Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

Khalid Khan; Amir Ali; Manuel De la Sen; Muhammad Irfan; Khalid Khan; Amir Ali; Manuel De la Sen; Muhammad Irfan

doi:10.3934/math.2022092

AIMS Mathematics

2022, Volume 7, Issue 2: 1580-1602. doi: 10.3934/math.2022092

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Research article

Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels

1.
Department of Mathematics, University of Malakand, Chakdara, Dir (L), Pakistan
2.
Department of Electricity and Electronics, Institute of Research and Development of Processes Faculty of Science and Technology, University of the Basque Country Campus of Leioa, Leioa 48940, Spain
3.
Department of Physics, University of Malakand, Chakdara, Dir (L), Pakistan

Received: 28 July 2021 Accepted: 21 October 2021 Published: 29 October 2021
MSC : 35Bxx, 35Qxx, 37Mxx, 41Axx, 65Mxx

In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 < \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.
- modified coupled KdV equation,
- Caputo and Caputo-Fabrizio operators,
- double Laplace transform,
- decomposition method
Citation: Khalid Khan, Amir Ali, Manuel De la Sen, Muhammad Irfan. Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels[J]. AIMS Mathematics, 2022, 7(2): 1580-1602. doi: 10.3934/math.2022092

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Abstract

In this article, the modified coupled Korteweg-de Vries equation with Caputo and Caputo-Fabrizio time-fractional derivatives are considered. The system is studied by applying the modified double Laplace transform decomposition method which is a very effective tool for solving nonlinear coupled systems. The proposed method is a composition of the double Laplace and decomposition method. The results of the problems are obtained in the form of a series solution for $ 0 < \alpha\leq 1 $, which is approaching to the exact solutions when $ \alpha = 1 $. The precision and effectiveness of the considered method on the proposed model are confirmed by illustrated with examples. It is observed that the proposed model describes the nonlinear evolution of the waves suffered by the weak dispersion effects. It is also observed that the coupled system forms the wave solution which reveals the evolution of the shock waves because of the steeping effect to temporal evolutions. The error analysis is performed, which is comparatively very small between the exact and approximate solutions, which signifies the importance of the proposed method.

References

[1]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[2]	R. Hilfer, Application of fractional calculus in physics, Singapore: World Scientific, 2000.
[3]	G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press, 2005.
[4]	R. L. Magin, Fractional calculus in bioengineering, Redding: Begell House, 2006.
[5]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[6]	J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus, Dordrecht: Springer, 2007. doi: 10.1007/978-1-4020-6042-7.
[7]	D. Baleanu, J. A. T. Machado, Fractional differentiation and its applications, Phys. Scr., 136 (2009). doi: 10.1088/0031-8949/2008/t136/011001.
[8]	D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, 1 Ed., Dordrecht: Springer, 2010. doi: 10.1007/978-90-481-3293-5.
[9]	D. J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422–443. doi: 10.1080/14786449508620739. doi: 10.1080/14786449508620739
[10]	O. Darrigol, Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, 2005.
[11]	Y. Benia, B. K. Sadallah, Existence of solution to Korteweg-de Vries equation in domains that can be transformed into rectangles, Math. Method. Appl. Sci., 41 (2018), 2684–2698. doi: 10.1002/mma.4773. doi: 10.1002/mma.4773
[12]	Y. Benia, A. Scapellato, Existence of solution to Korteweg-de Vries equation in a non-parabolic domain, Nonlinear Anal.-Theor., 195 (2020), 111758. doi: 10.1016/j.na.2020.111758. doi: 10.1016/j.na.2020.111758
[13]	K. Gustafson, D. del-Castillo-Negrete, W. Dorland, Finite Larmor radius effects on nondiffusive tracer transport in a zonal flow, Phys. Plasmas, 15 (2008), 102309. doi: 10.1063/1.3003072. doi: 10.1063/1.3003072
[14]	D. Henry, J. P. Treguier, Propagation of electronic longitudinal modes in a non-Maxwellian plasma, J. Plasma Phys., 8 (1972), 311–319. doi: 10.1017/S0022377800007169. doi: 10.1017/S0022377800007169
[15]	S. A. El-Wakil, E. M. Abulwafa, E. K. El-Shewy, A. A. Mahmoud, Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion, Phys. Plasmas, 18 (2011). doi: 10.1063/1.3640533.
[16]	A. A. Halim, S. P. Kshevetskii, S. B. Leble, Numerical integration of a coupled Korteweg-de Vries system, Comput. Math. Appl., 45 (2003), 581–591. doi: 10.1016/S0898-1221(03)00018-X. doi: 10.1016/S0898-1221(03)00018-X
[17]	R. Hirota, J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407–408. doi: 10.1016/0375-9601(81)90423-0. doi: 10.1016/0375-9601(81)90423-0
[18]	J. M. Sanz-Serna, I. Christie, Petrov-Galerkin methods for nonlinear dispersive waves, J. Comput. Phys., 39 (1981), 94–102. doi: 10.1016/0021-9991(81)90138-8. doi: 10.1016/0021-9991(81)90138-8
[19]	C. Dhaigude, V. Nikam, Solution of fractional partial differential equations using iterative method, Fract. Calc. Appl. Anal., 15 (2012), 684–699. doi: 10.2478/s13540-012-0046-8. doi: 10.2478/s13540-012-0046-8
[20]	L. Akinyemi, O. S. Iyiola, A reliable technique to study nonlinear time-fractional coupled Korteweg-de Vries equations, Adv. Differ. Equ., 2020 (2020), 1–27. doi: 10.1186/s13662-020-02625-w. doi: 10.1186/s13662-020-02625-w
[21]	A. A. Halim, S. B. Leble, Analytical and numerical solution of a coupled KdV-mKdV system, Chaos, Soliton. Fract., 19 (2004), 99–108. doi: 10.1016/S0960-0779(03)00085-7. doi: 10.1016/S0960-0779(03)00085-7
[22]	A. Atangana, J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Meth. Part. D. E., 34 (2018), 1502–1523. doi: 10.1002/num.22195. doi: 10.1002/num.22195
[23]	K. M. Furati, M. D. Kassim, N. T. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. doi: 10.1016/j.camwa.2012.01.009. doi: 10.1016/j.camwa.2012.01.009
[24]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29 (2019). doi: 10.1063/1.5074099.
[25]	M. Caputo, Linear model of dissipation whose $Q$ is almost frequency independent-II, Geophys. J. Int., 13 (1967), 529–539. doi: 10.1111/j.1365-246X.1967.tb02303.x. doi: 10.1111/j.1365-246X.1967.tb02303.x
[26]	M. Yavuz, T. A. Sulaiman, A. Yusuf, T. Abdeljawad, The Schrödinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel, Alex. Eng. J., 60 (2021), 2715–2724. doi: 10.1016/j.aej.2021.01.009. doi: 10.1016/j.aej.2021.01.009
[27]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. doi: 10.12785/pfda/010201. doi: 10.12785/pfda/010201
[28]	A. Atangana, J. F. Gómez-Aguilar, A new derivative with normal distribution kernel: Theory, methods and applications, Physica A, 476 (2017), 1–14. doi: 10.1016/j.physa.2017.02.016. doi: 10.1016/j.physa.2017.02.016
[29]	T. Bashiri, S. M. Vaezpour, J. J. Nieto, Approximating solution of Fabrizio-Caputo Volterra's model for population growth in a closed system by homotopy analysis method, J. Funct. Space., 2018 (2018), 1–10. doi: 10.1155/2018/3152502. doi: 10.1155/2018/3152502
[30]	M. A. Dokuyucu, E. Celik, H. Bulut, H. M. Baskonu, Cancer treatment model with the Caputo-Fabrizio fractional derivative, Eur. Phys. J. Plus, 133 (2018), 1–6. doi: 10.1140/epjp/i2018-11950-y. doi: 10.1140/epjp/i2018-11950-y
[31]	J. F. Gómez-Aguilar, A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus, 132 (2017), 1–21. doi: 10.1140/epjp/i2017-11293-3. doi: 10.1140/epjp/i2017-11293-3
[32]	J. F. Gómez-Aguilar, L. Torres, H. Yépez-Martínez, D. Baleanu, J. M. Reyes, I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Adv. Differ. Equ., 2016 (2016), 1–13. doi: 10.1186/s13662-016-0908-1. doi: 10.1186/s13662-016-0908-1
[33]	J. F. Gómez-Aguilar, H. Yépez-Martínez, C. Calderón-Ramón, I. Cruz-Orduõa, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel, Entropy, 17 (2015), 6289–6303. doi: 10.3390/e17096289. doi: 10.3390/e17096289
[34]	X. J. Yang, H. M. Srivastava, J. A. T. Machado, A new fractional derivative without singular kernel, Therm. Sci., 20 (2016), 753–756. doi: 10.2298/TSCI151224222Y
[35]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. doi: 10.12785/pfda/010202. doi: 10.12785/pfda/010202
[36]	N. Damil, M. Potier-Ferry, A. Najah, R. Chari, H. Lahmam, An iterative method based upon Padé approximamants, Commun. Numer. Meth. En., 15 (1999), 701–708. doi: 10.1002/(SICI)1099-0887(199910)15:10<701::AID-CNM283>3.0.CO;2-L. doi: 10.1002/(SICI)1099-0887(199910)15:10<701::AID-CNM283>3.0.CO;2-L
[37]	G. L. Liu, New research directions in singular perturbation theory: Artificial parameter approach and inverse-perturbation technique, In: Proceedings of conference of 7th modern mathematics and mechanics, Shanghai, 1997, 47–53.
[38]	J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Nonlin. Mech., 35 (2000), 37–43. doi: 10.1016/S0020-7462(98)00085-7. doi: 10.1016/S0020-7462(98)00085-7
[39]	J. M. Cadou, N. Moustaghfir, E. H. Mallil, N. Damil, M. Potier-Ferry, Linear iterative solvers based on pertubration techniques, C. R. Acad. Sci. II B-Mec., 329 (2001), 457–462. doi: 10.1016/S1620-7742(01)01357-5. doi: 10.1016/S1620-7742(01)01357-5
[40]	E. Mallil, H. Lahmam, N. Damil, M. Potier-Ferry, An iterative process based on homotopy and perturbation techniques, Comput. Method. Appl. M., 190 (2000), 1845–1858. doi: 10.1016/S0045-7825(00)00198-5. doi: 10.1016/S0045-7825(00)00198-5
[41]	J. H. He, An approximate solution technique depending on an artificial parameter: A special example, Commun. Nonlinear Sci., 3 (1998), 92–97. doi: 10.1016/S1007-5704(98)90070-3. doi: 10.1016/S1007-5704(98)90070-3
[42]	J. H. He, Newton-like iteration method for solving algebraic equations, Commun. Nonlinear Sci., 3 (1998), 106–109. doi: 10.1016/S1007-5704(98)90073-9. doi: 10.1016/S1007-5704(98)90073-9
[43]	G. Adomian, Solving frontier problems of physics: The decomposition method, Dordrecht: Springer, 1994. doi: 10.1007/978-94-015-8289-6.
[44]	A. Ali, Z. Gul, W. A. Khan, S. Ahmad, S. Zeb, Investigation of fractional order sine-Gordon equation using Laplace Adomian decomposition method, Fractals, 29 (2021). doi: 10.1142/S0218348X21501218.
[45]	M. Khan, M. Hussain, H. Jafari, Y. Khan, Application of Laplace decomposition method to solve nonlinear coupled partial differential equations, World Appl. Sci. J., 9 (2010), 13–19.
[46]	K. Majid, A. G. Muhammed, Application of Laplace decomposition to solve nonlinear partial differential equations, Int. J. Adv. Res. Comput. Sci. Appl., 2 (2010), 52–62.
[47]	H. Hosseinzadeh, H. Jafari, M. Roohani, Application of Laplace decomposition method for solving Klein-Gordon equation, World Appl. Sci. J., 8 (2010), 809–813.
[48]	M. S. Ismail, H. A. Ashi, A numerical solution for Hirota-Satsuma coupled KdV equation, Abstr. Appl. Anal., 2014 (2014), 1–9. doi: 10.1155/2014/819367. doi: 10.1155/2014/819367
[49]	H. Gündoğdu, Ö. F. Gözükızıl, Double Laplace decomposition method and exact solutions of Hirota, Schrödinger and complex mKdV equations, Konuralp J. Math., 7 (2019), 7–15.
[50]	G. Adomian, Modification of the decomposition approach to heat equation, J. Math. Anal. Appl., 124 (1987), 290–291. doi: 10.1016/0022-247X(87)90040-0. doi: 10.1016/0022-247X(87)90040-0
[51]	G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. doi: 10.1016/0022-247X(88)90170-9. doi: 10.1016/0022-247X(88)90170-9
[52]	K. Abbaoui, Y. Cherruault, V. Seng, Practical formulae for the calculus of multivariate Adomian polynomials, Math. Comp. Model., 22 (1995), 89–93. doi: 10.1016/0895-7177(95)00103-9. doi: 10.1016/0895-7177(95)00103-9
[53]	P. Veeresha, A numerical approach to the coupled atmospheric ocean model using a fractional operator, Math. Model. Numer. Simul. Appl., 1 (2021), 1–10. doi: 10.53391/mmnsa.2021.01.001. doi: 10.53391/mmnsa.2021.01.001
[54]	A. Yokus, Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schrödinger equation, Math. Model. Numer. Simul. Appl., 1 (2021), 24–31. doi: 10.53391/mmnsa.2021.01.003. doi: 10.53391/mmnsa.2021.01.003
[55]	M. Yavuz, N. Sene, Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model, J. Ocean Eng. Sci., 6 (2021), 196–205. doi: 10.1016/j.joes.2020.10.004. doi: 10.1016/j.joes.2020.10.004
[56]	E. K. Akgül, A. Akgül, M. Yavuz, New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos Soliton. Fract., 146 (2021), 110877. doi: 10.1016/j.chaos.2021.110877. doi: 10.1016/j.chaos.2021.110877
[57]	R. M. Jena, Two-hybrid techniques coupled with an integral transform for Caputo time-fractional Navier-Stokes equations, Prog. Fract. Differ. Appl., 6 (2020), 201–213. doi: 10.18576/pfda/060304
[58]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[59]	E. J. Moore, S. Sirisubtawee, S. Koonprasert, A Caputo-Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment, Adv. Differ. Equ., 2019 (2019), 1–20. doi: 10.1186/s13662-019-2138-9. doi: 10.1186/s13662-019-2138-9
[60]	I. N. Sneddon, The use of integral transforms, New York: McGraw-Hill, 1972.
[61]	A. Atangana, A. Akgül, Can transfer function and Bode diagram be obtained from Sumudu transform, Alex. Eng. J., 59 (2020), 1971–1984. doi: 10.1016/j.aej.2019.12.028. doi: 10.1016/j.aej.2019.12.028
[62]	A. M. O. Anwar, F. Jarad, D. Baleanu, F. Ayaz, Fractional Caputo heat equation within the double Laplace transform, Rom. J. Phys., 58 (2013), 15–22. doi: 10.5072/ZENODO.25498. doi: 10.5072/ZENODO.25498
[63]	I. L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Appl. Math. Lett., 21 (2008), 372–376. doi: 10.1016/j.aml.2007.05.008. doi: 10.1016/j.aml.2007.05.008
[64]	D. Kaya, I. E. Inan, Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation, Appl. Math. Comput., 151 (2004), 775–787. doi: 10.1016/S0096-3003(03)00535-6. doi: 10.1016/S0096-3003(03)00535-6

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