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:
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 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 (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
$ \begin{eqnarray}\begin{array}{l}{I}_{s}^{\alpha }f\left(t\right)=\displaystyle \frac{1}{\left(n-1\right)!}\displaystyle {\int }_{s}^{t}\displaystyle \frac{{\left(t-\tau \right)}^{n-1}f\left(\tau \right)}{{\left(\tau -{\rm{s}}\right)}^{n-\alpha }}{\rm{d}}\tau ,\\ t\gt \tau \geqslant s\geqslant 0.\end{array}\end{eqnarray}$
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
$ \begin{eqnarray}\begin{array}{lll}{T}_{t}^{\alpha }v\left(x,t\right) & = & \displaystyle \frac{{\partial }^{\alpha }v\left(x,t\right)}{\partial {t}^{\alpha }}\\ & = & \mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}\displaystyle \frac{{v}_{t}^{(n-1)}\left(x,t+\varepsilon {\left(t-s\right)}^{n-\alpha }\right)-{v}_{t}^{\left(n-1\right)}(x,t)}{\varepsilon },\\ & & \,t\gt s\geqslant 0.\end{array}\,\end{eqnarray}$
[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
$ \begin{eqnarray}{I}_{s}^{\alpha }v\left(x,t\right)=\left\{\begin{array}{ll}\displaystyle \frac{1}{\left(n-1\right)!}\displaystyle {\int }_{s}^{t}\displaystyle \frac{{\left(t-\tau \right)}^{n-1}v\left(x,\tau \right)}{{\left(\tau -{\rm{s}}\right)}^{n-\alpha }}{\rm{d}}\tau , & x\in I,\,t\gt \tau \geqslant s\geqslant 0,\,\\ v\left(x,t\right), & \alpha =0.\end{array}\right.\end{eqnarray}$
[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 (
$ \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 (
$ \begin{eqnarray}\left|{ {\mathcal R} }_{n}\left(x,t\right)\right|\leqslant \displaystyle \frac{M(x)}{{\alpha }^{n+1}\left(n+1\right)!}{\left(t-{t}_{0}\right)}^{\left(n+1\right)\alpha },\end{eqnarray}$
where ${ {\mathcal R} }_{n}\left(x,t\right)=\displaystyle {\sum }_{k=n+1}^{\infty }\displaystyle \frac{{T}_{t}^{k\alpha }v\left(x,{t}_{0}\right)}{{\alpha }^{k}\left(k\right)!}\left(t-{t}_{0}\right){}^{k\alpha }$ = $v\left(x,t\right)\,-\displaystyle {\sum }_{k=0}^{n}\displaystyle \frac{{T}_{t}^{k\alpha }v\left(x,{t}_{0}\right)}{{\alpha }^{k}\left(k\right)!}\left(t-{t}_{0}\right){}^{k\alpha }.$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 ( $ \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 (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 (
$ \begin{eqnarray}\begin{array}{l}{v}_{i}\left(x,t\right)=\displaystyle \sum _{k=0}^{\infty }{f}_{i,k}\left(x\right)\displaystyle \frac{{(t-{t}_{0})}^{k\alpha }}{{\alpha }^{k}k!},\\ x\in I,\,{t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha },i=1,2.\end{array}\end{eqnarray}$
According to the initial conditions (
$ \begin{eqnarray}\begin{array}{l}{v}_{i,n}\left(x,t\right)={\mu }_{i}\left(x\right)+\displaystyle \sum _{k=1}^{n}{f}_{i,k}\left(x\right)\displaystyle \frac{{(t-{t}_{0})}^{k\alpha }}{{\alpha }^{k}k!},\\ x\in I,\,{t}_{0}\leqslant t\lt {t}_{0}+{R}^{1/\alpha },i=1,2.\end{array}\end{eqnarray}$
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 (
$ \begin{eqnarray}\begin{array}{lll}{{\rm{Res}}}_{{v}_{i,n}}\left(x,t\right) & = & {T}_{t}^{\alpha }\left({\mu }_{i}\left(x\right)+\displaystyle \sum _{k=1}^{n}{f}_{i,k}\left(x\right)\displaystyle \frac{{(t-{t}_{0})}^{k\alpha }}{{\alpha }^{k}k!}\right)\\ & & -\,{{\ell }}_{i}\left[{\mu }_{i}\left(x\right)+\displaystyle \sum _{k=1}^{n}{f}_{i,k}\left(x\right)\displaystyle \frac{{(t-{t}_{0})}^{k\alpha }}{{\alpha }^{k}k!}\right]\\ & & -\,{{\mathscr{N}}}_{i}\left[{\mu }_{i}\left(x\right)+\displaystyle \sum _{k=1}^{n}{f}_{i,k}\left(x\right)\displaystyle \frac{{(t-{t}_{0})}^{k\alpha }}{{\alpha }^{k}k!}\right]\\ & & -\,{\varphi }_{i}\left(x,t\right),i=1,2.\end{array}\end{eqnarray}$
Set $n=1$ in equation (
For $n=2,3,\ldots ,N,$ applying the operator ${T}_{t}^{\alpha },$ ($n-1$)-times, on both sides of equation (
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 (
Let ${v}_{i}\left(x,t\right),i=1,2,$ be the exact solutions of FPDEs (
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
$ \begin{eqnarray}\begin{array}{lll}\parallel {v}_{i,n}\left(x,t\right)-{v}_{i,n+1}\left(x,t\right)\parallel & = & \parallel \displaystyle \sum _{k=n+1}^{\infty }{v}_{i,k}\left(x,t\right)\parallel \\ & & \,\leqslant \displaystyle \sum _{k=n+1}^{\infty }\parallel {v}_{i,k}\left(x,t\right)\,\parallel \,\\ & & \,\leqslant \parallel {f}_{i,0}\left(x\right)\parallel \displaystyle \sum _{k=n+1}^{\infty }{\rho }_{i}^{k}\\ & = & \,\displaystyle \frac{{\rho }_{i}^{n+1}}{1-{\rho }_{i}}\parallel {f}_{i,0}\left(x\right)\parallel ,\end{array}\end{eqnarray}$
which converges to zero as $n\to \infty .$ ■4. Numerical simulation of physical models
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 ( $ \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 by26 ), substitute the 1st-truncated series23 ) and (24 ) are23 ) and (24 ) are23 ) and (24 ) are26 ) were computed as follows23 ) 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.
$ \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 ( $ \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 ( $ \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 ( $ \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 ( $ \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 ( $ \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 (
Figure 1. Comparison of exact and 6th-CRPS approximate graphs of the coupled WBK system ( |
Figure 2. Surface waves plots of CRPS solution, ${v}_{6}\left(x,t\right),$ of equation ( |
Figure 3. Surface waves plots of CRPS solution, ${w}_{6}\left(x,t\right),$ of equation ( |
Table 1. Errors results for the 7th-CRPS of coupled system ( |
| $v\left(x,t\right)$ solution | $w\left(x,t\right)$ solution | |||
|---|---|---|---|---|
| ${t}_{i}$ | Absolute error | Relative error | Absolute error | Relative error |
| $0.2$ | $9.50351\times {10}^{-13}$ | $1.11807\times {10}^{-13}$ | $3.7905\times {10}^{-12}$ | $1.13844\times {10}^{-9}$ |
| $0.4$ | $2.54341\times {10}^{-10}$ | $2.99229\times {10}^{-11}$ | $1.01636\times {10}^{-9}$ | $2.04618\times {10}^{-8}$ |
| $0.6$ | $6.84333\times {10}^{-9}$ | $8.05115\times {10}^{-10}$ | $2.73447\times {10}^{-8}$ | $3.69024\times {10}^{-7}$ |
| $0.8$ | $7.18967\times {10}^{-8}$ | $8.45871\times {10}^{-9}$ | $2.87265\times {10}^{-7}$ | $2.59868\times {10}^{-6}$ |
Figure 4. Comparison of exact solutions with CRPS method and LADM [48] in 2D for system ( |
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 $\alpha =1$ are43 ) can be obtained in a similar manner. Thus, the CRPS approximate solutions in the series form are presented as40 ) 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.
$ \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 ( $ \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 (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 ( $ \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 ( $ \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 ( $ \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 ( $ \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 ( $ \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 ( $ \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. Graphs of exact and CRPS solution behavior of coupled MB system ( |
Table 2. Comparison of absolute errors, ${v}_{2}(x,t),$ of system ( |
| ${t}_{i}$ | ${x}_{i}$ | CRPS method | OHAM [49] | ADM [50] |
|---|---|---|---|---|
| $0.1$ | $0.1$ | $2.41474\times {10}^{-14}$ | $6.35267\times {10}^{-5}$ | $8.16297\times {10}^{-7}$ |
| $0.3$ | $2.23710\times {10}^{-14}$ | $1.90854\times {10}^{-4}$ | $7.64245\times {10}^{-7}$ | |
| $0.5$ | $2.07057\times {10}^{-14}$ | $3.18548\times {10}^{-4}$ | $7.16083\times {10}^{-7}$ | |
| $0.3$ | $0.1$ | $6.53311\times {10}^{-13}$ | $6.03098\times {10}^{-5}$ | $7.33445\times {10}^{-6}$ |
| $0.3$ | $6.04128\times {10}^{-13}$ | $1.81187\times {10}^{-4}$ | $6.86758\times {10}^{-6}$ | |
| $0.5$ | $5.59441\times {10}^{-13}$ | $3.02408\times {10}^{-4}$ | $6.43557\times {10}^{-6}$ | |
| $0.5$ | $0.1$ | $3.02491\times {10}^{-12}$ | $5.72865\times {10}^{-5}$ | $2.03415\times {10}^{-5}$ |
| $0.3$ | $2.79698\times {10}^{-12}$ | $1.72102\times {10}^{-4}$ | $1.90489\times {10}^{-5}$ | |
| $0.5$ | $2.59009\times {10}^{-12}$ | $2.87240\times {10}^{-4}$ | $1.78528\times {10}^{-5}$ |
Table 3. Comparison of absolute errors, ${w}_{2}(x,t),$ of system ( |
| ${t}_{i}$ | ${x}_{i}$ | CRPS method | OHAM [49] | ADM [50] |
|---|---|---|---|---|
| $0.1$ | $0.1$ | $9.57047\times {10}^{-15}$ | $1.65942\times {10}^{-5}$ | $5.88676\times {10}^{-5}$ |
| $0.3$ | $8.67188\times {10}^{-15}$ | $4.98691\times {10}^{-5}$ | $5.56914\times {10}^{-5}$ | |
| $0.5$ | $7.88952\times {10}^{-15}$ | $8.26491\times {10}^{-4}$ | $5.27169\times {10}^{-5}$ | |
| $0.3$ | $0.1$ | $2.58368\times {10}^{-13}$ | $1.55880\times {10}^{-5}$ | $1.78041\times {10}^{-4}$ |
| $0.3$ | $2.34352\times {10}^{-13}$ | $4.68439\times {10}^{-5}$ | $1.68429\times {10}^{-4}$ | |
| $0.5$ | $2.12989\times {10}^{-13}$ | $7.63646\times {10}^{-4}$ | $1.59428\times {10}^{-4}$ | |
| $0.5$ | $0.1$ | $1.19627\times {10}^{-12}$ | $1.46569\times {10}^{-5}$ | $2.99162\times {10}^{-4}$ |
| $0.3$ | $1.08511\times {10}^{-12}$ | $4.40448\times {10}^{-5}$ | $2.83001\times {10}^{-4}$ | |
| $0.5$ | $9.86156\times {10}^{-12}$ | $7.06678\times {10}^{-4}$ | $2.67868\times {10}^{-4}$ |
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. 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. 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. 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.