Abstract
The dynamical behaviour of the (4+1)-dimensional fractional Fokas equation is investigated in this paper. The modified auxiliary equation method and extended \((\frac{{G'}}{{{G^2}}})\)-expansion method, two reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions used the definition of the truncated M-derivative (TMD) to understand its dynamical behaviour. The hyperbolic, periodic, and trigonometric function solutions are used to derive the analytical solutions for the given model. As a result, dark, bright, and singular solitary wave solitons are obtained. We observe the fractional parameter impact of the above derivative on the physical phenomena. Each set of travelling wave solutions have a symmetrical mathematical form. Last but not least, we employ Mathematica to produce 2D and 3D figures of the analytical soliton solutions to emphasize the influence of TMD on the behaviour and symmetry of the solutions for the proposed problem. The physical importance of the solutions found for particular values of the combination of parameters during the representation of graphs as well as understanding of physical incidents.
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References
Ma, Y.L., Wazwaz, A.M., Li, B.Q.: A new \((3+ 1)\)-dimensional Sakovich equation in nonlinear wave motion: Painleve integrability, multiple solitons and soliton molecules. Qual. Theory Dyn. Syst. 21(4), 158 (2022)
Tala-Tebue, E., Rezazadeh, H., Javeed, S., Baleanu, D., Korkmaz, A.: Solitons of the \((1+ 1)\)-and \((2+ 1)\)-dimensional chiral nonlinear Schrodinger equations with the Jacobi elliptical function method. Qual. Theory Dyn. Syst. 22(3), 106 (2023)
Chen, S.J., Yin, Y.H., Lü, X.: Elastic collision between one lump wave and multiple stripe waves of nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 107205 (2023)
HamaRashid, H., Srivastava, H.M., Hama, M., Mohammed, P.O., Almusawa, M.Y., Baleanu, D.: Novel algorithms to approximate the solution of nonlinear integro-differential equations of Volterra–Fredholm integro type. AIMS Math. 8, 114572–14591 (2023)
HamaRashid, H., Srivastava, H.M., Hama, M., Mohammed, P.O., Al-Sarairah, E., Almusawa, M.Y.: New numerical results on existence of Volterra–Fredholm integral equation of nonlinear boundary integro-differential type. Symmetry 15, 1144 (2023)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)
Wu, G.C., Song, T.T., Wang, S.: Caputo-Hadamard fractional differential equations on time scales: numerical scheme, asymptotic stability, and chaos. Chaos: Interdiscip. J. Nonlinear Sci. 32(9), 093143 (2022)
Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
Mohammed, P.O., Machado, J.A.T., Guirao, J.L.G., Agarwal, R.P.: Adomian decomposition and fractional power series solution of a class of nonlinear fractional differential equations. Mathematics 9, 1070 (2021)
Abu-Shady, M., Kaabar, M.K.: A generalized definition of the fractional derivative with applications. Math. Probl. Eng. 1–9 (2021)
Martínez, F., Kaabar, M. K.: A novel theoretical investigation of the Abu–Shady–Kaabar fractional derivative as a modeling tool for science and engineering. Comput. Math. Methods Med. (2022)
Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam 24(2), 207–233 (1999)
Wang, K.: New perspective to the fractal Konopelchenko–Dubrovsky equations with M-truncated fractional derivative. Int. J. Geom. Methods Mod. Phys. 20(5), 2350072 (2023)
Abdeljawad, T.: On conformable fractional calculus. Comput. Appl. Math. 279, 57–66 (2015)
Owolabi, K.M., Atangana, A.: On the formulation of Adams–Bashforth scheme with Atangana–Baleanu–Caputo fractional derivative to model chaotic problems. Chaos: Interdiscip. J. Nonlinear Sci. 29(2), 023111 (2019)
Hilfer, R.: Fractional diffusion based on Riemann–Liouville fractional derivatives. J. Phys. Chem. B 104(16), 3914–3917 (2000)
Singh, R., Mishra, J., Gupta, V.K.: The dynamical analysis of a Tumor Growth model under the effect of fractal fractional Caputo–Fabrizio derivative. IJMCE (2023)
Jafari, H., Goswami, P., Dubey, R.S., Sharma, S., Chaudhary, A.: Fractional SIZR model of Zombie infection. IJMCE. 1(1), 91–104 (2023)
Tariq, H., Akram, G.: New traveling wave exact and approximate solutions for the nonlinear Cahn–Allen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88, 581–594 (2017)
Khalil, T.A., Badra, N., Ahmed, H.M., Rabie, W.B.: Optical solitons and other solutions for coupled system of nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index by Jacobi elliptic function expansion method. Optik 253, 168540 (2022)
Noureen, R., Naeem, M.N., Baleanu, D., Mohammed, P.O., Almusawa, M.Y.: Application of trigonometric B-spline functions for solving Caputo time fractional gas dynamics equation. AIMS Math. 8, 25343–25370 (2023)
Liu, J.G., Yang, X.J.: Symmetry group analysis of several coupled fractional partial differential equations. Chaos Solitons Fractals 173, 113603 (2023)
Huang, C., Jiang, Z., Huang, X., Zhou, X.: Bifurcation analysis of an SIS epidemic model with a generalized non-monotonic and saturated incidence rate. Int. J. Biomath., 2350033 (2023)
Gao, D., Lü, X., Peng, M.S.: Study on the (2+ 1)-dimensional extension of Hietarinta equation: soliton solutions and Bäcklund transformation. Phys. Scr. 98(9), 095225 (2023)
Yin, Y.H., Lü, X., Ma, W.X.: Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+ 1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 108(4), 4181–4194 (2022)
Biswas, A., Mirzazadeh, M., Triki, H., Zhou, Q., Ullah, M.Z., Moshokoa, S.P., Belic, M.: Perturbed resonant 1-soliton solution with anti-cubic nonlinearity by Riccati–Bernoulli sub-ODE method. Optik 156, 346–350 (2018)
Jiang, Z., Zhang, Z.G., Li, J.J., Yang, H.W.: Analysis of Lie symmetries with conservation laws and solutions of generalized \((4+ 1)\)-dimensional time-fractional Fokas equation. Fractal Fract. 6(2), 108 (2022)
Fokas, A.S.: Integrable nonlinear evolution partial differential equations in \(4+ 2\) and \(3+ 1\) dimensions. Phys. Rev. Lett. 96(19), 190201 (2006)
Chen, S.J., Lü, X., Yin, Y.H.: Dynamic behaviors of the lump solutions and mixed solutions to a \((2+ 1)\)-dimensional nonlinear model. Commun. Theor. Phys. 75(5), 055005 (2023)
Yin, Y.H., Lü, X.: Dynamic analysis on optical pulses via modified PINNs: soliton solutions, rogue waves and parameter discovery of the CQ-NLSE. CNSNS 126, 107441 (2023)
Liu, B., Zhang, X.E., Wang, B., Lü, X.: Rogue waves based on the coupled nonlinear Schrödinger option pricing model with external potential. Mod. Phys. Lett. B 36(15), 2250057 (2022)
Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C., Li, L.Q.: Lie group analysis for a \((2+ 1)\)-dimensional generalized modified dispersive water-wave system for the shallow water waves. Qual. Theory Dyn. Syst. 22(4), 129 (2023)
Rehman, H.U., Saleem, M.S., Zubair, M., Jafar, S., Latif, I.: Optical solitons with Biswas–Arshed model using mapping method. Optik 194, 163091 (2019)
Rehman, H.U., Younis, M., Jafar, S., Tahir, M., Saleem, M.S.: Optical solitons of biswas-arshed model in birefrigent fiber without four wave mixing. Optik 213, 164669 (2020)
Kim, H., Sakthivel, R.: New exact traveling wave solutions of some nonlinear higher-dimensional physical models. Rep. Math. Phys. 70(1), 39–50 (2012)
Wazwaz, A.M.: A variety of multiple-soliton solutions for the integrable \((4+ 1)\)-dimensional Fokas equation. Waves Random Complex Media. 31(1), 46–56 (2021)
Wang, M., Li, X.: Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A 343(1–3), 48–54 (2005)
Sarwar, S.: New soliton wave structures of nonlinear \((4+ 1)\)-dimensional Fokas dynamical model by using different methods. Alex. Eng. J. 60(1), 795–803 (2021)
Rehman, H.U., Jafar, S., Javed, A., Hussain, S., Tahir, M: New optical solitons of Biswas–Arshed equation using different techniques. Optik 206, 163670 (2020)
Arnous, A.H., Mirzazadeh, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Soliton solutions to resonant nonlinear Schrodinger’s equation with time-dependent coefficients by modified simple equation method. Optik 127(23), 11450–11459 (2016)
Abdulazeez, S.T., Modanli, M.: Analytic solution of fractional order pseudo-hyperbolic telegraph equation using modified double Laplace transform method. IJMCE 1(1), 105–114 (2023)
Nadeem, M., He, J.H., Sedighi, H.M.: Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana–Baleanu Caputo derivative. Math. Biosci. Eng. 20(5), 8190–8207 (2023)
Luo, X., Nadeem, M.: Laplace residual power series method for the numerical solution of time-fractional Newell–Whitehead–Segel model. Int. J. Numer. Methods Heat Fluid Flow 33(7), 2377–2391 (2023)
Luo, X., Nadeem, M.: Mohand homotopy transform scheme for the numerical solution of fractional Kundu–Eckhaus and coupled fractional Massive Thirring equations. Sci. Rep. 13(1), 3995 (2023)
Ansar, R., Abbas, M., Mohammed, P.O., Al-Sarairah, E., Gepreel, K.A., Soliman, M.S.: Dynamical study of coupled Riemann wave equation involving conformable, beta, and M-truncated derivatives via two efficient analytical methods. Symmetry 15, 1293 (2023)
Bekir, A.: Application of the \((\frac{{G^{\prime }}}{G})\)-expansion method for nonlinear evolution equations. Phys. Lett. A 372(19), 3400–3406 (2008)
Kudryashov, N.A.: A note on the \((\frac{{G^{\prime }}}{G})\)-expansion method. Comput. Appl. Math. 217(4), 1755–1758 (2010)
Mahak, N., Akram, G.: The modified auxiliary equation method to investigate solutions of the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity. Optik 207, 164467 (2020)
Akram, G., Sadaf, M., Zainab, I.: The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Optik 255, 168614 (2022)
Aljahdaly, N.H.: Some applications of the modified \((\frac{{G^{\prime }}}{{{G^2}}})\)-expansion method in mathematical physics. Results Phys. 13, 102272 (2019)
Zhao, Y.W., Xia, J.W., Lü, X.: The variable separation solution, fractal and chaos in an extended coupled \((2+ 1)\)-dimensional Burgers system. Nonlinear Dyn. 108(4), 4195–4205 (2022)
Sousa, J.V.D.C., de Oliveira, E.C.: A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties. Int. J. Anal. Appl. 16, 83–96 (2018)
Ozdemir, N., Esen, H., Secer, A., Bayram, M., Yusuf, A., Sulaiman, T.A.: Optical solitons and other solutions to the Hirota–Maccari system with conformable, M-truncated and beta derivatives. Mod. Phys. Lett. B 36(11), 2150625 (2022)
Hussain, A., Jhangeer, A., Abbas, N., Khan, I., Sherif, E.S.M.: Optical solitons of fractional complex Ginzburg–Landau equation with conformable, beta, and M-truncated derivatives: A comparative study. Adv. Differ. Equ. 612, 1–19 (2020)
Mohammed, W.W., Cesarano, C., Al-Askar, F.M.: Solutions to the \((4+ 1)\)-dimensional time-fractional Fokas equation with M-truncated derivative. Mathematics. 11(1), 194 (2022)
Akram, G., Sadaf, M., Abbas, M., Zainab, I., Gillani, S.R.: Efficient techniques for traveling wave solutions of time-fractional Zakharov–Kuznetsov equation. Math. Comput. Simul. 193, 607–622 (2022)
Akram, G., Gillani, S.R.: Sub pico-second soliton with Triki–Biswas equation by the extended \((\frac{{G^{\prime }}}{{{G^2}}})\)-expansion method and the modified auxiliary equation method. Optik 229, 166227 (2021)
Mohammed, W.W., Cesarano, C., Al-Askar, F.M.: Solutions to the \((4+ 1)\)-dimensional time-fractional Fokas equation with M-truncated derivative. Mathematics 11(1), 194 (2022)
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Researchers Supporting Project number (RSP2023R153), King Saud University, Riyadh, Saudi Arabia.
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Conceptualization, H.E., T.N. and N.C.; Data curation, H.E.; Funding acquisition, N.C. and D.B.; Investigation, H.E., M.A., T.N., P.O.M., N.C. and D.B.; Methodology, M.A., T.N. and D.B.; Project administration, P.O.M.; Resources, M.A.; Software, H.E. and P.O.M.; Supervision, M.A. and D.B.; Validation, N.C. and D.B.; Writing - original draft, H.E., T.N. and P.O.M.; Writing - review & editing, M.A. All authors have read and agreed to the published version of the manuscript.
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Ehsan, H., Abbas, M., Nazir, T. et al. Efficient Analytical Algorithms to Study Fokas Dynamical Models Involving M-truncated Derivative. Qual. Theory Dyn. Syst. 23, 49 (2024). https://doi.org/10.1007/s12346-023-00890-0
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DOI: https://doi.org/10.1007/s12346-023-00890-0
Keywords
- (4+1)-dimensional fractional Fokas equation
- Extended \((\frac{{G'}}{{{G^2}}})\)-expansion method
- Modified auxiliary equation method
- Truncated M-derivative
- Soliton solutions