A new analytical approach to the (1+1)-dimensional conformable Fisher equation
Keywords:Rational sine-Gordon expansion method, conformable derivative, travelling wave solutions, (1 1)-dimensional Fisher equation
In this paper, we use an effective method which is the rational sine-Gordon expansion method to present new wave simulations of a governing model. We consider the (1+1)-dimensional conformable Fisher equation which is used to describe the interactive relation between diffusion and reaction. Various types of solutions such as multi-soliton, kink, and anti-kink wave soliton solutions are obtained. Finally, the physical behaviours of the obtained solutions are shown by 3D, 2D, and contour surfaces.
Cooper, F., Khare, A., Mihaila, B., & Saxena, A. Exact solitary wave solutions for a discrete λφ4 field theory in (1+1)-dimensions. Physical Review E, 72(3), 036605, (2005).
Hirota, R. Exact solution of the Korteweg–de Vries equation for multiple collision of solitons. Physical Review Letters, 27(18), 1192–1194, (1971).
Malfliet, W. Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650–654, (1992).
Fan, E. Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4-5), 212–218, (2000).
Ma, W.X., Huang T., & Zhang, Y. A multiple exp-function method for nonlinear differential equations and its application. Physica Scripta, 82(6), 065003, (2010).
Ma, W.X., & Lee, J.H. A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo–Miwa equation. Chaos, Solitons & Fractals, 42(3), 1356–1363, (2009).
Feng, Z.S., & Wang, X.H. The first integral method to the two dimensional Burgers–Korteweg–de Vries equation. Physics Letters A, 308(2-3), 173–178, (2003).
Jawad, A.J.A.M., Petkovic, M.D., & Biswas, A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation, 217(2), 869–877, (2010).
Aksan, E.N., Bulut, H., & Kayhan, M. Some wave simulation properties of the (2+1) dimensional breaking soliton equation. ITM Web of Conferences, 13, 01014, (2017).
Bulut, H., Aksan, E.N., Kayhan, M., & Sulaiman, T.A. New solitary wave structures to the (3+1) dimensional Kadomtsev–Petviashvili and Schrödinger equation. Journal of Ocean Engineering and Science, 4(4), 373-378, (2019).
Yel, G., Baskonus, H.M., & Bulut, H. Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method. Optical and Quantum Electronics, 49(9), 285, (2017).
Yan, L., Baskonus, H.M., Cattani, C., & Gao, W. Extractions of the gravitational potential and high-frequency wave Perturbation properties of nonlinear (3+1)- dimensional Vakhnenko-Parkes equation via novel approach. Mathematical Methods in the Applied Sciences, 1-10, (2022).
Chen, Q., Baskonus, H.M., Gao, W., & Ilhan, E. Soliton theory and modulation instability analysis: The Ivancevic option pricing model in economy. Alexandria Engineering Journal, 61(10), 7843-7851, (2022).
Veeresha, P., Yavuz, M., & Bhaishya, C. A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators. An International Journal of Optimization and Control: Theories & Applications, 11(3), 52-67, (2021).
Jayaprakasha, P.C., & Bhaishya, C. Numerical analysis of predator-prey model in presence of toxicant by a novel approach. Mathematics in Computer Science, 11(4), 3963-3983, (2021).
Bhaishya, C. A new application of Hermite collocation method. International Journal of Mathematical, Engineering and Management Sciences, 14(1), 182-190, (2019).
Bhaishya, C., & Jaipala. Comparative study of homotopy perturbation method and Genocchi polynomial method for first order fractional differential equation. Journal of Computer and Mathematical Sciences, 10(1), 197-206, (2019).
Bhaishya, C., & Veeresha, P. Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel. Proceedings of the Royal Society A, 477(2253), 20210438, (2021).
Zhou, Q., Ekici M., Sonmezoglu, A., Manafian, J., Khaleghizadeh, S., & Mirzazadeh, M. Exact solitary wave solutions to the generalized Fisher equation. Optik, 127(24), 12085-12092, (2016).
Fisher, R.A. The advance of advantageous genes. Annals of Eugenics, 7(4), 355-369, (1937).
Wazwaz, A.M. The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. Applied Mathematics and Computation, 187(2), 1131-1142, (2007).
Triki, H., & Wazwaz, A.M. Trial equation method for solving the generalized Fisher equation with variable coefficients. Physics Letters A, 380(13), 1260-1262, (2016).
Matinfar, M., Bahar, S.R., & Ghasemi, M. Solving the Generalized Fisher’s equation by differential transform method. Journal of Applied Mathematics and Informatics, 30(3-4), 555–560, (2012).
Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65–70, (2014).
Atangana, A., Baleanu, D., & Alsaedi, A. New properties of conformable derivative. Open Mathematics, 13(1), 889–898, (2015).
Yan, L., Yel, G., Baskonus, H.M., Bulut, H., & Gao, W. Newly developed analytical method and its applications of some mathematical models. International Journal of Modern Physics B, 36(04), 2250040, (2022).
Yan, L., Yel, G., Kumar, A., Baskonus, H.M., & Gao, W. Newly developed analytical scheme and its applications to the some nonlinear partial differential equations with the conformable derivative. Fractal and Fractional, 5(4), 238, 1-15, (2021).
Yamgoué, S.B., Deffo, G.R., & Pelap, F.B. A new rational sine-Gordon expansion method and application to nonlinear wave equations arising in mathematical physics. The European Physical Journal Plus, 134(8), 380, (2019).
Tyson, J.J., & Brazhnik, P.K. On travelling wave solutions of Fisher’s equation in two spatial dimensions. SIAM Journal on Applied Mathematics, 60(2), 371-391, (2000).
Murray, J.D. Mathematical Biology: I. An Introduction (3rd Edition). Springer (2002).
Veeresha, P., Prakasha, D.G., & Baskonus, H.M. Novel simulations to the time fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42, (2019).
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Copyright (c) 2022 Gülnur Yel, Miraç Kayhan, Armando Ciancio
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