1. Introduction
The essence of calculus and its consequence in the science and technology is to study the processes or phenomena associated with the rate of change. The concept of calculus is begun in the seventeenth century and then day by day it becomes an essential and efficient tool to examine all the phenomena and predict the future and needs of living beings in nature. However, as research goes deeper and peak, researchers and investigators pointed out some limitations with respect to the calculus of classical calculus, particularly while analysing the phenomena associated to hereditary consequences, long rang processes, non-Markovian processes, memory-based mechanisms, Brownian motion, random walk and others. On another hand, the calculus with respect to fractional order was debuted soon after the traditional calculus, but recently it familiarised due to the limitations of integer order calculus and also it helps to incorporate all the above-cited consequences and also offers some interesting and simulating results.
Many senior pioneers and researchers prescribed the importance of the fractional calculus (FC) and also suggested diverse definitions and properties related to differential and integral operators of FC [1-6]. The concept and theory of fractional differential and integral operators are widely used to model many real-world problems and physical mechanisms. Further, it aids us to capture some essential and important consequences and more information about the corresponding phenomena. There are diverse definitions and notions for the differential and integral operators of non-integer order. The most familiar and extensively used operators are Riemann-Liouville (RL), Caputo [3], Caputo-Fabrizio [7] and Atangana-Baleanu (AB) derivatives [8]. Each previously defined notion has its own limitations while examining a particular phenomenon with the specific circumference. For instance, RL operator fails satisfies the universal truth of derivative (i.e. the derivative of constant is zero) and also while using this operator, we cannot use initial conditions in the classical form. However, three operators have exemplified some interesting consequences in diverse areas including fluid mechanics, optical physics, chaos theory, biological models, disease analysis, circuit analysis and others [9-30].
In 2006, Wazwaz studied a special class of bi-Hamiltonian and have an associated isospectral problem called modified Camassa-Holm (CH) and Degasperis-Procesi (DP) equations [31, 32]. Further, he employed sine-cosine and tanh technique to investigate peakon solutions and transformed to bell-shaped solitons. These equations are an important family of equations familiarly known as modified $b$-equation [33]1 ) is integrable only at either $\beta =2$ or $\beta =3,$ by Degasperis and Procesi [34]. Further, at $\beta =2$ and $\beta =3,$ equation (1 ) respectively signifies modified Camassa-Holm (mCH) and Degasperis-Procesi (mDP) equations and which are presented as follows
$\begin{eqnarray}{v}_{t}-{v}_{xxt}+\left(\beta +1\right){v}^{2}{v}_{x}=\beta {v}_{x}{v}_{xx}+v{v}_{xxx},\end{eqnarray}$
where $\beta $ is positive integer and $v\left(x,t\right)$ signifies the horizontal component of the fluid velocity. Clearly, it has been proved that equation ( $\begin{eqnarray}{v}_{t}-{v}_{xxt}+3{v}^{2}{v}_{x}=2{v}_{x}{v}_{xx}+v{v}_{xxx},\end{eqnarray}$
and $\begin{eqnarray}{v}_{t}-{v}_{xxt}+4{v}^{2}{v}_{x}=3{v}_{x}{v}_{xx}+v{v}_{xxx}.\end{eqnarray}$
The CH equation is initially proposed for the approximation of the Euler equation with incompressibility and it signified as shallow water equitation. Moreover, it is completely integrable with a Lax pair [34]. Similarly, equation (3 ) is exemplifies phenomena of shallow water and also completely integrable.
The projected solution procedure is a mixture of techniques based on homotopy [35] and Laplace transform (LT) [36]. The novelty of $q$-HATM is it does not requires linearization, perturbations, extraction of any polynomial and inclusion of any additional assumptions. Further, it can employ for diverse class with high nonlinearity and the algorithm of the considered scheme is simple and essay to handle nonlinear part of the equations. Due to the accuracy and efficiency of the projected technique is applied to examine the wide classes of nonlinear and complex models and problems and also for the system of equations [37-44].
In the present investigation, we considered three fractional operators within the frame of Caputo sense and investigate two equations to illustrate the applicability and exactness of the considered model. The considered equations are analysed and also find the solution via numerical and analytical schemes by many researchers. For instance, authors in [45] find the numerical solution for these equations with the help of variational iteration scheme, homotopy perturbation algorithm is employed [33] and find the solution and presented some interesting consequences, researchers in [46] hired homotopy perturbation technique to analyse the projected equations within the frame of FC and others.
The rest of the paper is presented as follows; the basic notion of FC and LT are presented in in section 2 . In section 3 , we presented solution procedure of the $q$-HATM with respect to considered problem. The numerical results and discussion are presented in section 4 and in the lost section; we presented some concluding remarks on achieved results.
2. Preliminaries
The Caputo fractional derivative of $f\in {C}_{-1}^{n}$ is presented as
$\begin{eqnarray}\begin{array}{l}{{D}}_{t}^{\alpha }f\left(t\right)\,=\left\{\begin{array}{ll}\displaystyle \frac{{{\rm{d}}}^{n}f\left(t\right)}{{\rm{d}}{t}^{n}}, & \alpha =n\in {\mathbb{N}},\\ \displaystyle \frac{1}{{\rm{\Gamma }}(n-\alpha )}\displaystyle {\int }_{0}^{t}{\left(t-\vartheta \right)}^{n-\alpha -1}{f}^{(n)}\left(\vartheta \right){\rm{d}}\vartheta , & n-1\lt \alpha \lt n\,,n\in {\mathbb{N}}.\end{array}\right.\end{array}\end{eqnarray}$
The LT for a Caputo fractional derivative ${{D}}_{t}^{\alpha }f\left(t\right)$ is presented as below
$\begin{eqnarray}\begin{array}{l} {\mathcal L} \left[{{D}}_{t}^{\alpha }f\left(t\right)\right]={s}^{\alpha }F\left(s\right)-\displaystyle \sum _{r=0}^{n-1}{s}^{\alpha -r-1}{f}^{\left(r\right)}\left({0}^{+}\right),\\ \,\left(n-1\lt \alpha \leqslant n\right),\end{array}\end{eqnarray}$
where $F\left(s\right)$ is LT of $f(t).$3. Solution for FEFK equation
In this paper, we consider familiar and most used fractional operator in order to find the solution for the considered model. Here, we present the algorithm of the projected scheme for the modified $b$-equation with Caputo fractional operator.
Consider the equation defined in equation (1 ) by incorporating time derivative by time-fractional derivative using Caputo operator as follows
$\begin{eqnarray}{{D}}_{t}^{\alpha }v\left(x,t\right)-{v}_{xxt}+\left(\beta +1\right){v}^{2}{v}_{x}-\beta {v}_{x}{v}_{xx}-v{v}_{xxx}=0,\end{eqnarray}$
with initial conditions $\begin{eqnarray}v\left(x,0\right)={v}_{0}\left(x,t\right).\end{eqnarray}$
Taking LT on equation (6 ) and then using equation (7 ), we get
$\begin{eqnarray}\begin{array}{l} {\mathcal L} \left[v\left(x,t\right)\right]=\displaystyle \frac{1}{s}\left({v}_{0}\left(x,t\right)\right)+\displaystyle \frac{1}{{s}^{\alpha }} {\mathcal L} \left\{-{v}_{xxt}\right.\\ \,+\left.\left(\beta +1\right){v}^{2}{v}_{x}-\beta {v}_{x}{v}_{xx}-v{v}_{xxx}\right\}.\end{array}\end{eqnarray}$
Using the solution procedure of HAM, ${\mathscr{N}}$ is presented as
$\begin{eqnarray}\begin{array}{l}{\mathscr{N}}\left[\varphi \left(x,t;q\right)\right]= {\mathcal L} \left[\varphi \left(x,t;q\right)\right]-\displaystyle \frac{1}{s}\left({v}_{0}\left(x,t\right)\right)\\ \,+\,\displaystyle \frac{1}{{s}^{\alpha }} {\mathcal L} \left\{-\displaystyle \frac{{\partial }^{3}}{{\partial }^{2}x\partial t}\varphi \left(x,t;q\right)+\left(\beta +1\right){\varphi }^{2}\right.\\ \,\times \left(x,t;q\right)\displaystyle \frac{\partial }{\partial x}\varphi \left(x,t;q\right)-\,\beta \displaystyle \frac{\partial }{\partial x}\varphi \left(x,t;q\right)\displaystyle \frac{{\partial }^{2}}{{\partial }^{2}x}\\ \,\times \left.\varphi \left(x,t;q\right)-\varphi \left(x,t;q\right)\displaystyle \frac{{\partial }^{3}}{{\partial }^{3}x}\varphi \left(x,t;q\right)\right\}.\end{array}\end{eqnarray}$
At $ {\mathcal H} (x,t)=1,$ the deformation equation defined as
$\begin{eqnarray} {\mathcal L} \left[{v}_{m}\left(x,t\right)-{k}_{m}{v}_{m-1}\left(x,t\right)\right]=\hslash {\Re }_{m}\left[{\vec{v}}_{m-1}\right],\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{\Re }_{m}\left[{\vec{v}}_{m-1}\right]= {\mathcal L} \left[{v}_{m-1}\left(x,t\right)\right]-\left(1-\displaystyle \frac{{k}_{m}}{n}\right)\\ \,\times \left\{\displaystyle \frac{1}{s}\left({v}_{0}\left(x,t\right)\right)\right\}+\,\displaystyle \frac{1}{{s}^{\alpha }} {\mathcal L} \left\{-\displaystyle \frac{{\partial }^{3}}{{\partial }^{2}x\partial t}{v}_{m-1}\right.\\ \,+\left(\beta +1\right)\displaystyle \sum _{j=0}^{i}\displaystyle \sum _{i=0}^{m-1}{v}_{j}{v}_{i-j}\displaystyle \frac{\partial }{\partial x}{v}_{m-1-i}\\ \,-\,\left.\beta \displaystyle \sum _{i=0}^{m-1}\displaystyle \frac{\partial }{\partial x}{v}_{i}\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}{v}_{m-1-i}-\displaystyle \sum _{i=0}^{m-1}{v}_{i}\displaystyle \frac{{\partial }^{3}}{\partial {x}^{3}}{v}_{m-1-i}\right\}.\end{array}\end{eqnarray}$
On employing inverse LT on equation (10 ), one can get
$\begin{eqnarray}{v}_{m}\left(x,t\right)={k}_{m}{v}_{m-1}\left(x,\,t\right)+\hslash { {\mathcal L} }^{-1}\left\{{\Re }_{m}\left[{\vec{v}}_{m-1}\right]\right\}.\end{eqnarray}$
On simplifying the above equations systematically by using ${v}_{0}\left(x,t\right)$ we can evaluate
$\begin{eqnarray}v\left(x,t\right)={v}_{0}\left(x,t\right)+\displaystyle \sum _{m=1}^{\infty }{v}_{m}\left(x,t\right){\left(\displaystyle \frac{1}{n}\right)}^{m}.\end{eqnarray}$
4. Numerical results and discussion
Here, we consider two different cases for the projected model for the specific value of $\beta $ and capture the behaviour of the achieved result with respect to fractional operators.
In this case, we consider the fractional mCH equation and defined as follows
$\begin{eqnarray}{{D}}_{t}^{\alpha }v\left(x,t\right)-{v}_{xxt}+3{v}^{2}{v}_{x}-2{v}_{x}{v}_{xx}-v{v}_{xxx}=0,\end{eqnarray}$
with initial condition $\begin{eqnarray}v\left(x,0\right)=-2{{\rm{sech}} }^{2}\left(\displaystyle \frac{x}{2}\right).\end{eqnarray}$
The analytical solution for the classical mCH equation is
$\begin{eqnarray}v\left(x,t\right)=-2{{\rm{sech}} }^{2}\left(\displaystyle \frac{x-2t}{2}\right).\end{eqnarray}$
In this case, we consider the fractional mDP equation and defined as follows
$\begin{eqnarray}{{D}}_{t}^{\alpha }v\left(x,t\right)-{v}_{xxt}+4{v}^{2}{v}_{x}-3{v}_{x}{v}_{xx}-v{v}_{xxx}=0,\end{eqnarray}$
with initial condition $\begin{eqnarray}v\left(x,0\right)=-\displaystyle \frac{15}{8}{{\rm{sech}} }^{2}\left(\displaystyle \frac{x}{2}\right).\end{eqnarray}$
The analytical solution for the classical mDP equation is
$\begin{eqnarray}v\left(x,t\right)=-\displaystyle \frac{15}{8}{{\rm{sech}} }^{2}\left(\displaystyle \frac{x-(5/2)t}{2}\right).\end{eqnarray}$
Here, with the help of the above-achieved results and consequences, we demonstrate the nature of achieved results for the change of space and time within the frame of fractional operator and parameters of the scheme. In the present investigation, we used third order series solution to present the associated nature. The surfaces of the solution attained for mCH and mDP equations are respectively presented in figures 1 and 4 is compared with the exact solution. The response of the solution for the equations cited in Case I and Case II with fractional order are respectively illustrated in figures 2 and 5 and it cleared form the plots that, in the range of [−2, 2] both the equations shows some stimulating behaviour for fractional-order and it may help to understand more consequences of the projected equations. The homotopy parameter curves have been plotted for mCH and mDP equations with diverse fractional-order for $n=1$ and 2 and respectively cited in figures 3 and 6. This all plots show that the considered method vastly depends on fractional-order and dissipates some interesting behaviours.
Figure 1. Behaviour of the achieved solution for (a) obtained, (b) exact and (c) absolute error for the mCH equation at $\hslash =-1,\,n=1$ and $\alpha =1.$ |
Figure 2. Response of the achieved solution for equation considered in Case I at $n=1,\hslash =-1$ and $t=0.1.$ |
Figure 3. $\hslash $-curves for $v\left(x,\,t\right)$ defined in Case I with distinct $\alpha $ at $x=5$ and $t=0.01$ with $\left({\rm{a}}\right)\,n=1$ and $\left({\rm{b}}\right)\,n=2.$ |
Figure 4. Surface of the achieved solution for (a) obtained, (b) exact and (c) absolute error for the mDP equation at $\hslash =-1,\,n=1$ and $\alpha =1.$ |
Figure 5. Response of the achieved solution for equation considered in Case II at $n=1,\hslash =-1$ and $t=0.01.$ |
Figure 6. $\hslash $-curves for $v\left(x,\,t\right)$ defined in Case II with distinct $\alpha $ at $x=2$ and $t=0.01$ with (a) n =1 and $\left({\rm{b}}\right)\,n=2.$ |
The numerical study is conducted for both cases with respect to results available and confirmed that the considered method is accurate and which is illustrated in tables 1 and 2, respectively. Even though for small accuracy help to make a huge change in science and technology form the current study we can perceive about the applicability, methodical, efficiency and accuracy of the considered method and fractional operator.
Table 1. Comparison study of obtained results for mCH equation and results achieved by HPM [19] at $\hslash =-1$ and $n=1.$ |
| $t$ | $x$ | ${v}_{\left|\mathrm{HPM} \mbox{-} \mathrm{Exact}\right|}$ | ${v}_{\left|q \mbox{-} \mathrm{HATM} \mbox{-} \mathrm{Exact}\right|}$ |
|---|---|---|---|
| $0.05$ | $8$ | $2.80771\times {10}^{-4}$ | $2.78618\times {10}^{-4}$ |
| $9$ | $1.03633\times {10}^{-4}$ | $5.86830\times {10}^{-4}$ | |
| $10$ | $3.8170\times {10}^{-5}$ | $9.27737\times {10}^{-4}$ | |
| $0.1$ | $8$ | $5.91138\times {10}^{-4}$ | $1.30477\times {10}^{-3}$ |
| $9$ | $2.18174\times {10}^{-4}$ | $1.03341\times {10}^{-4}$ | |
| $10$ | $8.03570\times {10}^{-5}$ | $2.17590\times {10}^{-4}$ | |
| $0.15$ | $8$ | $9.34203\times {10}^{-4}$ | $3.43893\times {10}^{-4}$ |
| $9$ | $3.44769\times {10}^{-4}$ | $4.83516\times {10}^{-4}$ | |
| $10$ | $1.26982\times {10}^{-4}$ | $3.81314\times {10}^{-5}$ | |
| $0.2$ | $8$ | $1.31339\times {10}^{-3}$ | $8.02785\times {10}^{-5}$ |
| $9$ | $4.84685\times {10}^{-4}$ | $1.26863\times {10}^{-4}$ | |
| $10$ | $1.78511\times {10}^{-4}$ | $1.78353\times {10}^{-4}$ |
Table 2. Comparison study of obtained results with different fractional operators for mDP equation and results achieved by HPM [19] at $\hslash =-1$ and $n=1.$ |
| $t$ | $x$ | ${v}_{\left|\mathrm{HPM} \mbox{-} \mathrm{Exact}\right|}$ | ${v}_{\left|q \mbox{-} \mathrm{HATM} \mbox{-} \mathrm{Exact}\right|}$ |
|---|---|---|---|
| $0.05$ | $8$ | $3.33255\times {10}^{-4}$ | $3.30732\times {10}^{-4}$ |
| $9$ | $1.23003\times {10}^{-4}$ | $7.05929\times {10}^{-4}$ | |
| $10$ | $4.53050\times {10}^{-5}$ | $1.13149\times {10}^{-3}$ | |
| $0.1$ | $8$ | $7.10978\times {10}^{-4}$ | $1.61409\times {10}^{-3}$ |
| $9$ | $2.62396\times {10}^{-4}$ | $1.22660\times {10}^{-4}$ | |
| $10$ | $9.66440\times {10}^{-5}$ | $2.61711\times {10}^{-4}$ | |
| $0.15$ | $8$ | $1.13907\times {10}^{-3}$ | $4.19332\times {10}^{-4}$ |
| $9$ | $4.20359\times {10}^{-4}$ | $5.97992\times {10}^{-4}$ | |
| $10$ | $1.54820\times {10}^{-4}$ | $4.52587\times {10}^{-5}$ | |
| $0.2$ | $8$ | $1.62421\times {10}^{-3}$ | $9.65514\times {10}^{-5}$ |
| $9$ | $5.99362\times {10}^{-4}$ | $1.54681\times {10}^{-4}$ | |
| $10$ | $2.20743\times {10}^{-4}$ | $2.20558\times {10}^{-4}$ |
5. Conclusion
In the present work, $q$-HATM is employed successfully for the fractional mCH and mDP equations. We derived a third series solution to present the behaviour of the corresponding results. The behaviour has been captured with different orders associated with an AB operator. We consider the Caputo fractional operator and it is widely used to study the diverse class of complex and nonlinear models. To demonstrate the accuracy of the projected solution procedure, the numerical simulation is presented in comparison with results derived by HPM and confirms that the projected scheme is more accurate. The current investigation shows the applicability and importance of incorporating the fractional operators into physical and dynamical problems as well as models. Finally, we can say that the projected method can be used to examine complex and nonlinear models exemplifying real-world issues.