Abstract
The nonlinear fractional differential equations (FDEs) are produced by mathematical modelling of some nonlinear physical systems. The study of such nonlinear physical models through wave solutions analysis corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, we are going to explore the conformable time-fractional KdV equations using the expa function method. The way to reach explicit exact wave solutions is to transform the fractional order PDE into a nonlinear ODE of discrete order through travelling wave transforms. The subsequent equation has been explored by utilizing the exp a function approach. Consequently, some new explicit exact wave solutions of the said equations are effectively formulated and graphically conveyed with the help of numerical simulation.
1 Introduction
The Korteweg– de Vries (KdV) equations have been observed in a broad variety of material science phenomena as a model for the development and communication of nonlinear waves. The KdV equation was introduced by Korteweg and de Vries to describe shallow water waves of long wavelength with small amplitude. Subsequently the KdV equation has been merged into a number of other physical contexts as collision-free hydromagnetic waves, stratified internal waves, particle acoustic waves, plasma physics, and so on [1, 2, 3, 4, 5, 6, 7].
Various fields of science and engineering are influenced by Leibnitz’s work on fractional calculus and having substantially increasing impact on these sciences during last twenty years ostensibly [8, 9, 10]. From different minds for different applications, many definitions of fractional derivatives, Like Hilfer, Riemann-Liouville, Caputo form and so on, have been introduced in the literature [11, 12], but the most fascinating definition and the geometrical explanation for complex fractional transform and fractional derivative are given in [13, 14].
The explicit exact wave solutions have always been a particular importance among the researchers in many fields of nonlinear sciences. The availability of symbolic computation softwares is a direct help to minimize the manual labor for finding problematic solutions to nonlinear evolution equations. Various significant methods have been presented to explore exact solutions in different research articles, for example, the ansatz method, modified simple equation method, the extended trail equation, the first integral, the improved tan(ϕ(η)/2)-expansion, the exp-function and the exp(–ϕ(ϵ)) methods [15, 16, 17, 18, 19, 20, 21]. Also some other reliable techniques, like homotopy method [22, 23, 24, 25], generalized Kudryashov method [26, 27, 28, 29], modified Kudryashov and the sine-Gordon expansion approach [30, 31, 32, 33] have been done by different researchers. Furthermore, the auxiliary equation method, the extended tanh-function method, generalized projective Riccati equation method, functional variable method, and Jacobi elliptic function expansion method have been explored in [34, 35, 36, 37, 38, 39] for integral and fractional order as well. The expa function method is a new and an efficient technique which has been acknowledged by the scholars rapidly. In particular, Ali and Hassan, Hosseini et al., Zayed and Al-Nowehy all have utilized the expa function method in [40, 41, 42, 43] respectively.
The paper’s aim is to apply the expa function approach to nonlinear conformable time-fractional KdV equations. The investigation for exact solutions is as follow. The description of conformable derivative and expa function approach is provided in section 2. The section 3 represents the method utilization for extracting new exact solutions with their graphical representation. The conclusion is provided in the last section.
2 A new approach for fractional derivatives
A simple but a fascinating definition of the fractional derivative called conformable fractional derivative has been introduced by Khalil et al. [13]. The Liebniz rule and the chain rule both are obeyed by this derivative. Recall the definition of the conformable derivative and with some properties.
Definition 1
Suppose h : ℝ+ → ℝ be a function. Then, for all t > 0,
is known as the conformable fractional derivative of h of order y, 0 < y ≤ 1. Some useful properties are being listed as follows:
Let h : ℝ+ → ℝ be a differentiable and y-differentiable function, g be a differentiable function defined in the range of h.
On the top of that, the following rules hold.
Conjointly, if h is differentiable, then
2.1 A transitory explanation of expa function method
The present subsection provides a concise explanation for the expa [40, 42, 44] in generating new explicit exact solutions of nonlinear conformable time-fractional DE. For this purpose, suppose that we have a nonlinear conformable time-fractional DE that can be presented in the form
The FDE (1) can be changed into the following nonlinear ODE of integer order
with the use of following transformation
where k, l are nonzero arbitrary constants.
Let us try to search a non-trivial solution for the Eq. (2) in the following form
where ai and bi for (0 ≤ i ≤ N) are found later and N is a free positive integer.
Replacing the Eq.(4) in the nonlinear Eq.(2), yields
Setting li(0 ≤ i ≤ τ) in Eq.(4) to be zero, results in a set of nonlinear equations as below
which by solving the generated set (6), we approach to non-trivial solutions of the nonlinear FDE (1).
3 Explicit exact wave solutions for time fractional KdV equations
In this section, the following time fractional KdV and modified KdV equations [7], are going to be considered for solution via expa method.
Using the transformation (3), integrating w.r.t. ξ and taking constant of integration is zero, we get
Through balancing, we select N = 1, the nontrivial solution (4) reduces to:
By setting the above solution in reduced equation Eq. (8) and equating factors of each power of aξ in the resulting equation, we reach a nonlinear algebraic set as
which its solution yields
Thus, the following new explicit exact solutions to the conformable time fractional third order modified KdV equation can be written as
where
Solution profile of U1,2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
We now consider the time fractional conformable KdV Equation as follows.
Using the transformation (3), integrating w.r.t. ξ and taking constant of integration is zero, we get
Through balancing the terms we select N = 2, the non-trivial solution (4) becomes
By setting the above non-trivial solution in reduced equation Eq. (12) and equating factors of each power of αξ in the resulting equation, we reach a nonlinear algebraic set as
which its solution yields
Thus, the following new explicit exact solution to the conformable time fractional third order KdV equation can be written as
where
where
Solution profile of U1 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
Solution profile of U2 corresponding to p = 3, q = 2, β0 = 1 = β1 and α = 3
4 Conclusion
We have succeeded in exploring the explicit exact wave solutions of conformable time fractional KdV equations via expa function method. These solutions are given explicitly and verified by setting back in the reduced equations with the aid of Mathematica. Furthermore, the numerical simulation of some but different kind of solutions through MATLAB has been given for the reader to visualize the basic phenomena.
References
[1] Dr. D. J. Korteweg and Dr. G. de Vries. Xli. on the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39(240):422–443, 1895.10.1080/14786449508620739Search in Google Scholar
[2] Hans Schamel. A modified korteweg-de vries equation for ion acoustic wavess due to resonant electrons. Journal of Plasma Physics, 9(3):377–387, 1973.10.1017/S002237780000756XSearch in Google Scholar
[3] Zuntao Fu, Shida Liu, and Shikuo Liu. New solutions to mkdv equation. Physics Letters A, 326(5):364 – 374, 2004.10.1016/j.physleta.2004.04.059Search in Google Scholar
[4] Abdul-Majid Wazwaz. New sets of solitary wave solutions to the kdv, mkdv, and the generalized kdv equations. Communications in Nonlinear Science and Numerical Simulation, 13(2):331 – 339, 2008.10.1016/j.cnsns.2006.03.013Search in Google Scholar
[5] SR Mousavian, H Jafari, CM Khalique, and SA Karimi. New exact-analytical solutions for the mkdv equation. TJMCS, 2(3):413–416, 2011.10.22436/jmcs.02.03.02Search in Google Scholar
[6] S. Saha Ray. Soliton solutions for time fractional coupled modified kdv equations using new coupled fractional reduced differential transform method. Journal of Mathematical Chemistry, 51(8):2214–2229, 2013.10.1007/s10910-013-0210-3Search in Google Scholar
[7] S. Sahoo and S. Saha Ray. Solitary wave solutions for time fractional third order modified kdv equation using two reliable techniques
[8] G. Samko, A.A. Kilbas, and Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, 1993.Search in Google Scholar
[9] A. Kilbas, M. H. Srivastava, and J. J. Trujillo. Theory and application of fractional differential equations. North Holland Mathematics Studies, 204, 2006.Search in Google Scholar
[10] Kenneth S Miller and Bertram Ross. An introduction to the fractional calculus and fractional differential equations. 1993.Search in Google Scholar
[11] Guy Jumarie. Modified riemann-liouville derivative and fractional taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications, 51(9-10):1367–1376, 2006.10.1016/j.camwa.2006.02.001Search in Google Scholar
[12] Michele Caputo and Mauro Fabrizio. A new definition of fractional derivative without singular kernel. Progress in fractional differentiation and applications, 1(2):73 – 85, 2015.Search in Google Scholar
[13] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh. A new deflation of fractional derivative. J.Comput.Appl.Math., 264:65–70, 2014.10.1016/j.cam.2014.01.002Search in Google Scholar
[14] Ji-Huan He, S.K. Elagan, and Z.B. Li. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A, 376(4):257 – 259, 2012.10.1016/j.physleta.2011.11.030Search in Google Scholar
[15] Kamyar Hosseini, Peyman Mayeli, and Reza Ansari. Bright and singular soliton solutions of the conformable time-fractional klein–gordon equations with different nonlinearities. Waves in Random and Complex Media, 28(3):426–434, 2018.10.1080/17455030.2017.1362133Search in Google Scholar
[16] M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, and A. Biswas. Optical solitons perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Optik, 127(22):10659–10669, 2016.10.1016/j.ijleo.2016.08.076Search in Google Scholar
[17] Yücel Çenesiz, Dumitru Baleanu, Ali Kurt, and Orkun Tasbozan. New exact solutions of burgers’ type equations with conformable derivative. Waves in Random and Complex Media, 27(1):103–116, 2017.10.1080/17455030.2016.1205237Search in Google Scholar
[18] Mostafa Eslami and Hadi Rezazadeh. The first integral method for wu–zhang system with conformable time-fractional derivative. Calcolo, 53(3):475–485, 2016.10.1007/s10092-015-0158-8Search in Google Scholar
[19] Kamyar Hosseini, Jalil Manafian, Farzan Samadani, Mohammadreza Foroutan, Mohammad Mirzazadeh, and Qin Zhou. Resonant optical solitons with perturbation terms and fractional temporal evolution using improved tan(ϕ(η)/2)-expansion method and exp function approach. Optik, 158:933 – 939, 2018.10.1016/j.ijleo.2017.12.139Search in Google Scholar
[20] Orkun Tasbozan, Yücel Çenesiz, Ali Kurt, and Dumitru Baleanu. New analytical solutions for conformable fractional pdes arising in mathematical physics by exp-function method. Open Physics, 15(1):647–651.10.1515/phys-2017-0075Search in Google Scholar
[21] K. Hosseini, D. Kumar, M. Kaplan, and E.Y. Bejarbaneh. New exact traveling wave solutions of the unstable nonlinear schrödinger equations. Commun. Theor. Phys., 68:761–767, 2017.10.1088/0253-6102/68/6/761Search in Google Scholar
[22] Ali Kurt, Orkun Tasbozan, and Yucel Cenesiz. Homotopy analysis method for conformable burgers–korteweg-de vries equation. Bull. Math. Sci. Appl, 17:17–23, 2016.10.18052/www.scipress.com/BMSA.17.17Search in Google Scholar
[23] A Kurt, O Tasbozan, and D Baleanu. Newsolutions for conformable fractional nizhnik-novikov-veselov system via
[24] OS lyiola, O Tasbozan, A Kurt, and Y Çenesiz. On the analytical solutions of the system of conformable time-fractional robertson equations with 1 - d diffusion. Chaos, Solitons & Fractals, 94:1–7, 2017.10.1016/j.chaos.2016.11.003Search in Google Scholar
[25] Ali Kurt, Yücel Çenesiz, and Orkun Tasbozan. On the solution of burgers’ equation with the new fractional derivative. Open Physics, 13(1), 2015.10.1515/phys-2015-0045Search in Google Scholar
[26] Y. Pandir, Y. Gurefe, and E. Misirli. A new approach to kudryashov’s method for solving some nonlinear physical models. Int.J. Phys. Sci., 7:2860–2866, 2012.10.5897/IJPS12.071Search in Google Scholar
[27] M. Kaplan, A. Bekir, and A. Akbulut. A generalized kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dyn., 85:2843–2850, 2016.10.1007/s11071-016-2867-1Search in Google Scholar
[28] K. Hosseini, P. Mayeli, and D. Kumar. New exact solutions of the coupled sine-gordon equations in nonlinear optics using the modified kudryashov method. Journal of Modern Optics, 65(3):361–364, 2018.10.1080/09500340.2017.1380857Search in Google Scholar
[29] K. Hosseini, E. Yazdani Bejarbaneh, A. Bekir, and M. Kaplan. New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Optical and Quantum Electronics, 49(7):241, 2017.10.1007/s11082-017-1070-zSearch in Google Scholar
[30] Alper Korkmaz and Kamyar Hosseini. Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods. Optical and Quantum Electronics, 49(8):278, 2017.10.1007/s11082-017-1116-2Search in Google Scholar
[31] D. Kumar, K. Hosseini, and F. Samadani. The sine-gordon expansion method to look for the traveling wave solutions of the tzitzéica type equations in nonlinear optics. Optik - International Journal for Light and Electron Optics, 149:439 – 446, 2017.10.1016/j.ijleo.2017.09.066Search in Google Scholar
[32] K Hosseini, A Bekir, M Kaplan, and Ö Güner. On a new technique for solving the nonlinear conformable time-fractional differential equations. Optical and Quantum Electronics, 49(11):343, 2017.10.1007/s11082-017-1178-1Search in Google Scholar
[33] Orkun Tasbozan, Mehmet Şenol, Ali Kurt, and Ozan Özkan. Newsolutions of fractional drinfeld-sokolov-wilson system in shallow water waves. Ocean Engineering, 161:62–68, 2018.10.1016/j.oceaneng.2018.04.075Search in Google Scholar
[34] A. Akbulut and K. Kaplan. Auxiliary equation method for time fractional differential equation with conformable derivatives. Computers and Mathematics with Applications, 75:876–882, 2018.10.1016/j.camwa.2017.10.016Search in Google Scholar
[35] A. Bekir. Application of the extended tanh method for coupled nonlinear evolution equation. Commun. Nonlinear Sci., 13:1742–1751, 2008.10.1016/j.cnsns.2007.05.001Search in Google Scholar
[36] Hadi Rezazadeh, Alper Korkmaz, Mostafa Eslami, Javad Vahidi, and Rahim Asghari. Traveling wave solution of conformable fractional generalized reaction duffing model by generalized projective riccati equation method. Optical and Quantum Electronics, 50(3):150, 2018.10.1007/s11082-018-1416-1Search in Google Scholar
[37] Yücel Çenesiz, Orkun Tasbozan, and Ali Kurt. Functional variable method for conformable fractional modified kdv-zk equation and maccari system. Tbilisi Mathematical Journal, 10(1):118–126, 2017.10.1515/tmj-2017-0010Search in Google Scholar
[38] Orkun Tasbozan, Yücel Çenesiz, and Ali Kurt. New solutions for conformable fractional boussinesq and combined kdv-mkdv equations using jacobi elliptic function expansion method. The European Physical Journal Plus, 131(7):244, 2016.10.1140/epjp/i2016-16244-xSearch in Google Scholar
[39] Ali Kurt, Yücel Cenesiz, and Orkun Tasbozan. Exact solution for the conformable burgers’ equation by the hopf-cole transform. Cankaya University Journal of Science and Engineering, 13(2).Search in Google Scholar
[40] A. T. Ali and E. R. Hassan. General expa function method for nonlinear evolution equations. Appl. Math. Comput., 217:451–459, 2010.10.1016/j.amc.2010.06.025Search in Google Scholar
[41] E. M. E. Zayed and A. G. Al-Nowehy. Generalized kudryashov method and general exp a function method for solving a high order nonlinear schrödinger equation. J. Space Explor., 6:1–26, 2017.Search in Google Scholar
[42] K. Hosseini, Z. Ayati, and R. Ansari. New exact solution of the tzitzéica type equations in nonlinear optics using the expa function method. J. Mod. Opt., 65(7):847–851, 2018.10.1080/09500340.2017.1407002Search in Google Scholar
[43] K. Hosseini, A. Zabihi, F. Samadani, and R. Ansari. New explicit exact solutions of the unstable nonlinear schrödinger’s equation using the expa and hyperbolic function methods. Optical and Quantum Electronics, 50(2):82, 2018.10.1007/s11082-018-1350-2Search in Google Scholar
[44] Asim Zafar. Rational exponential solutions of conformable space-time fractional equal-width equations. Nonlinear Engineering, 2018.10.1515/nleng-2018-0076Search in Google Scholar
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