删除或更新信息,请邮件至freekaoyan#163.com(#换成@)

Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional

本站小编 Free考研考试/2022-01-02

P Veeresha,1, D G Prakasha,2,31Department of Mathematics, Karnatak University, Dharwad-580003, Karnataka, India
2Department of Mathematics, Faculty of Science, Davangere University, Shivagangothri, Davangere-577007, Karnataka, India

First author contact: 3 Author to whom any correspondence should be addressed.
Received:2020-05-02Revised:2020-05-28Accepted:2020-06-23Online:2020-09-16


Abstract
In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa-Holm and Degasperis-Procesi equations using the Caputo fractional operator. Both the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in $\hslash $-curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.
Keywords: Laplace transform;Camassa-Holm equation;q-homotopy analysis method;Degasperis-Procesi equation;fractional derivative


PDF (978KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
P Veeresha, D G Prakasha. Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator. Communications in Theoretical Physics, 2020, 72(10): 105002- doi:10.1088/1572-9494/aba24b

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]$\begin{eqnarray}{v}_{t}-{v}_{xxt}+\left(\beta +1\right){v}^{2}{v}_{x}=\beta {v}_{x}{v}_{xx}+v{v}_{xxx},\end{eqnarray}$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 (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$\begin{eqnarray}{v}_{t}-{v}_{xxt}+3{v}^{2}{v}_{x}=2{v}_{x}{v}_{xx}+v{v}_{xxx},\end{eqnarray}$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

Here, we present some fundamental notions of FC and LT [1-6]:

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.

New window|Download| PPT slide
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.

New window|Download| PPT slide
Figure 2.Response of the achieved solution for equation considered in Case I at $n=1,\hslash =-1$ and $t=0.1.$


Figure 3.

New window|Download| PPT slide
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.

New window|Download| PPT slide
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.

New window|Download| PPT slide
Figure 5.Response of the achieved solution for equation considered in Case II at $n=1,\hslash =-1$ and $t=0.01.$


Figure 6.

New window|Download| PPT slide
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.
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}$

New window|CSV


Table 2.
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}$

New window|CSV

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.

Reference By original order
By published year
By cited within times
By Impact factor

Liouville J 1832 Memoire surquelques questions de geometrieet de mecanique, et sur un nouveau genre de calcul pour resoudreces questions
J. Ecole. Polytech. 13 1 69

[Cited within: 2]

Riemann G F B 1896 Versuch Einer Allgemeinen Auffassung der Integration und Differentiation, Gesammelte Mathematische Werke Leipzig Teubner


Caputo M 1969 Elasticita e Dissipazione Bologna Zanichelli Editore
[Cited within: 1]

Miller K S Ross B 1993 An Introduction to Fractional Calculus and Fractional Differential Equations New York Wiley


Podlubny I 1999 Fractional Differential Equations New York Academic


Kilbas A A Srivastava H M Trujillo J J 2006 Theory and Applications of Fractional Differential Equations Amsterdam Elsevier
[Cited within: 2]

Caputo M Fabrizio M 2015 A new definition of fractional derivative without singular kernel
Prog. Fract. Differ. Appl. 1 73 85

DOI:10.12785/pfda/010201 [Cited within: 1]

Atangana A Baleanu D 2016 New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model
Therm. Sci. 20 763 769

DOI:10.2298/TSCI160111018A [Cited within: 1]

Baleanu D Guvenc Z B Machado J A T 2010 New Trends in Nanotechnology and Fractional Calculus Applications Dordrecht Springer
[Cited within: 1]

Esen A Sulaiman T A Bulut H Baskonus H M 2018 Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation
Optik 167 150 156

DOI:10.1016/j.ijleo.2018.04.015

Veeresha P Prakasha D G 2020 An efficient technique for two-dimensional fractional order biological population model
Int. J. Modelling Simul. Sci. Comput. 11 2050005

DOI:10.1142/S1793962320500051

Gao W Veeresha P Prakasha D G Baskonus H M Yel G 2020 New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function
Chaos Solitons Fractals 134 109696

DOI:10.1016/j.chaos.2020.109696

Baleanu D Wu G C Zeng S D 2017 Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations
Chaos Solitons Fractals 102 99 105

DOI:10.1016/j.chaos.2017.02.007

Veeresha P Prakasha D G Baskonus H M 2019 New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives
Chaos 29 013119

DOI:10.1063/1.5074099

Nuruddeen R I Aboodh K S Ali K K 2018 Analytical investigation of soliton solutions to three quantum Zakharov-Kuznetsov equations
Commun. Theor. Phys. 70 405 412

DOI:10.1088/0253-6102/70/4/405

Gao W Yel G Baskonus H M Cattani C 2020 Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation
AIMS Math. 5 507 521

DOI:10.3934/math.2020034

Gao W Veeresha P Prakasha D G Baskonus H M Yel G 2020 New numerical results for the time-fractional Phi-four equation using a novel analytical approach
Symmetry 12 1 16

DOI:10.3390/sym12030478

Zhang Y Cattani C Yang X-J 2015 Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains
Entropy 17 6753 6764

DOI:10.3390/e17106753

Kumar D Singh J Baleanu D 2018 Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel
Physica A 492 155 167

DOI:10.1016/j.physa.2017.10.002 [Cited within: 2]

Yokus A Gulbahar S 2019 Numerical solutions with linearization techniques of the fractional Harry Dym equation
Appl. Math. Nonlinear Sci. 4 35 42

DOI:10.2478/AMNS.2019.1.00004

Cattani C 2018 A review on harmonic wavelets and their fractional extension
J. Adv. Eng. Comput. 2 224 238

DOI:10.25073/jaec.201824.225

Ali K K Nuruddeen R I Aboodh K S 2018 New structures for the space-time fractional simplified MCH and SRLW equations
Chaos Solitons Fractals 106 304 309

DOI:10.1016/j.chaos.2017.11.038

Sulaiman T A Bulut H Atas S S 2019 Optical solitons to the fractional Schrödinger-Hirota equation
Appl. Math. Nonlinear Sci. 4 535 542

DOI:10.2478/AMNS.2019.2.00050

Raslan K R EL-Danaf T S Ali K K 2017 Exact solution of the space-time fractional coupled EW and coupled MEW equations
Eur. Phys. J. Plus 132 1 11

DOI:10.1140/epjp/i2017-11590-9

Gao W Veeresha P Prakasha D G Baskonus H M 2020 Novel dynamical structures of 2019-nCoV with nonlocal operator via powerful computational technique
Biology 9 107

DOI:10.3390/biology9050107

Raslan K R Ali K K Shallal M A 2017 The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations
Chaos Solitons Fractals 103 404 409

DOI:10.1016/j.chaos.2017.06.029

Ali K K Nuruddeen R I Raslan K R 2018 New hyperbolic structures for the conformable time-fractional variant Bussinesq equations
Opt. Quantum Electron. 50 60

DOI:10.1007/s11082-018-1330-6

Sulaiman T A Yel G Bulut H 2019 M-fractional solitons and periodic wave solutions to the Hirota Maccari system
Mod. Phys. Lett. B 33 1950052

DOI:10.1142/S0217984919500520

Baskonus H M Sulaiman T A Bulut H 2019 On the new wave behavior to the Klein-Gordon-Zakharov equations in plasma physics
Indian J. Phys. 93 393 399

DOI:10.1007/s12648-018-1262-9

Prakasha D G Veeresha P 2020 Analysis of Lakes pollution model with Mittag-Leffler kernel
J. Ocean Eng. Sci. 1 13

DOI:10.1016/j.joes.2020.01.004 [Cited within: 1]

Wazwaz A M 2006 Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations
Phys. Lett. A 352 500 504

DOI:10.1016/j.physleta.2005.12.036 [Cited within: 1]

Wazwaz A M 2007 New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations
Appl. Math. Comput. 186 130 141

DOI:10.1016/j.amc.2006.07.092 [Cited within: 1]

Zhang B Li S Liu Z 2008 Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations
Phys. Lett. A 372 1867 1872

DOI:10.1016/j.physleta.2007.10.072 [Cited within: 2]

Degasperis A Procesi M 2002 Asymptotic Integrability Symmetry and Perturbation Theory Singapore World Scientific 23 37
[Cited within: 2]

Liao S J 1997 Homotopy analysis method and its applications in mathematics
J. Basic Sci. Eng. 5 111 125

[Cited within: 1]

Singh J Kumar D Swroop R 2016 Numerical solution of time- and space-fractional coupled Burgers' equations via homotopy algorithm
Alexandria Eng. J. 55 1753 1763

DOI:10.1016/j.aej.2016.03.028 [Cited within: 1]

Srivastava H M Kumar D Singh J 2017 An efficient analytical technique for fractional model of vibration equation
Appl. Math. Model. 45 192 204

DOI:10.1016/j.apm.2016.12.008 [Cited within: 1]

Veeresha P Baskonus H M Prakasha D G Gao W Yel G 2020 Regarding new numerical solution of fractional Schistosomiasis disease arising in biological phenomena
Chaos Solitons Fractals 133 109661

DOI:10.1016/j.chaos.2020.109661

Bulut H Kumar D Singh J Swroop R Baskonus H M 2018 Analytic study for a fractional model of HIV infection of CD4+T lymphocyte cells
Math. Nat. Sci. 2 33 43

DOI:10.22436/mns.02.01.04

Veeresha P Prakasha D G 2020 Solution for fractional generalized Zakharov equations with Mittag-Leffler function
Results Eng. 5 1 12

DOI:10.1016/j.rineng.2019.100085

Kumar D Agarwal R P Singh J 2018 A modified numerical scheme and convergence analysis for fractional model of Lienard's equation
J. Comput. Appl. Math. 399 405 413

DOI:10.1016/j.cam.2017.03.011

Singh J Kumar D Hammouch Z Atangana A 2018 A fractional epidemiological model for computer viruses pertaining to a new fractional derivative
Appl. Math. Comput. 316 504 515

DOI:10.1016/j.amc.2017.08.048

Prakasha D G Veeresha P Baskonus H M 2019 Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative
Eur. Phys. J. Plus 134 1 11

DOI:10.1140/epjp/i2019-12590-5

Prakasha D G Veeresha P Singh J 2019 Fractional approach for equation describing the water transport in unsaturated porous media with Mittag--Leffler kernel
Front. Phys. 7 193

DOI:10.3389/fphy.2019.00193 [Cited within: 1]

Jafari H Zabihi M Salehpoor E 2010 Application of variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations, Numer
Methods Partial Differ. Equ. 26 1033 1039

DOI:10.1002/num.20472 [Cited within: 1]

Gupta P K Singh M Yildirim A 2016 Approximate analytical solution of the time-fractional Camassa-Holm, modified Camassa-Holm, and Degasperis-Procesi equations by homotopy perturbation method
Sci. Iranica A 23 155 165

[Cited within: 1]

相关话题/Novel approach modified