On the exact solutions of nonlinear extended Fisher-Kolmogorov equation by using the He's variational approach

Kottakkaran Sooppy Nisar; Shami Ali Mohammed Alsallami; Mustafa Inc; Muhammad Sajid Iqbal; Muhammad Zafarullah Baber; Muhammad Akhtar Tarar; Kottakkaran Sooppy Nisar; Shami Ali Mohammed Alsallami; Mustafa Inc; Muhammad Sajid Iqbal; Muhammad Zafarullah Baber; Muhammad Akhtar Tarar

doi:10.3934/math.2022766

AIMS Mathematics

2022, Volume 7, Issue 8: 13874-13886. doi: 10.3934/math.2022766

Previous Article Next Article

Research article Special Issues

On the exact solutions of nonlinear extended Fisher-Kolmogorov equation by using the He's variational approach

1.
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia
2.
Department of Mathematical Sciences, College of Applied Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
3.
Biruni University, Department of Computer Engineering, Istanbul, Turkey
4.
Department of Medical Research, China Medical University Hospital, 40402 Taichung, Taiwan
5.
Department of Humanities & Basic Science, Military College of Signals, National University of Sciences and Technology Islamabad, Humayun Road 46000 Rawalpindi, Pakistan
6.
Department of Mathematics, The University of Lahore, Lahore, Pakistan
7.
Civil Engineering Department, The University of Lahore, Lahore, Pakistan

Received: 01 February 2022 Revised: 18 April 2022 Accepted: 09 May 2022 Published: 24 May 2022
MSC : 35C08, 37K05, 49K20

In this article, we investigate existence and the exact solutions of the extended Fisher-Kolmogorov (EFK) equation. This equation is used in the population growth dynamics and wave propagation. The fourth-order term in this model describes the phase transitions near critical points which are also known as Lipschitz points. He's variational method is adopted to construct the soliton solutions as well as the periodic wave solutions successfully for the extended (higher-order) EFK equation. This approach is simple and has the greatest advantages because it can reduce the order of our equation and make the equation more simple. So, the results that are obtained by this approach are very simple and straightforward. The graphics behavior of these solutions are also sketched in 3D, 2D, and corresponding contour representations by the different choices of parameters.
- soliton solutions,
- extended Fisher-Kolmogorov equation,
- He's variational methods,
- semi-inverse method,
- variational principle
Citation: Kottakkaran Sooppy Nisar, Shami Ali Mohammed Alsallami, Mustafa Inc, Muhammad Sajid Iqbal, Muhammad Zafarullah Baber, Muhammad Akhtar Tarar. On the exact solutions of nonlinear extended Fisher-Kolmogorov equation by using the He's variational approach[J]. AIMS Mathematics, 2022, 7(8): 13874-13886. doi: 10.3934/math.2022766

Related Papers:

Abstract

In this article, we investigate existence and the exact solutions of the extended Fisher-Kolmogorov (EFK) equation. This equation is used in the population growth dynamics and wave propagation. The fourth-order term in this model describes the phase transitions near critical points which are also known as Lipschitz points. He's variational method is adopted to construct the soliton solutions as well as the periodic wave solutions successfully for the extended (higher-order) EFK equation. This approach is simple and has the greatest advantages because it can reduce the order of our equation and make the equation more simple. So, the results that are obtained by this approach are very simple and straightforward. The graphics behavior of these solutions are also sketched in 3D, 2D, and corresponding contour representations by the different choices of parameters.

References

[1]	R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
[2]	A. Kolmogorov, I. Petrovskii, N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1–25.
[3]	C. Elphick, P. Coullet, D. Repaux, Nature of spatial chaos. Phys. Rev. Lett., 58 (1987), 431. https://doi.org/10.1103/PhysRevLett.58.431 doi: 10.1103/PhysRevLett.58.431
[4]	G. T. Dee, W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., 60 (1988), 2641. https://doi.org/10.1103/PhysRevLett.60.2641 doi: 10.1103/PhysRevLett.60.2641
[5]	W. van Saarloos, Dynamical velocity selection: Marginal stability, Phys. Rev. Lett., 58 (1987), 2571. https://doi.org/10.1103/PhysRevLett.58.2571 doi: 10.1103/PhysRevLett.58.2571
[6]	P. Veeresha, D. G. Prakasha, J. Singh, I. Khan, D. Kumar, Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel, Adv. Differ. Equ., 2020 (2020), 174. https://doi.org/10.1186/s13662-020-02617-w doi: 10.1186/s13662-020-02617-w
[7]	N. Khiari, K. Omrani, Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions, Comput. Math. Appl., 62 (2011), 4151–4160. https://doi.org/10.1016/j.camwa.2011.09.065 doi: 10.1016/j.camwa.2011.09.065
[8]	K. Ismail, N. Atouani, K. Omrani, A three-level linearized high-order accuracy difference scheme for the extended Fisher-Kolmogorov equation, Eng. Comput., 2021. https://doi.org/10.1007/s00366-020-01269-4 doi: 10.1007/s00366-020-01269-4
[9]	P. Danumjaya, A. K. Pani, Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation, J. Comput. Appl. Math., 174 (2005), 101–117. https://doi.org/10.1016/j.cam.2004.04.002 doi: 10.1016/j.cam.2004.04.002
[10]	A. Bashan, Y. Ucar, M. N. Yagmurlu, A. Esen, Numerical solutions for the fourth order extended Fisher-Kolmogorov equation with high accuracy by differential quadrature method, Sigma J. Eng. Nat. Sci., 9 (2018), 273–284.
[11]	S. Kumar, R. Jiwari, R. C. Mittal, Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher-Kolmogorov model, Wave Motion, 109 (2022), 102863. https://doi.org/10.1016/j.wavemoti.2021.102863 doi: 10.1016/j.wavemoti.2021.102863
[12]	M. Younis, S. T. Rizvi, Q. Zhou, A. Biswas, M. Belic, Optical solitons in dual-core fibers with G'/G-expansion scheme, J. Optoelectron. Adv. M., 17 (2015), 505–510.
[13]	M. Younis, A. R. Seadawy, M. Z. Baber, S. Husain, M. S. Iqbal, S. T. Rizvi, et al., Analytical optical soliton solutions of the Schrodinger-Poisson dynamical system, Results Phys., 27 (2021), 104369. https://doi.org/10.1016/j.rinp.2021.104369 doi: 10.1016/j.rinp.2021.104369
[14]	M. Younis, A. R. Seadawy, M. Z. Baber, M. W. Yasin, S. T. Rizvi, M. S. Iqbal, Abundant solitary wave structures of the higher dimensional Sakovich dynamical model, Math. Method. Appl. Sci., 2021. https://doi.org/10.1002/mma.7919 doi: 10.1002/mma.7919
[15]	A. R. Seadawy, M. Younis, M. Z. Baber, S. T. Rizvi, M. S. Iqbal, Diverse acoustic wave propagation to confirmable time-space fractional KP equation arising in dusty plasma, Commun. Theor. Phys., 73 (2021), 115004. https://doi.org/10.1088/1572-9494/ac18bb doi: 10.1088/1572-9494/ac18bb
[16]	M. Younis, A. R. Seadawy, I. Sikandar, M. Z. Baber, N. Ahmed, S. T. Rizvi, et al., Nonlinear dynamical study to time fractional Dullian-Gottwald-Holm model of shallow water waves, Int. J. Mod. Phys. B, 36 (2022), 2250004. https://doi.org/10.1142/S0217979222500047 doi: 10.1142/S0217979222500047
[17]	M. W. Yasin, M. S. Iqbal, A. R. Seadawy, M. Z. Baber, M. Younis, S. T. Rizvi, Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model, Int. J. Nonlin. Sci. Num., 2021. https://doi.org/10.1515/ijnsns-2021-0113 doi: 10.1515/ijnsns-2021-0113
[18]	U. Younas, M. Younis, A. R. Seadawy, S. T. Rizvi, S. Althobaiti, S. Sayed, Diverse exact solutions for modified nonlinear Schrodinger equation with conformable fractional derivative, Results Phys., 20 (2021), 103766. https://doi.org/10.1016/j.rinp.2020.103766 doi: 10.1016/j.rinp.2020.103766
[19]	B. Ghanbari, Abundant soliton solutions for the Hirota-Maccari equation via the generalized exponential rational function method, Mod. Phys. Lett. B, 33 (2019), 1950106. https://doi.org/10.1142/S0217984919501069 doi: 10.1142/S0217984919501069
[20]	A. R. Seadawy, M. Bilal, M. Younis, S. T. Rizvi, S. Althobaiti, M. M. Makhlouf, Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique, Chaos Soliton. Fract., 144 (2021), 110669. https://doi.org/10.1016/j.chaos.2021.110669 doi: 10.1016/j.chaos.2021.110669
[21]	E. M. E. Zayed, A. G. Al-Nowehy, New generalized $\phi^{6}$-model expansion method and its applications to the (3+1) dimensional resonant nonlinear Schrodinger equation with parabolic law nonlinearity, Optik, 214 (2020), 164702. https://doi.org/10.1016/j.ijleo.2020.164702 doi: 10.1016/j.ijleo.2020.164702
[22]	S. Kumar, R. Jiwari, R. C. Mittal, J. Awrejcewicz, Dark and bright soliton solutions and computational modeling of nonlinear regularized long wave model, Nonlinear Dyn., 104 (2021), 661–682. https://doi.org/10.1007/s11071-021-06291-9 doi: 10.1007/s11071-021-06291-9
[23]	R. Jiwari, V. Kumar, S. Singh, Lie group analysis, exact solutions and conservation laws to compressible isentropic Navier-Stokes equation, Eng. Comput., 2020. https://doi.org/10.1007/s00366-020-01175-9 doi: 10.1007/s00366-020-01175-9
[24]	O. P. Yadav, R. Jiwari, Some soliton-type analytical solutions and numerical simulation of nonlinear Schrodinger equation, Nonlinear Dyn., 95 (2019), 2825–2836. https://doi.org/10.1007/s11071-018-4724-x doi: 10.1007/s11071-018-4724-x
[25]	V. Kumar, R. K. Gupta, R. Jiwari, Painleve analysis, Lie symmetries and exact solutions for variable coefficients Benjamin-Bona-Mahony-Burger (BBMB) equation, Commun. Theor. Phys., 60 (2013), 175.
[26]	M. Tatari, M. Dehghan, On the convergence of He's variational iteration method, J. Comput. Appl. Math., 207 (2007), 121–128. https://doi.org/10.1016/j.cam.2006.07.017 doi: 10.1016/j.cam.2006.07.017
[27]	A. Jabbari, M. Kheiri, A. Bekir, Exact solutions of the coupled Higgs equation and the Maccari system using He's semi-inverse method and (G'/G)-expansion method, Comput. Math. Appl., 62 (2011), 2177–2186. https://doi.org/10.1016/j.camwa.2011.07.003 doi: 10.1016/j.camwa.2011.07.003
[28]	M. Akbari, Exact solutions of the coupled Higgs equation and the Maccari system using the modified simplest equation method. Inf. Sci. Lett., 2 (2013), 155–158. https://doi.org/10.12785/isl/020304 doi: 10.12785/isl/020304
[29]	K. J. Wang, G. D. Wang, Solitary and periodic wave solutions of the generalized fourth-order Boussinesq equation via He's variational methods, Math. Method. Appl. Sci., 44 (2021), 5617–5625. https://doi.org/10.1002/mma.7135 doi: 10.1002/mma.7135
[30]	A. Biswas, Q. Zhou, M. Z. Ullah, H. Triki, S. P. Moshokoa, M. Belic, Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle, Optik, 143 (2017), 131–134. https://doi.org/10.1016/j.ijleo.2017.06.087 doi: 10.1016/j.ijleo.2017.06.087
[31]	A. Biswas, M. Asma, P. Guggilla, L. Mullick, L. Moraru, M. Ekici, et al., Optical soliton perturbation with Kudryashov's equation by semi-inverse variational principle, Phys. Lett. A, 384 (2020), 126830. https://doi.org/10.1016/j.physleta.2020.126830 doi: 10.1016/j.physleta.2020.126830
[32]	I. Mehdipour, D. D. Ganji, M. Mozaffari, Application of the energy balance method to nonlinear vibrating equations, Curr. Appl. Phys., 10 (2010), 104–112. https://doi.org/10.1016/j.cap.2009.05.016 doi: 10.1016/j.cap.2009.05.016
[33]	S. S. Ganji, D. D. Ganji, S. Karimpour, He's energy balance and He's variational methods for nonlinear oscillations in engineering, Int. J. Mod. Phys. B, 23 (2009), 461–471. https://doi.org/10.1142/S0217979209049644 doi: 10.1142/S0217979209049644
[34]	M. S. Iqbal, Solutions of boundary value problems for nonlinear partial differential equations by fixed point methods, 2011.
[35]	J. H. He, Variational principle and periodic solution of the Kundu-Mukherjee-Naskar equation, Results Phys., 17 (2020), 103031. https://doi.org/10.1016/j.rinp.2020.103031 doi: 10.1016/j.rinp.2020.103031

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)