Let $v\left(x,t\right)$ be a function of two variables that can be expressed as the MFPS of equation (9), that is, $v\left(x,t\right)={f}_{0}\left(x\right)$$+{f}_{1}\left(x\right){\left(t-{t}_{0}\right)}^{\alpha }$$+{f}_{2}\left(x\right){\left(t-{t}_{0}\right)}^{2\alpha }+\ldots .$ By putting $t={t}_{0},$ one can obtain ${f}_{0}\left(x\right)=v\left(x,{t}_{0}\right).$ Applying the operator ${T}_{t}^{\alpha }$ one time to $v\left(x,t\right)$ leads to
$ \begin{eqnarray}{T}_{t}^{\alpha }v\left(x,t\right)=\alpha {f}_{1}\left(x\right)+2\alpha {f}_{2}\left(x\right){\left(t-{t}_{0}\right)}^{\alpha }+\ldots .\end{eqnarray}$Evaluating the result at $t={t}_{0}$ leads to ${T}_{t}^{k\alpha }v\left(x,{t}_{0}\right)=\alpha {f}_{1}(x),$ and hence ${f}_{1}(x)=\tfrac{{T}_{t}^{\alpha }v\left(x,{t}_{0}\right)}{\alpha }.$ Again, by applying the operator ${T}_{t}^{\alpha }$ two times to $v\left(x,t\right),$ one can obtain$ \begin{eqnarray}\begin{array}{lll}{T}_{t}^{2\alpha }v\left(x,t\right) & = & \left(2\alpha \right)\alpha {f}_{2}\left(x\right)\\ & & +\,\left(2\alpha \right)\left(3\alpha \right){f}_{3}\left(x\right){\left(t-{t}_{0}\right)}^{\alpha }+\ldots .\end{array}\end{eqnarray}$While, the substitution of $t={t}_{0}$ leads to ${f}_{2}(x)=\tfrac{{T}_{t}^{2\alpha }v\left(x,t\right)}{2{\alpha }^{2}}.$ If we follow this approach, apply the operator ${T}_{t}^{\alpha }$k-times to $v\left(x,t\right)$ and evaluate $t={t}_{0}$ in the resulting formula, then it can easy to see that ${f}_{k}\left(x\right)=\tfrac{{T}_{t}^{k\alpha }v\left(x,{t}_{0}\right)}{{\alpha }^{k}\left(k\right)!}.\,$ ■
We mention here that, the nth-partial sum of the MFPS (9) that expressed the approximate numerical solution can be generated from$ \begin{eqnarray}\begin{array}{l}{v}_{n}\left(x,t\right)=\displaystyle \sum _{k=0}^{n}{f}_{k}(x)\left(t-{t}_{0}\right){}^{k\alpha },\,\\ x\in I,\,{t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha },R\gt 0.\end{array}\end{eqnarray}$
Let $\alpha \in (n-1,n],$${T}_{t}^{k\alpha }v\left(x,t\right)$ exist at a neighborhood of a point ${t}_{0}$ for $k=0,1,2,\ldots ,n+1,$ and $v\left(x,t\right)$ can be expressed by the MFPS (9) about $t={t}_{0}$ such that $\left|{T}_{t}^{(n+1)\alpha }v\left(x,t\right)\right|\leqslant M(x),$ for some $n\in {\mathbb{N}}.$ Then, for all $({t}_{0},{t}_{0}+{R}^{1/\alpha }),$ the reminder ${ {\mathcal R} }_{n}\left(t\right)$ of the MFPS satisfies
From the assumption that $\left|{T}_{t}^{(n+1)\alpha }v\left(x,t\right)\right|\leqslant M(x),$ it follows
$ \begin{eqnarray}-M(x)\leqslant {T}_{t}^{\left(n+1\right)\alpha }v\left(x,t\right)\leqslant M(x).\end{eqnarray}$Now, by applying the operator ${I}_{t}^{(n+1)\alpha }$ to both sides of the inequality (14), we can get that$ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{-M(x)}{{\alpha }^{n+1}\left(n+1\right)!}{\left(t-{t}_{0}\right)}^{(n+1)\alpha }\leqslant {I}_{t}^{\left(n+1\right)\alpha }{T}_{t}^{\left(n+1\right)\alpha }\\ \,\times \,v\left(x,t\right)\leqslant \displaystyle \frac{M\left(x\right)}{{\alpha }^{n+1}\left(n+1\right)!}{\left(t-{t}_{0}\right)}^{\left(n+1\right)\alpha }.\end{array}\end{eqnarray}$Hence, from equation (12), we complete the proof. ■
3. Detailed steps of CRPS method
This section is devoted to a description of the RPS method for solving nonlinear coupled FPDEs in a conformable sense. To achieve our goal, we consider a general nonlinear coupled FPDEs of the following form:$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }{v}_{i}\left(x,t\right)={{\ell }}_{i}\left[{v}_{i}\right]+{{\mathscr{N}}}_{i}\left[{v}_{i}\right]+{\varphi }_{i}\left(x,t\right),0\lt \alpha \leqslant 1,\\ i=1,2,t\geqslant {t}_{0},\,x\in I,\end{array}\end{eqnarray}$subject to the initial conditions$ \begin{eqnarray}{v}_{i}\left(x,0\right)={\mu }_{i}\left(x\right),i=1,2,\end{eqnarray}$where ${T}_{t}^{\alpha }$ is the conformable fraction derivative of order $\alpha ,$${{\mathscr{N}}}_{i}$ is the nonlinear differential operator, ${{\ell }}_{i}$ is the linear differential operator, and ${\varphi }_{i}$ is the source term.
Next, the basic phases of the CRPS method are explained as follows:
Suppose that the solutions of the nonlinear FPDEs (16) and (17) have the MFPS about $t={t}_{0}:$
Define the nth-truncated residual functions as follows:
$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{{v}_{i,n}}\left(x,t\right) & = & {T}_{t}^{\alpha }{v}_{i,n}\left(x,t\right)-{{\ell }}_{i}\left[{v}_{i,n}\right]\\ & & -\,{{\mathscr{N}}}_{i}\left[{v}_{i,n}\right]-{\varphi }_{i}\left(x,t\right),i=1,2,\end{array}\end{eqnarray}$whereas the residual function is ${{\rm{Res}}}_{\infty }\left(x,t\right)={\mathrm{lim}}_{n\to \infty }{{\rm{Res}}}_{{v}_{i,n}}\left(x,t\right),$$x\in I,\,{t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha }.$
Here, it can be noted that ${{\rm{Res}}}_{\infty }\left(x,t\right)$ is vanished for each $x\in I$ and $t\in \left[{t}_{0},{t}_{0}+{R}^{1/\alpha }\right],$ where ${R}^{1/\alpha }$ is the convergence radius for the MFPS (19). Consequently, ${T}_{t}^{n\alpha }{{\rm{Res}}}_{\infty }\left(x,t\right)=0$ and ${T}_{t}^{(n-1)\alpha }{{\rm{Res}}}_{\infty }{\left(x,t\right)}_{\left|t={t}_{0}\right.}=0$ for each $n=1,2,\ldots .$
Substitute the nth-truncated MFPS solution (19) into the nth-truncated residual function (20) such that
Set $n=1$ in equation (21), then by using ${{\rm{Res}}}_{{v}_{i,1}}{\left(x,t\right)}_{\left|t={t}_{0}\right.}=0,$ the 1st-unknown coefficients ${f}_{i,1}\left(x\right),i\,=1,2,$ can be obtained. Therefore, the 1st-approximate MFPS solutions ${v}_{i,1}\left(x,t\right),i=1,2,$ is obtained so that ${v}_{i,1}\left(x,t\right)\,={\mu }_{i}\left(x\right)$$+{f}_{i,1}\left(x\right)\tfrac{{(t-{t}_{0})}^{\alpha }}{\alpha }.$
For $n=2,3,\ldots ,N,$ applying the operator ${T}_{t}^{\alpha },$ ($n-1$)-times, on both sides of equation (21) such that ${T}_{t}^{(n-1)\alpha }{{\rm{Res}}}_{{v}_{i,n}}\left(x,t\right),$ evaluating the resulting equations at $t={t}_{0}$ with equality to zero such that ${T}_{t}^{(n-1)\alpha }{{\rm{Res}}}_{{v}_{i,n}}\left(x,{t}_{0}\right)=0,$ and using the fact that ${T}_{t}^{\alpha }{t}^{p}=0$ for $p\gt \alpha $ at $t={t}_{0},$ then the unknown coefficients ${f}_{i,k}\left(x\right),$$k=2,3,\ldots ,n,i=1,2,$ of the nth-truncated MFPS solution (20) can be determined.
Collect the obtained coefficients ${f}_{i,n}\left(x\right)$ and ${v}_{i,n}\left(x,t\right)$ for each $n=0,1,2,\ldots ,N$ in term of expanded MFPS and try to find a general pattern with the term infinite series so that the exact solution ${v}_{i}\left(x,t\right)$ of FPDEs (16) and (17) is obtained; otherwise, the pattern obtained in the sense of given series coefficients represents the $N$th-approximate MFPS solutions. Furthermore, if the number of series terms increases, the error is reduced. In this regard, calculating additional MFPS coefficients will give us a more accurate solution.
Let ${v}_{i}\left(x,t\right),i=1,2,$ be the exact solutions of FPDEs (16) and (17). If there exists a fixed constant ${\rho }_{i}\in [0,1]$ such that $\parallel {v}_{i,n+1}\left(x,t\right)\parallel \leqslant {\rho }_{i}\parallel {v}_{i,n}\left(x,t\right)\parallel ,i=1,2$ for all $n\in {\mathbb{N}},$$x\in I,$ and $t\in \left[{t}_{0},{t}_{0}+{R}^{1/\alpha }\right],$ then the $n$-term approximate solutions ${v}_{i,n}\left(x,t\right)$ converges to ${v}_{i}\left(x,t\right)$ as $n\to \infty .$
For all $x\in I$ and $t\in \left[{t}_{0},{t}_{0}+{R}^{1/\alpha }\right],$ let ${f}_{i,0}\left(x\right)={v}_{i}\left(x,0\right),$ then from $\parallel {v}_{i,n+1}\left(x,t\right)\parallel \leqslant {\rho }_{i}\parallel {v}_{i,n}\left(x,t\right)\parallel ,$ we have $\parallel {v}_{i,1}\left(x,t\right)\parallel \leqslant {\rho }_{i}\parallel {v}_{i,0}\left(x,t\right)\parallel ={\rho }_{i}\parallel {f}_{i,0}\left(x\right)\parallel .$ Similarly, $\parallel {v}_{i,2}\left(x,t\right)\parallel $$\leqslant {\rho }_{i}\parallel {v}_{i,1}\left(x,t\right)\parallel \leqslant {{\rho }_{i}}^{2}\parallel {f}_{i,0}\left(x\right)\parallel .$ Thus, $\parallel {v}_{i,k}\left(x,t\right)\parallel \,\leqslant {\rho }_{i}^{k}\parallel {f}_{i,0}\left(x\right)\parallel $ and $\displaystyle {\sum }_{k=n+1}^{\infty }\parallel {v}_{i,k}\left(x,t\right)\parallel \leqslant \displaystyle {\sum }_{k=n+1}^{\infty }{\rho }_{i}^{k}\parallel {f}_{i,0}\left(x\right)\parallel $ = ${f}_{i,0}\left(x\right)\displaystyle {\sum }_{k=n+1}^{\infty }{\rho }_{i}^{k}.$ Consequently, we have
This section presents some numerical applications of nonlinear coupled FPDEs in conformable sense, including coupled WBK system, coupled MB system, and coupled JM system associated with Schrödinger's potential energy. The numerical results are discussed graphically and quantitatively to verify the validity and applicability of the proposed method in obtaining accurate MFPS solutions for those coupled systems. Mathematica 10 software package is used in all computational process.
Consider the nonlinear fractional coupled WBK system [48]:
$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }v+v{v}_{x}+{w}_{x}+{v}_{xx}=0,\,0\lt \alpha \leqslant 1,\\ {T}_{t}^{\alpha }w+{(vw)}_{x}-{w}_{xx}+3{v}_{xxx}=0,\end{array}\end{eqnarray}$along with the initial conditions$ \begin{eqnarray}\begin{array}{l}v\left(x,0\right)=\displaystyle \frac{1}{2}\left(1-16\,\tanh \left(-2x\right)\right),\\ w\left(x,0\right)=16\left(1-{\tanh }^{2}\left(-2x\right)\right).\end{array}\end{eqnarray}$The exact solutions of FPDEs (23) and (24) at $\alpha =1$ are$ \begin{eqnarray}\begin{array}{l}\begin{array}{c}v\left(x,t\right)\end{array}=\displaystyle \frac{1}{2}-8\,\tanh \left(-2x+t\right),\\ \begin{array}{c}w\left(x,t\right)\end{array}=16-16{\tanh }^{2}\left(-2x+t\right).\end{array}\end{eqnarray}$This fractional model was developed in classical fluids flow to depict the propagation of shallow-water waves with different dispersion relations [28].
According to procedure of the CRPS steps, the nth-truncated series solutions are given by$ \begin{eqnarray}\begin{array}{l}{v}_{n}\left(x,t\right)=\displaystyle \frac{1}{2}\left(1-16\,\tanh \left(-2x\right)\right)+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\\ {w}_{n}\left(x,t\right)=16\left(1-{\tanh }^{2}\left(-2x\right)\right)+\displaystyle \sum _{k=1}^{n}{b}_{k}(x)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\end{array}\end{eqnarray}$and the $n$th-truncated residual functions are given by$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{v}^{n}\left(x,t\right) & = & {T}_{t}^{\alpha }{v}_{n}\left(x,t\right)+{v}_{n}\left(x,t\right){\left({v}_{n}\left(x,t\right)\right)}_{x}\\ & & +\,{\left({w}_{n}\left(x,t\right)\right)}_{x}+{\left({v}_{n}\left(x,t\right)\right)}_{xx},\\ {{\rm{Res}}}_{w}^{n}\left(x,t\right) & = & {T}_{t}^{\alpha }{w}_{n}\left(x,t\right)+{\left({v}_{n}\left(x,t\right){w}_{n}\left(x,t\right)\right)}_{x}\\ & & -\,{\left({w}_{n}\left(x,t\right)\right)}_{xx}+3{\left({v}_{n}\left(x,t\right)\right)}_{xxx}.\end{array}\end{eqnarray}$Now, to find the first unknown coefficients, ${a}_{1}\left(x\right)$ and ${b}_{1}\left(x\right),$ of the MFPS (26), substitute the 1st-truncated series$ \begin{eqnarray}\begin{array}{l}{v}_{1}\left(x,t\right)=\displaystyle \frac{1}{2}\left(1-16\,\tanh \left(-2x\right)\right)+{a}_{1}\left(x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\\ {w}_{1}\left(x,t\right)=16\left(1-{\tanh }^{2}\left(-2x\right)\right)+{b}_{1}\left(x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\end{array}\end{eqnarray}$into the 1st-truncated residual functions, $\left\{{{\rm{Res}}}_{v}^{1}\left(x,t\right),\right.$$\left.{{\rm{Res}}}_{w}^{1}\left(x,t\right)\right\},$ such that$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{v}^{1}\left(x,t\right) & = & {a}_{1}\left(x\right)+\left(\displaystyle \frac{1}{2}\left(1-16\,\tanh \left(-2x\right)\right)+\displaystyle \frac{{a}_{1}\left(x\right){t}^{\alpha }}{\alpha }\right)\\ & & \times \,\left(16{{\rm{sech}} }^{2}\left(2x\right)+\displaystyle \frac{{a}_{1}^{{\prime} }(x){t}^{\alpha }}{\alpha }\right)\\ & & +\,\left(-64{{\rm{sech}} }^{2}\left(2x\right)\tanh (2x)+\displaystyle \frac{{b}_{1}^{{\prime} }(x){t}^{\alpha }}{\alpha }\right)\\ & & +\,\left(-64{{\rm{sech}} }^{2}\left(2x\right)\tanh (2x)+\displaystyle \frac{{a}_{1}^{{\prime\prime} }(x){t}^{\alpha }}{\alpha }\right),\\ {{\rm{Res}}}_{w}^{1}\left(x,t\right) & = & {b}_{1}\left(x\right)+\left(\displaystyle \frac{1}{2}\left(1-16\,\tanh \left(-2x\right)\right)+\displaystyle \frac{{a}_{1}\left(x\right){t}^{\alpha }}{\alpha }\right)\\ & & \times \,\left(-64{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right)+\displaystyle \frac{{b}_{1}^{{\prime} }(x){t}^{\alpha }}{\alpha }\right)\\ & & +\,\left(16{{\rm{sech}} }^{2}\left(2x\right)+\displaystyle \frac{{a}_{1}^{{\prime} }(x){t}^{\alpha }}{\alpha }\right)\\ & & \times \,\left(16\left(1-{\tanh }^{2}\left(-2x\right)\right)+\displaystyle \frac{{b}_{1}(x){t}^{\alpha }}{\alpha }\right)\\ & & -\,\left(128\left(\left(\cosh \left(4x\right)-2\right)\right.{{\rm{sech}} }^{4}\left(2x\right)+\displaystyle \frac{{b}_{1}^{{\prime\prime} }(x){t}^{\alpha }}{\alpha }\right)\\ & & +\,3\left(128\left(\left(\cosh \left(4x\right)-2\right)\right.{{\rm{sech}} }^{4}\left(2x\right)+\displaystyle \frac{{a}_{1}^{{\prime\prime} ^{\prime} }(x){t}^{\alpha }}{\alpha }\right).\end{array}\end{eqnarray}$By using ${{\rm{Res}}}_{v}^{1}\left(x,0\right)=0$ and ${{\rm{Res}}}_{w}^{1}\left(x,0\right)=0,$ one can get ${a}_{1}\left(x\right)=-8{{\rm{sech}} }^{2}\left(2x\right)$ and ${b}_{1}\left(x\right)=32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right).$ Therefore, the 1st-CPS approximation of FPDEs (23) and (24) are$ \begin{eqnarray}\begin{array}{lll}{v}_{1}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-8{{\rm{sech}} }^{2}\left(2x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\\ {w}_{1}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }.\end{array}\end{eqnarray}$For $n=2,$ substituting the 2nd-truncated series solutions$ \begin{eqnarray}\begin{array}{lll}{v}_{2}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-8{{\rm{sech}} }^{2}\left(2x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }\\ & & +\,{a}_{2}\left(x\right)\displaystyle \frac{{t}^{2\alpha }}{2{\alpha }^{2}},\\ {w}_{2}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }\\ & & +\,{b}_{2}\left(x\right)\displaystyle \frac{{t}^{2\alpha }}{2{\alpha }^{2}},\end{array}\end{eqnarray}$into the 2nd-truncated residual functions $\left\{{{\rm{Res}}}_{v}^{2}\left(x,t\right),{{\rm{Res}}}_{w}^{2}\left(x,t\right)\right\},$ operating ${T}_{t}^{\alpha }$ on both sides of the resulting equations $\left\{{T}_{t}^{\alpha }{{\rm{Res}}}_{v}^{2}\left(x,t\right),{T}_{t}^{\alpha }{{\rm{Res}}}_{w}^{2}\left(x,t\right)\right\}$ and then using the fact ${T}_{t}^{\alpha }{t}^{p}=0$ for $p\gt \alpha $ at $t=0$ to get$\begin{aligned}T_{t}^{\alpha} \operatorname{Res}_{v}^{2}(x, t)_{\mid t=0}=& a_{2}(x)+16 \operatorname{sech}^{2}(2 x) \tanh (2 x) \\&+\left(\frac{16 a_{2}(x) \operatorname{sech}^{2}(2 x) t^{\alpha}}{\alpha}\right.\\&-\frac{512 \operatorname{sech}^{4}(2 x) \tanh (2 x) t^{\alpha}}{\alpha} \\&+\frac{a_{2}^{\prime}(x) t^{\alpha}}{2 \alpha}+\frac{8 a_{2}^{\prime}(x) \tanh (2 x) t^{\alpha}}{\alpha} \\&\left.+\frac{b_{2}(x) t^{\alpha}}{\alpha}+\frac{a_{2}^{\prime \prime}(x) t^{\alpha}}{\alpha}\right)\left.\right|_{t=0} ^{\alpha} \\&+\left(\frac{48 a_{2}(x) \operatorname{sech}^{2}(2 x) \tanh (2 x) t^{2 \alpha}}{\alpha^{2}}\right.\\&\left.-\frac{12 a_{2}^{\prime}(x) \operatorname{sech}^{2}(2 x) t^{2 \alpha}}{\alpha^{2}}\right)\left.\right|_{t=0} ^{\infty} \\&+\left.\left(\frac{a_{2}(x) a_{2}^{\prime}(x) t^{3 \alpha}}{\alpha^{3}}\right)\right|_{t=0}=0,\end{aligned}$$ {T}_{t}^{\alpha }{{\rm{Res}}}_{w}^{2} {(x,t)}_{|t=0} = {b}_{2}(x)+3{{\rm{sech}} }^{4}(2x)\\ -64{{\rm{sech}} }^{2}(2x){\tanh }^{2}(2x)\\ (\displaystyle \frac{16{b}_{2}(x){{\rm{sech}} }^{2}(2x){t}^{\alpha }}{\alpha }\\ -\,\displaystyle \frac{1024{{\rm{sech}} }^{6}(2x){t}^{\alpha }}{\alpha }\\ -\,\displaystyle \frac{64{a}_{2}(x){{\rm{sech}} }^{2}(2x)\tanh (2x){t}^{\alpha }}{\alpha }\\ +\,\displaystyle \frac{4096{{\rm{sech}} }^{4}(2x){\tanh }^{2}(2x){t}^{\alpha }}{\alpha }\\ +\,\displaystyle \frac{16{a}_{2}^{{\prime} }(x){t}^{\alpha }}{\alpha }-\displaystyle \frac{16{a}_{2}^{{\prime} }(x){\tanh }^{2}(2x){t}^{\alpha }}{\alpha }\\ +\,\displaystyle \frac{{b}_{2}^{{\prime} }(x){t}^{\alpha }}{2\alpha }+\displaystyle \frac{8{b}_{2}^{{\prime} }(x)\tanh (2x){t}^{\alpha }}{\alpha }\\ -\,\displaystyle \frac{{b}_{2}^{{\prime} ^{\prime} }(x){t}^{\alpha }}{\alpha }+\displaystyle \frac{3{a}_{2}^{{\prime} ^{\prime\prime} }(x){t}^{\alpha }}{\alpha })\\ +\,(\displaystyle \frac{96{a}_{2}(x){{\rm{sech}} }^{4}(2x){t}^{2\alpha }}{{\alpha }^{2}}\\ +\,\displaystyle \frac{48{b}_{2}(x){{\rm{sech}} }^{2}(2x)\tanh (2x){t}^{2\alpha }}{{\alpha }^{2}}\\ -\,\displaystyle \frac{192{a}_{2}(x){{\rm{sech}} }^{2}(2x){\tanh }^{2}(2x){t}^{2\alpha }}{{\alpha }^{2}}\\ +\,\displaystyle \frac{48{a}_{2}^{{\prime} }(x){{\rm{sech}} }^{2}(2x)\tanh (2x){t}^{2\alpha }}{{\alpha }^{2}}\\ {{-\displaystyle \frac{12{b}_{2}^{{\prime} }(x){{\rm{sech}} }^{2}(2x){t}^{2\alpha }}{{\alpha }^{2}})}_{}|}_{t=0}\\ {+{(\displaystyle \frac{{b}_{2}(x){a}_{2}^{{\prime} }(x){t}^{3\alpha }}{{\alpha }^{3}}+\displaystyle \frac{{a}_{2}(x){b}_{2}^{{\prime} }(x){t}^{3\alpha }}{{\alpha }^{3}})}_{}|}_{t=0}=0. $That is, ${a}_{2}\left(x\right)=-16{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right)$ and ${b}_{2}\left(x\right)=32(-2+\,\cosh (4x)){{\rm{sech}} }^{4}\left(2x\right).$ Therefore, the 2nd-MFPS approximate solution of PDFEs (23) and (24) are$ \begin{eqnarray}\begin{array}{lll}{v}_{2}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right){t}^{\alpha }}{\alpha }\\ & & -\,\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{2\alpha }}{{\alpha }^{2}},\\ {w}_{2}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,\displaystyle \frac{32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{\alpha }}{\alpha }\\ & & +\,\displaystyle \frac{16(-2+\,\cosh (4x)){{\rm{sech}} }^{4}\left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}.\end{array}\end{eqnarray}$For $n=3,$ substituting the 3rd-truncated series solutions,$ \begin{eqnarray}\begin{array}{lll}{v}_{3}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right){t}^{\alpha }}{\alpha }\\ & & -\,\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}+{a}_{3}\left(x\right)\displaystyle \frac{{t}^{3\alpha }}{3!{\alpha }^{3}},\\ {w}_{3}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,\displaystyle \frac{32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{\alpha }}{\alpha }\\ & & +\,\displaystyle \frac{16\left(-2+\,\cosh \left(4x\right)\right){{\rm{sech}} }^{4}\left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}\\ & & +\,{b}_{3}\left(x\right)\displaystyle \frac{{t}^{3\alpha }}{3!{\alpha }^{3}},\end{array}\end{eqnarray}$into the 3rd-truncated residual $\left\{{{\rm{Res}}}_{v}^{3}\left(x,t\right),{{\rm{Res}}}_{w}^{3}\left(x,t\right)\right\},$ operating ${T}_{t}^{\alpha }$ twice on both sides of the resulting equations $\left\{{T}_{t}^{2\alpha }{{\rm{Res}}}_{v}^{3}\left(x,t\right),{T}_{t}^{2\alpha }{{\rm{Res}}}_{w}^{3}\left(x,t\right)\right\},$ and using $\left\{{T}_{t}^{2\alpha }{{\rm{Res}}}_{v}^{3}{\left(x,t\right)}_{\left|t=0\right.}=0,{T}_{t}^{2\alpha }{{\rm{Res}}}_{w}^{3}{\left(x,t\right)}_{\left|t=0\right.}=0\right\},$ one can get the 3rd-unknown coefficients as$ \begin{eqnarray}\begin{array}{l}{a}_{3}\left(x\right)=-16\left(-2+\,\tanh \left(4x\right)\right){{\rm{sech}} }^{4}\left(2x\right),\\ {b}_{3}\left(x\right)=32\left(-11\,\sinh \left(2x\right)+\,\sinh \left(6x\right)\right){{\rm{sech}} }^{5}\left(2x\right).\end{array}\end{eqnarray}$Therefore, the 3rd-MFPS approximate solutions are$ \begin{eqnarray}\begin{array}{lll}{v}_{3}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right){t}^{\alpha }}{\alpha }\\ & & -\,\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}\\ & & -\,\displaystyle \frac{8\left(-2+\,\tanh \left(4x\right)\right){{\rm{sech}} }^{4}\left(2x\right){t}^{3\alpha }}{3{\alpha }^{3}},\\ {w}_{3}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,\displaystyle \frac{32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{\alpha }}{\alpha }\\ & & +\,\displaystyle \frac{16\left(-2+\,\cosh \left(4x\right)\right){{\rm{sech}} }^{4}\left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}\\ & & +\,\displaystyle \frac{16(-11\,\sinh (2x)+\,\sinh (6x)){{\rm{sech}} }^{5}\left(2x\right){t}^{3\alpha }}{3{\alpha }^{3}}.\end{array}\end{eqnarray}$As the former, the 4th-unknown coefficients ${a}_{4}\left(x\right)$ and ${b}_{4}\left(x\right)$ can be obtained using the same manner of CRPS with the help of the fact that ${T}_{t}^{3\alpha }{{\rm{Res}}}_{v}^{4}\left(x,0\right)=$${T}_{t}^{3\alpha }{{\rm{Res}}}_{w}^{4}\left(x,0\right)=0$ such that$ \begin{eqnarray}\begin{array}{l}{a}_{4}\left(x\right)=-16\left(-11\,\sinh \left(2x\right)+\,\sinh \left(6x\right)\right){{\rm{sech}} }^{5}\left(2x\right),\\ {b}_{4}\left(x\right)=32\left(33-26\,\cosh (4x)+\,\cosh (8x)\right){{\rm{sech}} }^{6}\left(2x\right).\end{array}\end{eqnarray}$Thus, the 4th-MFPS approximate solution of FPDEs (23) and (24) are$ \begin{eqnarray}\begin{array}{lll}{v}_{4}\left(x,t\right) & = & \displaystyle \frac{1}{2}-8\,\tanh \left(-2x\right)-\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right){t}^{\alpha }}{\alpha }\\ & & -\,\displaystyle \frac{8{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}\\ & & -\,\displaystyle \frac{8(-2+\,\tanh \left(4x\right)){{\rm{sech}} }^{4}\left(2x\right){t}^{3\alpha }}{3{\alpha }^{3}}\\ & & -\,\displaystyle \frac{2(-11\,\sinh (2x)+\,\sinh (6x)){{\rm{sech}} }^{5}\left(2x\right){t}^{4\alpha }}{3{\alpha }^{4}},\\ {w}_{4}\left(x,t\right) & = & 16-16{\tanh }^{2}\left(-2x\right)\\ & & +\,\displaystyle \frac{32{{\rm{sech}} }^{2}\left(2x\right)\tanh \left(2x\right){t}^{\alpha }}{\alpha }\\ & & +\,\displaystyle \frac{16\left(-2+\,\cosh \left(4x\right)\right){{\rm{sech}} }^{4}\left(2x\right){t}^{2\alpha }}{{\alpha }^{2}}\\ & & +\,\displaystyle \frac{16\left(-11\,\sinh \left(2x\right)+\,\sinh \left(6x\right)\right){{\rm{sech}} }^{5}\left(2x\right){t}^{3\alpha }}{3{\alpha }^{3}}\\ & & +\,\displaystyle \frac{4\left(33-26\,\cosh (4x)+\,\cosh (8x)\right){{\rm{sech}} }^{6}\left(2x\right){t}^{4\alpha }}{3{\alpha }^{4}}.\end{array}\end{eqnarray}$By continuing this procedure, the rest of the components up to arbitrary order n can be computed and finally the multiple CRPS solution in terms of infinite series can be obtained. However, few terms of the unknown coefficients of equation (26) were computed as follows$ \begin{eqnarray}\begin{array}{lll}{a}_{5}\left(x\right) & = & -16\left(33-26\,\cosh (4x)\right.\\ & & \left.+\,\cosh (8x)\right){{\rm{sech}} }^{6}\left(2x\right),\\ {b}_{5}\left(x\right) & = & 32\left(302\,\sinh (2x)-57\,\sinh (6x)\right.\\ & & +\left.\,\sinh (10x)\right){{\rm{sech}} }^{7}\left(2x\right),\\ {a}_{6}\left(x\right) & = & -\,16\left(302\,\sinh (2x)-57\,\sinh (6x)\right.\\ & & \left.+\,\sinh (10x)\right){{\rm{sech}} }^{7}\left(2x\right),\\ {b}_{6}\left(x\right) & = & 32\left(-1208+1191\,\cosh (4x)-120\,\cosh (8x)\right.\\ & & \left.+\,\cosh (12x)\right){{\rm{sech}} }^{8}\left(2x\right).\end{array}\end{eqnarray}$To show the efficiency and reliability of the CRPS method, the surface plots of the 6th-approximate and exact solutions of coupled WBK system (23) and (24) are illustrated in figure 1 for $\alpha =1,$$-1\leqslant t\leqslant 1$ and $-\pi \leqslant x\leqslant \pi .$ Due to the high accuracy of the CRPS method, there are nearly no differences between the graphs of approximate solutions and exact solutions.The absolute and relative errors is obtained at $\alpha =1,$$n=7,$ and summarized in table 1 for a fixed value of $x=\pi $ and some selected grid points of $t\in [0,1]$ with step size $0.2,$ which displays excellent solutions within a few iterations. Figures 2 and 3 show surface wave plots of the 6th-CPRS solutions of equation (26) for both dependent variables $v$6(x, t) and $w$6(x, t), respectively, such that $\alpha \,\in $ {0.95, 0.75, 0.5, 0.25} for each $t\in [0,1]\,{\rm{a}}{\rm{n}}{\rm{d}}\,{\rm{x}}\in [-\pi ,\pi ]$. Figures 5 and 6 show surface waves plots of the 6th-CRPS solutions of coupled WBK system (23) and (24) at different values of fractional-order levels such that $\alpha \in \{0.95,0.75,0.5,0.25\}$ for each $t\in \left[0,1\right]$ and $x\in [-\pi ,\pi ].$ From these figures, it can be noted that the behavior of the approximate solution almost coincides with the nature of the exact solution for the coupled WBK system (23) and (24) considering integer derivative of $\alpha .$ While figure 4 show the 2D behavior of the exact solutions for the coupled WBK system (23) and (24) compared with CRPS method and Laplace Adomian decomposition method (LADM) [48] at different fractional levels $\alpha =1$ and $\alpha =0.25$ with $n=2,$$t=1,$ and $-6\leqslant x\leqslant 6.$ From figure 4, it can be observed that the behavior of 2nd CRPS solutions is closer to the exact solutions than those obtained by LADM corresponding to fractional levels of conformable meaning. Generally, from aforesaid results, the CRPS method provides excellent solutions from a few iterations as compared to other numerical solvers, which illustrate the reliability and efficiency of the considered method.
Figure 1.
New window|Download| PPT slide Figure 1.Comparison of exact and 6th-CRPS approximate graphs of the coupled WBK system (23) and (24) for $-1\leqslant t\leqslant 1$ and $-\pi \leqslant x\leqslant \pi :$ (a) exact solution $v\left(x,t\right);$ (b) CRPS solution ${v}_{6}\left(x,t\right)$ at $\alpha =1;$ (c) exact solution $w\left(x,t\right);$ and (d) CRPS solution ${w}_{6}\left(x,t\right)$ at $\alpha =1.$
Figure 2.
New window|Download| PPT slide Figure 2.Surface waves plots of CRPS solution, ${v}_{6}\left(x,t\right),$ of equation (26) at different fractional-order levels for each $0\leqslant t\leqslant 1$ and $-\pi \leqslant x\leqslant \pi :$ (a) $\alpha =0.95;$ (b) $\alpha =0.75;$ and (c) $\alpha =0.5;$ (d) $\alpha =0.25.$
Figure 3.
New window|Download| PPT slide Figure 3.Surface waves plots of CRPS solution, ${w}_{6}\left(x,t\right),$ of equation (26) at different fractional-order levels for each $0\leqslant t\leqslant 1$ and $-\pi \leqslant x\leqslant \pi :$ (a) $\alpha =0.95;$ (b) $\alpha =0.75;$ (c) $\alpha =0.5;$ and (d) $\alpha =0.25.$
Table 1. Table 1.Errors results for the 7th-CRPS of coupled system (23) and (24) at $\alpha =1.$
New window|Download| PPT slide Figure 4.Comparison of exact solutions with CRPS method and LADM [48] in 2D for system (23) and (24) for $t=1$ and $-6\leqslant x\leqslant 6,$ in which exact (red); CRPS (blue); LADM (green): (a) $v\left(x,t\right)$ solutions at $\alpha =1;$ (b) $w\left(x,t\right)$ solutions at $\alpha =1;$ (c) $v\left(x,t\right)$ solutions at $\alpha =0.25;$ and (d) $w\left(x,t\right)$ solutions at $\alpha =0.25.$
As a special case, consider the coupled fractional WBK equations with $\eta =1$ and $\xi =0:$$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }v+v{v}_{x}+{w}_{x}=0,\\ {T}_{t}^{\alpha }w+{(vw)}_{x}+{v}_{xxx}=0,\end{array}\end{eqnarray}$along with the following initial conditions:$ \begin{eqnarray}\begin{array}{l}v\left(x,0\right)=\mu -2k\,\coth \left(k\left(x+h\right)\right),\\ w\left(x,0\right)=-2{k}^{2}{\mathrm{csch}}^{2}\left(k\left(x+h\right)\right),\end{array}\end{eqnarray}$where $\mu ,$$k,$$h$ are real finite constants.
This fractional model converts the fractional coupled WBK system into fractional coupled equations of MB, which is an appropriate estimation tool for weak nonlinear and LWs occurring in fluid mechanics described a dynamic system in Newtonian fluids flow. The exact solutions at $\alpha =1$ are$ \begin{eqnarray}\begin{array}{l}\begin{array}{c}v\left(x,t\right)\end{array}=\mu -2k\,\coth \left(k\left(\left(x+h\right)-\mu t\right)\right),\\ \begin{array}{c}w\left(x,t\right)\end{array}=-2{k}^{2}{\mathrm{csch}}^{2}\left(k\left(\left(x+h\right)-\mu t\right)\right).\end{array}\end{eqnarray}$Define the nth-truncated series solutions by$ \begin{eqnarray}\begin{array}{l}{v}_{n}\left(x,t\right)=\mu -2k\,\coth (k\left(x+h\right))+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\\ {w}_{n}\left(x,t\right)=-2{k}^{2}{\mathrm{csch}}^{2}\left(k\left(x+h\right)\right)+\displaystyle \sum _{k=1}^{n}{b}_{k}(x)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\end{array}\end{eqnarray}$and the nth-truncated residual functions by$ \begin{eqnarray}\begin{array}{l}{{\rm{Res}}}_{v}^{n}\left(x,t\right)={T}_{t}^{\alpha }{v}_{n}\left(x,t\right)+{v}_{n}\left(x,t\right){\left({v}_{n}\left(x,t\right)\right)}_{x}+{\left({w}_{n}\left(x,t\right)\right)}_{x},\\ {{\rm{Res}}}_{w}^{n}\left(x,t\right)={T}_{t}^{\alpha }{w}_{n}\left(x,t\right)+{\left({v}_{n}\left(x,t\right){w}_{n}\left(x,t\right)\right)}_{x}\\ \,+{\left({v}_{n}\left(x,t\right)\right)}_{xxx}.\end{array}\end{eqnarray}$Following the procedure of the CRPS steps, one can get$ \begin{eqnarray}\begin{array}{rcl}{v}_{0}\left(x,t\right) & = & \mu -2k\,\coth (k\left(x+h\right)),\\ {w}_{0}\left(x,t\right) & = & -2{k}^{2}{\mathrm{csch}}^{2}\left(k\left(x+h\right)\right),\\ {v}_{1}\left(x,t\right) & = & \displaystyle \frac{-2{k}^{2}\mu {\mathrm{csch}}^{2}\left(k\left(x+h\right)\right)}{\alpha }{t}^{\alpha },\\ {w}_{1}\left(x,t\right) & = & \displaystyle \frac{-4{k}^{3}\mu \,\coth \left(k\left(x+h\right)\right){\mathrm{csch}}^{2}\left(k\left(x+h\right)\right)}{\alpha }{t}^{\alpha },\\ {v}_{2}\left(x,t\right) & = & \displaystyle \frac{-2{k}^{3}{\mu }^{2}\,\coth \left(k\left(x+h\right)\right){\mathrm{csch}}^{2}\left(k\left(x+h\right)\right)}{{\alpha }^{2}}{t}^{2\alpha },\\ {w}_{2}\left(x,t\right) & = & \displaystyle \frac{-2{k}^{4}{\mu }^{2}(2+\,\cosh \left(2k\left(x+h\right)\right)){\mathrm{csch}}^{4}\left(k\left(x+h\right)\right)}{{\alpha }^{2}}{t}^{2\alpha },\\ {v}_{3}\left(x,t\right) & = & \displaystyle \frac{-2{k}^{4}{\mu }^{3}(2+\,\cosh \left(2k\left(x+h\right)\right)){\mathrm{csch}}^{4}\left(k\left(x+h\right)\right)}{3{\alpha }^{3}}{t}^{3\alpha },\\ {w}_{3}\left(x,t\right) & = & \displaystyle \frac{-2{k}^{5}{\mu }^{3}\left(11\,\cosh \left(k\left(x+h\right)\right)+\,\cosh \left(3k\left(x+h\right)\right)\right){\mathrm{csch}}^{5}\left(k\left(x+h\right)\right)}{3{\alpha }^{3}}{t}^{3\alpha },\\ {v}_{4}\left(x,t\right) & = & \displaystyle \frac{-{k}^{5}{\mu }^{4}\left(11\,\cosh \left(k\left(x+h\right)\right)+\,\cosh \left(3k\left(x+h\right)\right)\right){\mathrm{csch}}^{5}\left(k\left(x+h\right)\right)}{6{\alpha }^{4}}{t}^{4\alpha },\\ {w}_{4}\left(x,t\right) & = & \displaystyle \frac{-{k}^{6}{\mu }^{4}\left(33+26\,\cosh \left(2k\left(x+h\right)\right)+\,\cosh \left(4k\left(x+h\right)\right)\right){\mathrm{csch}}^{6}\left(k\left(x+h\right)\right)}{6{\alpha }^{4}}{t}^{4\alpha },\end{array}\end{eqnarray}$and so on, the other components of CFPS (43) can be obtained in a similar manner. Thus, the CRPS approximate solutions in the series form are presented as$ \begin{eqnarray}\begin{array}{rcl}v\left(x,t\right) & = & {v}_{0}\left(x,t\right)+{v}_{1}\left(x,t\right)+{v}_{2}\left(x,t\right)+{v}_{3}\left(x,t\right)+\ldots ,\\ w\left(x,t\right) & = & {w}_{0}\left(x,t\right)+{w}_{1}\left(x,t\right)+{w}_{2}\left(x,t\right)+{w}_{3}\left(x,t\right)+\ldots .\end{array}\end{eqnarray}$To illustrate the efficiency and reliability of the proposed method in resolving the coupled MB system, numerical simulations are performed and the results are reported in figure 5 and tables 2 and 3. For specific, figure 5 explores the 2D behavior of the exact and CRPS solutions for the coupled MB system (40) and (41) at different values of fractional order $\alpha ,$ where $\alpha \in \left\{0.6,0.7,0.8,0.9,1.0\right\}$ with $x=0.5,$$\mu =0.005,$$k=0.1,$$h=10,$ and $t\in [0,1].$ Here, it can be noted that the CRPS pattern solutions are in good agreement with each other and with the exact solution. In tables 2 and 3, the numerical solutions obtained by the CRPS method are compared with two computational methods, the optimal homotopy asymptotic method (OHAM) [49] and Adomian decomposition method (ADM) [50], in terms of absolute error. From these Tables, we can see that the absolute errors obtained by the considered method are better than OHAM and ADM.
Consider the nonlinear fractional coupled JM system [28]:
$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }v+{v}_{xxx}+\displaystyle \frac{3}{2}w{w}_{xxx}+\displaystyle \frac{9}{2}{w}_{x}{w}_{xx}-6v{v}_{x}\\ \,-\,6vw{w}_{x}-\displaystyle \frac{3}{2}{v}_{x}{w}^{2}=0,\\ {T}_{t}^{\alpha }w+{w}_{xxx}-6{v}_{x}w-6v{w}_{x}-\displaystyle \frac{15}{2}{w}_{x}{w}^{2}=0,\end{array}\end{eqnarray}$along with the initial conditions$ \begin{eqnarray}\begin{array}{l}v\left(x,0\right)=\displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right),\\ w\left(x,0\right)=c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right).\end{array}\end{eqnarray}$The exact solutions of FPDEs (47) and (48) at $\alpha =1$ are$ \begin{eqnarray}\begin{array}{l}\begin{array}{c}v\left(x,t\right)\end{array}=\displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{c}{2}\left(x+\displaystyle \frac{1}{2}{c}^{2}t\right)\right)\right),\\ \begin{array}{c}w\left(x,t\right)\end{array}=c\,{\rm{sech}} \left(\displaystyle \frac{c}{2}\left(x+\displaystyle \frac{1}{2}{c}^{2}t\right)\right).\end{array}\end{eqnarray}$To apply the CFPS steps, let us assume that the $n$th-truncated series solutions of system (47) and (48) have the following multiple CFPS about ${t}_{0}=0:$$ \begin{eqnarray}\begin{array}{l}{v}_{n}\left(x,t\right)=\displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right)+\displaystyle \sum _{k=1}^{n}{a}_{k}\left(x\right)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\\ {w}_{n}\left(x,t\right)=c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)+\displaystyle \sum _{k=1}^{n}{b}_{k}(x)\displaystyle \frac{{t}^{k\alpha }}{{\alpha }^{k}k!},\end{array}\end{eqnarray}$and the $n$th-truncated residual functions are defined by$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{v}^{n}\left(x,t\right) & = & {T}_{t}^{\alpha }{v}_{n}\left(x,t\right)+{\left({v}_{n}\left(x,t\right)\right)}_{xxx}\\ & & +\,\displaystyle \frac{3}{2}{w}_{n}\left(x,t\right){\left({w}_{n}\left(x,t\right)\right)}_{xxx}\\ & & +\,\displaystyle \frac{9}{2}{\left({w}_{n}\left(x,t\right)\right)}_{x}{\left({w}_{n}\left(x,t\right)\right)}_{xx}\\ & & -\,6{v}_{n}\left(x,t\right){\left({v}_{n}\left(x,t\right)\right)}_{x}\\ & & -\,6{v}_{n}\left(x,t\right){w}_{n}\left(x,t\right){\left({w}_{n}\left(x,t\right)\right)}_{x}\\ & & -\,\displaystyle \frac{3}{2}{\left({v}_{n}\left(x,t\right)\right)}_{x}{({w}_{n}\left(x,t\right))}^{2},\\ {{\rm{Res}}}_{w}^{n}\left(x,t\right) & = & {T}_{t}^{\alpha }{w}_{n}\left(x,t\right)+{\left({w}_{n}\left(x,t\right)\right)}_{xxx}\\ & & -\,6{\left({v}_{n}\left(x,t\right)\right)}_{x}{w}_{n}\left(x,t\right)\\ & & -\,6{v}_{n}\left(x,t\right){\left({w}_{n}\left(x,t\right)\right)}_{x}\\ & & -\,\displaystyle \frac{15}{2}{\left({w}_{n}\left(x,t\right)\right)}_{x}{({w}_{n}\left(x,t\right))}^{2}.\end{array}\end{eqnarray}$Now, to find the first unknown coefficients ${a}_{1}\left(x\right)$ and ${b}_{1}\left(x\right)$ of MFPS (50), substitute the 1st-truncated series$ \begin{eqnarray}\begin{array}{l}{v}_{1}\left(x,t\right)=\displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right)+{a}_{1}\left(x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\\ {w}_{1}\left(x,t\right)=c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)+{b}_{1}\left(x\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\end{array}\end{eqnarray}$into the 1st-truncated residual functions $\left\{{{\rm{Res}}}_{v}^{1}\left(x,t\right),{{\rm{Res}}}_{w}^{1}\left(x,t\right)\right\},$ and then calculate the resulting equations at $t=0$ such that$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{v}^{1}\left(x,t\right) & = & {a}_{1}\left(x\right)-\displaystyle \frac{1}{4}{c}^{5}{{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right)\tanh \left(\displaystyle \frac{cx}{2}\right)\\ & & -\,\displaystyle \frac{1}{4}{c}^{5}{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right){\tanh }^{3}\left(\displaystyle \frac{cx}{2}\right),\\ {{\rm{Res}}}_{w}^{1}\left(x,t\right) & = & {b}_{1}\left(x\right)+\displaystyle \frac{3}{8}{c}^{4}\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)\tanh \left(\displaystyle \frac{cx}{2}\right)\\ & & -\,\displaystyle \frac{1}{8}{c}^{4}{{\rm{sech}} }^{3}\left(\displaystyle \frac{cx}{2}\right)\tanh \left(\displaystyle \frac{cx}{2}\right)\\ & & -\,\displaystyle \frac{1}{8}{c}^{4}\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right){\tanh }^{3}\left(\displaystyle \frac{cx}{2}\right).\end{array}\end{eqnarray}$By using ${{\rm{Res}}}_{v}^{1}\left(x,0\right)=0$ and ${{\rm{Res}}}_{w}^{1}\left(x,0\right)=0,$ one can get$ \begin{eqnarray}\begin{array}{l}{a}_{1}\left(x\right)=2{c}^{5}{\mathrm{csch}}^{3}\left(cx\right)\,{\sinh }^{4}\left(\displaystyle \frac{cx}{2}\right),\\ {b}_{1}\left(x\right)=-{c}^{4}{\mathrm{csch}}^{2}\left(cx\right)\,{\sinh }^{3}\left(\displaystyle \frac{cx}{2}\right).\end{array}\end{eqnarray}$Therefore, the 1st-CPS approximation of system (47) and (48) are$ \begin{eqnarray}\begin{array}{lll}{v}_{1}\left(x,t\right) & = & \displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right)\\ & & +\,2{c}^{5}{\mathrm{csch}}^{3}\left(cx\right)\,{\sinh }^{4}\left(\displaystyle \frac{cx}{2}\right)\displaystyle \frac{{t}^{\alpha }}{\alpha },\\ {w}_{1}\left(x,t\right) & = & c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)-{c}^{4}{\mathrm{csch}}^{2}\left(cx\right)\,{\sinh }^{3}\left(\displaystyle \frac{cx}{2}\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }.\end{array}\end{eqnarray}$For $n=2,$ we substitute the 2nd-truncated series solutions$ \begin{eqnarray}\begin{array}{lll}{v}_{2}\left(x,t\right) & = & \displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right)\\ & & +\,2{c}^{5}{\mathrm{csch}}^{3}\left(cx\right)\,{\sinh }^{4}\left(\displaystyle \frac{cx}{2}\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }\\ & & +\,{a}_{2}\left(x\right)\displaystyle \frac{{t}^{2\alpha }}{2{\alpha }^{2}},\\ {w}_{2}\left(x,t\right) & = & c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)-{c}^{4}{\mathrm{csch}}^{2}\left(cx\right)\,{\sinh }^{3}\left(\displaystyle \frac{cx}{2}\right)\displaystyle \frac{{t}^{\alpha }}{\alpha }\\ & & +\,{b}_{2}\left(x\right)\displaystyle \frac{{t}^{2\alpha }}{2{\alpha }^{2}},\end{array}\end{eqnarray}$and into the 2nd-truncated residual functions $\left\{{{\rm{Res}}}_{v}^{2}\left(x,t\right),{{\rm{Res}}}_{w}^{2}\left(x,t\right)\right\},$ and operating ${T}_{t}^{\alpha }$ on both sides of the resulting equations $\left\{{T}_{t}^{\alpha }{{\rm{Res}}}_{v}^{2}\left(x,t\right),{T}_{t}^{\alpha }{{\rm{Res}}}_{w}^{2}\left(x,t\right)\right\}$ with the help of ${T}_{t}^{\alpha }{t}^{p}=0$ for $p\gt \alpha $ at $t=0,$ it yields:$ \begin{eqnarray}\begin{array}{l}{T}_{t}^{\alpha }{{\rm{Res}}}_{v}^{2}{\left(x,t\right)}_{\left|t=0\right.}={a}_{2}\left(x\right)-\displaystyle \frac{1}{16}{c}^{8}\left(2-\,\cosh \left(cx\right)\right)\\ \,\times {{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right)=0,\\ {T}_{t}^{\alpha }{{\rm{Res}}}_{w}^{2}{\left(x,t\right)}_{\left|t=0\right.}={b}_{2}\left(x\right)+\displaystyle \frac{1}{32}{c}^{7}\left(3-\,\cosh \left(cx\right)\right)\\ \,\times {{\rm{sech}} }^{3}\left(\displaystyle \frac{cx}{2}\right)=0.\end{array}\end{eqnarray}$Therefore$ \begin{eqnarray}\begin{array}{l}{a}_{2}\left(x\right)=\displaystyle \frac{1}{16}{c}^{8}(2-\,\cosh \left(cx\right)){{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right),\\ {b}_{2}\left(x\right)=\displaystyle \frac{-1}{32}{c}^{7}(3-\,\cosh \left(cx\right)){{\rm{sech}} }^{3}\left(\displaystyle \frac{cx}{2}\right)\end{array}\end{eqnarray}$and the 2nd-CPS approximate solution of coupled system (47) and (48) are$ \begin{eqnarray}\begin{array}{lll}{v}_{2}\left(x,t\right) & = & \displaystyle \frac{1}{8}{c}^{2}\left(1-4{{\rm{sech}} }^{2}\left(\displaystyle \frac{cx}{2}\right)\right)\\ & & +\,\displaystyle \frac{2}{\alpha }{c}^{5}{\mathrm{csch}}^{3}\left(cx\right)\,{\sinh }^{4}\left(\displaystyle \frac{cx}{2}\right){t}^{\alpha }\\ & & +\,\displaystyle \frac{1}{32{\alpha }^{2}}{c}^{8}\left(2-\,\cosh \left(cx\right)\right){{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right){t}^{2\alpha },\\ {w}_{2}\left(x,t\right) & = & c\,{\rm{sech}} \left(\displaystyle \frac{cx}{2}\right)\\ & & -\,\displaystyle \frac{1}{\alpha }{c}^{4}{\mathrm{csch}}^{2}\left(cx\right)\,{\sinh }^{3}\left(\displaystyle \frac{cx}{2}\right){t}^{\alpha }\\ & & -\,\displaystyle \frac{1}{64{\alpha }^{2}}{c}^{7}\left(3-\,\cosh \left(cx\right)\right){{\rm{sech}} }^{3}\left(\displaystyle \frac{cx}{2}\right){t}^{2\alpha }.\end{array}\end{eqnarray}$By continuing this procedure, the rest of the components of equation (50) up to arbitrary order $n$ can be computed and finally the multiple CFPS solution in terms of infinite series can be obtained. However, the following CRPS approximations are given$ \begin{eqnarray}\begin{array}{l}{v}_{3}\left(x,t\right)={v}_{2}\left(x,t\right)-\displaystyle \frac{1}{192{\alpha }^{3}}{c}^{11}\left(5-\,\cosh (cx)\right){{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right)\\ \,\times \,\tanh \left(\displaystyle \frac{cx}{2}\right){t}^{3\alpha },\\ {w}_{3}\left(x,t\right)={w}_{2}\left(x,t\right)+\displaystyle \frac{1}{1536{\alpha }^{3}}{c}^{10}\left(23{\rm{Ssinh}}\left(\displaystyle \frac{cx}{2}\right)\right.\\ \,-\left.\sinh \left(\displaystyle \frac{3cx}{2}\right)\right){{\rm{sech}} }^{4}\left(\displaystyle \frac{cx}{2}\right){t}^{3\alpha },\\ {v}_{4}\left(x,t\right)={v}_{3}\left(x,t\right)-\displaystyle \frac{1}{6144{\alpha }^{4}}{c}^{14}\left(33-26\,\cosh (cx)\right.\\ \,+\left.\cosh (2cx)\right){{\rm{sech}} }^{6}\left(\displaystyle \frac{cx}{2}\right){t}^{4\alpha },\\ {w}_{4}\left(x,t\right)={w}_{3}\left(x,t\right)+\displaystyle \frac{1}{49152{\alpha }^{4}}{c}^{13}\left(115-76\,\cosh \left(cx\right)\right.\\ \,+\left.\cosh \left(2cx\right)\right){{\rm{sech}} }^{5}\left(\displaystyle \frac{cx}{2}\right){t}^{4\alpha },\\ {v}_{5}\left(x,t\right)={v}_{4}\left(x,t\right)+\displaystyle \frac{1}{1024(5!){\alpha }^{5}}{c}^{17}\\ \left(302\,\sinh \left(\displaystyle \frac{cx}{2}\right)-57\,\sinh \left(\displaystyle \frac{3cx}{2}\right)+\,\sinh \left(\displaystyle \frac{5cx}{2}\right)\right)\\ \,\times {{\rm{sech}} }^{7}\left(\displaystyle \frac{cx}{2}\right){t}^{5\alpha },\\ {w}_{5}\left(x,t\right)={w}_{4}\left(x,t\right)-\displaystyle \frac{1}{8192\left(5!\right){\alpha }^{5}}{c}^{16}\\ \left(723-236\,\cosh \left(cx\right)+\,\cosh \left(2cx\right)\right){{\rm{sech}} }^{5}\left(\displaystyle \frac{cx}{2}\right)\\ \,\times \,\tanh \left(\displaystyle \frac{cx}{2}\right){t}^{5\alpha }.\end{array}\end{eqnarray}$Thus, the CRPS approximate solutions in the series form are presented by$ \begin{eqnarray}\begin{array}{l}v\left(x,t\right)={v}_{0}\left(x,t\right)+{v}_{1}\left(x,t\right)+{v}_{2}\left(x,t\right)+{v}_{3}\left(x,t\right)+\ldots ,\\ w\left(x,t\right)={w}_{0}\left(x,t\right)+{w}_{1}\left(x,t\right)+{w}_{2}\left(x,t\right)+{w}_{3}\left(x,t\right)+\ldots .\end{array}\end{eqnarray}$To demonstrate the effectiveness of the CRPS method for Example 2 and to avoid difficult fractional differentiation computation, the coupled surfaces of the 4th-CRPS approximation geometrically has been plotted in 3D space graphs in figure 6 with $c=0.5,$$\alpha =1,$$-50\leqslant x\leqslant 50,$ and $0\leqslant t\leqslant 10.$ While the 2D graphical representation of the exact and 4th-approximated solutions of Example 2 are plot in figure 7 for $=1,$$c=0.5,$$t=1,$ and $x\in \left[-50,50\right],$ which guides us to a deep understanding of the behavior of the coupled JM system. Agreement between numerical results obtained by CRPS with exact solutions appears very appreciable by means of illustrative results in figure 7. Finally, figures 8 and 9 demonstrate the surface behavior of CRPS approximations of coupled JM system for distinct values of fractional levels $\alpha $ for $c=0.25,$$0\leqslant t\leqslant 10,$ and $-10\leqslant x\leqslant 10.$ From such 3D graphics, we conclude that the graphs almost match, similar in their behaviors, and in good agreement with each other, especially when considering the integer-order derivative. Indeed, the fractional orders have strong effects on the model profiles, which tend to lead to unusual behaviors in the event of a significant departure from the integer value $\alpha =1.$
Figure 5.
New window|Download| PPT slide Figure 5.Graphs of exact and CRPS solution behavior of coupled MB system (40) and (41) at different values of $\alpha $ with $x=0.5$ and $0\leqslant t\leqslant 1,$ where red for the exact; blue $\alpha =1;$ green $\alpha =0.9;$ yellow $\alpha =0.8;$ orange $\alpha =0.7;$ gray $\alpha =0.6:$ (a) $v\left(x,t\right)$ solutions; (b) $w\left(x,t\right)$ solutions.
Table 2. Table 2.Comparison of absolute errors, ${v}_{2}(x,t),$ of system (40) and (41) at $\alpha =1.$
New window|Download| PPT slide Figure 6.Surface graphs of exact and 4th-CRPS solutions for example 2 at $\alpha =1,$$c=0.5,$$0\leqslant t\leqslant 10,$ and $-50\leqslant x\leqslant 50:$ (a) exact solution $v(x,t);$ (b) CRPS solution ${v}_{4}\left(x,t\right);$ (c) exact solution $w(x,t);$ and (d) CRPS solution ${w}_{4}\left(x,t\right).$
Figure 7.
New window|Download| PPT slide Figure 7.Nature of exact and 4th-CRPS solutions in 2D for example 2 at $=1,$$c=0.5,$$t=1,$ and $-50\leqslant x\leqslant 50:$ (a) exact solutions $v(x,t);$ (b) exact solutions $w(x,t);$ (c) CRPS solutions ${v}_{4}(x,t);$ and (d) CRPS solutions ${w}_{4}(x,t).$
Figure 8.
New window|Download| PPT slide Figure 8.Surface graphs of 4th-CRPS solutions, ${v}_{4}\left(x,t\right),$ of Example 2 at different fractional-order levels for $c=0.25,$$0\leqslant t\leqslant 10,$ and $-10\leqslant x\leqslant 10:$ (a) $\alpha =1;$ (b) $\alpha =0.75;$ (c) $\alpha =0.5;$ and (d) $\alpha =0.25.$
Figure 9.
New window|Download| PPT slide Figure 9.Surface graphs of 4th-CRPS solutions, ${w}_{4}\left(x,t\right),$ of Example 2 at different fractional-order levels for $c=0.25,$$0\leqslant t\leqslant 10,$ and $-10\leqslant x\leqslant 10:$ (a) $\alpha =1;$ (b) $\alpha =0.75;$ (c) $\alpha =0.5;$ and (d) $\alpha =0.25.$
5. Closing comments
This article targets to extend the application of the CFPS method to explore analytic-approximate solutions of a coupled system of nonlinear FPDEs occurring in shallow water waves with conformable fractional derivative. This target has been successfully achieved using the suggested approach that when directly implemented provides suitable approximations of conformable multiple series solutions with easily determinable components without any need for transformation, discretization, or limitations by deriving the residual error functions. Some efficacious experiments were presented to validate the capacity and reliability of the CRPS approach, including a coupled system of conformable WBK model, conformable MB model, and conformable JM model with Schrödinger's potential energy. The CRPS results were compared with exact solutions at $\alpha =\,1,$ and with each other for different values of fractional levels as well as those obtained by other existing methods, which were found in good agreement with each other even after computing a few iterations. From the numerical and graphical results, it can be concluded that the proposed method is a systematic, powerful, and suitable tool for the FPDEs systems, valid for a long time with great potential in scientific applications, and easy to implement when compared to other numerical techniques.
Acknowledgments
Authors gratefully acknowledge Ajman University for providing facilities for our research under Grant Ref. No. 2019-IRG-HBS-11.
MillerK SRossB1993An Introduction to the Fractional Calculus and Fractional Differential Equations New York Wiley [Cited within: 1]
KilbasASrivastavaHTrujilloJ2006Theory and Applications of Fractional Differential Equations Amsterdam Elsevier
PodlubnyI1999Fractional Differential Equations San Diego, CA Academic
MainardiF2010Fractional Calculus and Waves in Linear Viscoelasticity London Imperial College Press
ZaslavskyG M2005Hamiltonian Chaos and Fractional Dynamics Oxford Oxford University Press [Cited within: 1]
Al-SmadiMAbu ArqubO2019 Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates Appl. Math. Comput. 342 280294 DOI:10.1016/j.amc.2018.09.020 [Cited within: 1]
OsmanM SGhanbariBMachadoJ A T2019 New complex waves in nonlinear optics based on the complex Ginzburg–Landau equation with Kerr law nonlinearity Eur. Phys. J. Plus 134 20 DOI:10.1140/epjp/i2019-12442-4
Abu ArqubOOdibatZAl-SmadiM2018 Numerical solutions of time-fractional partial integrodifferential equations of Robin functions types in Hilbert space with error bounds and error estimates Nonlinear Dyn. 94 18191834 DOI:10.1007/s11071-018-4459-8
WangYZhuL2017 Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method Adv. Differ. Equ. 2017 27 DOI:10.1186/s13662-017-1085-6
Abu ArqubOAl-SmadiM2018 Numerical algorithm for solving time‐fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions Numer. Methods Partial Differ. Equ. 34 15771597 DOI:10.1002/num.22209 [Cited within: 1]
Abu ArqubOShawagfehN2019 Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media J. Porous Media 22 411434 DOI:10.1615/JPorMedia.2019028970 [Cited within: 2]
OsmanM S2019 One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation Nonlinear Dyn. 96 14911496 DOI:10.1007/s11071-019-04866-1
Abu ArqubO2018 Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm Int. J. Numer. Methods Heat Fluid Flow 28 828856 DOI:10.1108/HFF-07-2016-0278
OsmanM S2017 Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2+1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients Nonlinear Dyn. 87 12091216 DOI:10.1007/s11071-016-3110-9
Abu ArqubO2017 Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions Comput. Math. Appl. 73 12431261 DOI:10.1016/j.camwa.2016.11.032
GengF ZCuiM2012 A reproducing kernel method for solving nonlocal fractional boundary value problems Appl. Math. Lett. 25 818823 DOI:10.1016/j.aml.2011.10.025
Abu ArqubO2018 Solutions of time‐fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space Numer. Methods Partial Differ. Equ. 34 17591780 DOI:10.1002/num.22236
HuangLLiX FZhaoYDuanX Y2011 Approximate solution of fractional integro-differential equations by Taylor expansion method Comput. Math. Appl. 62 11271134 DOI:10.1016/j.camwa.2011.03.037
Abu ArqubOAl-SmadiM2019 Numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media using iterative reproducing kernel algorithm J. Porous MediaAccepted
Abu ArqubOAl-SmadiM2020 An adaptive numerical approach for the solutions of fractional advection-diffusion and dispersion equations in singular case under Riesz's derivative operator Physica A 540 123257 DOI:10.1016/j.physa.2019.123257 [Cited within: 2]
Abu ArqubOAl-SmadiM2020 Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions Soft Comput. 2020 122 DOI:10.1007/s00500-020-04687-0
ThabetHKendreS2018 Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform Chaos Solitons Fractals 109 238245 DOI:10.1016/j.chaos.2018.03.001
FengQ2018 A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation Chin. J. Phys. 56 28172828 DOI:10.1016/j.cjph.2018.08.006
ChenCJiangY2018 Simplest equation method for some time-fractional partial differential equations with conformable derivative Comput. Math. Appl. 75 29782988 DOI:10.1016/j.camwa.2018.01.025 [Cited within: 1]
AtanganaABaleanuD2013 Nonlinear fractional Jaulent–Miodek and Whitham–Broer–Kaup equations within Sumudu transform Abstract Appl. Anal. 2013 18 DOI:10.1155/2013/160681 [Cited within: 3]
MomaniSAbu ArqubOFreihatAAl-SmadiM2016 Analytical approximations for Fokker–Planck equations of fractional order in multistep schemes Appl. Comput. Math. 15 319330 [Cited within: 1]
KhaterM M AKumarD2017 New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water J. Ocean Eng. Sci. 2 223228 DOI:10.1016/j.joes.2017.07.001 [Cited within: 1]
JaveedSSaifSWaheedABaleanuD2018 Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq–Burgers Results Phys. 9 12751281 DOI:10.1016/j.rinp.2018.04.026 [Cited within: 1]
WangJXuT-ZWangG-W2018 Numerical algorithm for time-fractional Sawada–Kotera equation and Ito equation with Bernstein polynomials Appl. Math. Comput. 338 111 DOI:10.1016/j.amc.2018.06.001 [Cited within: 1]
Abu ArqubOAl-SmadiM2018 Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space Chaos Solitons Fractals 117 161167 DOI:10.1016/j.chaos.2018.10.013 [Cited within: 1]
Abu ArqubOMaayahB2018 Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator Chaos Solitons Fractals 117 117124 DOI:10.1016/j.chaos.2018.10.007
Abu ArqubOMaayahB2019 Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana–Baleanu fractional sense Chaos Solitons Fractals 125 163170 DOI:10.1016/j.chaos.2019.05.025
Abu ArqubOMaayahB2019 Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC-Fractional Volterra integro-differential equations Chaos Solitons Fractals 126 394402 DOI:10.1016/j.chaos.2019.07.023
Abu ArqubO2019 Numerical algorithm for the solutions of fractional order systems of dirichlet function types with comparative analysis Fundam. Inform. 166 111137 DOI:10.3233/FI-2019-1796
Abu ArqubO2018 Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm Calcolo 55 128 DOI:10.1007/s10092-018-0274-3 [Cited within: 1]
LiangJTangLXiaYZhangY2020 Bifurcations and exact solutions for a class of MKdV equations with the conformable fractional derivative via dynamical system method Int. J. Bifurcation Chaos 30 2050004 DOI:10.1142/S0218127420500042 [Cited within: 1]
ZhangBZhuWXiaYBaiY2020 A unified analysis of exact traveling wave solutions for the fractional-order and integer-order Biswas–Milovic equation: via bifurcation theory of dynamical system Qualitative Theory Dyn. Syst. 19 11 DOI:10.1007/s12346-020-00352-x
ZhuWXiaYZhangBBaiY2019 Exact traveling wave solutions and bifurcations of the time-fractional differential equations with applications Int. J. Bifurcation Chaos 29 1950041 DOI:10.1142/S021812741950041X [Cited within: 1]
ArqubO A2013 Series solution of fuzzy differential equations under strongly generalized differentiability J. Adv. Res. Appl. Math. 5 3152 DOI:10.5373/jaram.1447.051912 [Cited within: 2]
Abu ArqubOEl-AjouAMomaniS2015 Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations J. Comput. Phys. 293 385399 DOI:10.1016/j.jcp.2014.09.034 [Cited within: 2]
El-AjouAAbu ArqubOMomaniSBaleanuDAlsaediA2015 A novel expansion iterative method for solving linear partial differential equations of fractional order Appl. Math. Comput. 257 119133 DOI:10.1016/j.amc.2014.12.121
El-AjouAAbu ArqubOMomaniS2015 Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: a new iterative algorithm J. Comput. Phys. 293 8195 DOI:10.1016/j.jcp.2014.08.004
Abu ArqubO2019 Application of residual power series method for the solution of time-fractional Schrodinger equations in one-dimensional space Fundam. Inform. 166 87110 DOI:10.3233/FI-2019-1795 [Cited within: 1]
AkgülAHashemiM SIncMRaheemS A2017 Constructing two powerful methods to solve the Thomas–Fermi equation Nonlinear Dyn. 87 14351444 DOI:10.1007/s11071-016-3125-2 [Cited within: 1]
AliAShahKKhanR A2018 Numerical treatment for traveling wave solutions of fractional Whitham–Broer–Kaup equations Alexandria Eng. J. 57 19911998 DOI:10.1016/j.aej.2017.04.012 [Cited within: 3]
RafeiMDanialiH2007 Application of the variational iteration method to the Whitham–Broer–Kaup equations Comput. Math. Appl. 54 10791085 DOI:10.1016/j.camwa.2006.12.054 [Cited within: 3]
,2, Samir Hadid31Department of Applied Science, Ajloun College, 
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